JOURDAIN Benjamin

We are interested in martingale rearrangement couplings. As introduced by Wiesel [37] in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. In reason of the lack of relative compactness of the set of couplings with given marginals for the adapted Wasserstein topology, the existence of such a projection is not clear at all. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Fr\'echet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. Here, we give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction we gave in [24] of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fr\'echet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglb\"ock and Juillet [9] and which involve the uniform distribution on [0, 1] in addition to the two marginals. We last discuss the stability in adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginal distributions.

Our main result is to establish stability of martingale couplings: suppose that $\pi$ is a martingale coupling with marginals $\mu, \nu$. Then, given approximating marginal measures $\tilde \mu \approx \mu, \tilde \nu\approx \nu$ in convex order, we show that there exists an approximating martingale coupling $\tilde\pi \approx \pi$ with marginals $\tilde \mu, \tilde \nu$. In mathematical finance, prices of European call / put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call / put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time, the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for a several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.

In this thesis, we are interested in the numerical solution of models governed by partial differential equations that admit a probabilistic interpretation. In a first step, we consider partial differential equations in high dimension. Based on a probabilistic interpretation of the solution which allows to obtain point estimates of the solution via Monte-Carlo methods, we propose an algorithm combining an adaptive interpolation method and a variance reduction method to approximate the solution on its whole definition domain. In a second step, we focus on reduced basis methods for parameterized partial differential equations. We propose two gluttonous algorithms based on a probabilistic interpretation of the error. We also propose a discrete optimization algorithm that is probably approximately correct in relative accuracy and that allows us, for these two gluttonous algorithms, to judiciously select a snapshot to add to the reduced basis based on the probabilistic representation of the approximation error.

While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.

We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q − L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and Röckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 − (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.

We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $\rho$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $\rho \ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.

This thesis is divided into four parts that can be read independently. In this manuscript, we make some contributions to the theoretical study and to the applications in finance of optimal quantization. In the first part, we recall the theoretical foundations of optimal quantization as well as the classical numerical methods to construct optimal quantizers. The second part focuses on the numerical integration problem in dimension 1, which arises when one wishes to compute numerically expectations, such as in the valuation of derivatives. We recall the existing strong and weak error results and extend the results of the second order convergences to other classes of less regular functions. In a second part, we present a weak error result in dimension 1 and a second development in higher dimension for a product quantizer. In the third part, we focus on a first numerical application. We introduce a stationary Heston model in which the initial condition of the volatility is assumed to be random with the stationary distribution of the EDS of the CIR governing the volatility. This variant of the original Heston model produces a more pronounced implied volatility smile for European options on short maturities than the standard model. We then develop a numerical method based on recursive quantization produced for the evaluation of Bermudian and barrier options. The fourth and last part deals with a second numerical application, the valuation of Bermudian options on exchange rates in a 3-factor model. These products are known in the markets as PRDCs. We propose two schemes to evaluate this type of options, both based on optimal product quantization and establish a priori error estimates.

The purpose of this paper is to study the existence and uniqueness of solutions to a Stochastic Differential Equation (SDE) coming from the eigenvalues of Wishart processes. The coordinates are non-negative, evolve as Cox-Ingersoll-Ross (CIR) processes and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision, and give a necessary and sufficient condition on the parameters of the SDE for this multiple collision not to occur in finite time.

By Gyongy's theorem, a local and stochastic volatility model is calibrated to the market prices of all call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labord\`ere (2011), provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to Figalli (2008), we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called Regime Switching Local Volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. We then deduce existence of the calibrated model by extending the results in Figalli (2008).

We are interested in the approximation in Wasserstein distance with index $\rho\ge 1$ of a probability measure $\mu$ on the real line with finite moment of order $\rho$ by the empirical measure of $N$ deterministic points. The minimal error converges to $0$ as $N\to+\infty$ and we try to characterize the order associated with this convergence. Apart when $\mu$ is a Dirac mass and the error vanishes, the order is not larger than $1$. We give a necessary condition and a sufficient condition for the order to be equal to this threshold $1$ in terms of the density of the absolutely continuous with respect to the Lebesgue measure part of $\mu$. We also check that for the order to lie in the interval $\left(1/\rho,1\right)$, the support of $\mu$ has to be a bounded interval, and that, when $\mu$ is compactly supported, the order is not smaller than $1/\rho$. Last, we give a necessary and sufficient condition in terms of the tails of $\mu$ for the order to be equal to some given value in the interval $\left(0,1/\rho\right)$.

No summary available.

It is known since [24] that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of $\vert x-y\vert$ is smaller than twice their $\mathcal W_1$-distance (Wasserstein distance with index $1$). We showed in [24] that replacing $\vert x-y\vert$ and $\mathcal W_1$ respectively with $\vert x-y\vert^\rho$ and $\mathcal W_\rho^\rho$ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing $\mathcal W_\rho^\rho$ with the product of $\mathcal W_\rho$ times the centred $\rho$-th moment of the second marginal to the power $\rho-1$. Then we study the generalisation of this new stability inequality to higher dimension.

No summary available.

We establish for dual quantization the counterpart of Kieffer's uniqueness result for compactly supported one dimensional probability distributions having a $\log$-concave density (also called strongly unimodal): for such distributions, $L^r$-optimal dual quantizers are unique at each level $N$, the optimal grid being the unique critical point of the quantization error. An example of non-strongly unimodal distribution for which uniqueness of critical points fails is exhibited. In the quadratic $r=2$ case, we propose an algorithm to compute the unique optimal dual quantizer. It provides a counterpart of Lloyd's method~I algorithm in a Voronoi framework. Finally semi-closed forms of $L^r$-optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.

This thesis is motivated by the study of the stability of the martingale optimal transport problem, and is naturally organized into two parts. In the first part, we exhibit a new family of martingale couplings between two one-dimensional probability measures μ and ν comparable in convex order. In particular, this family contains the inverse transform martingale coupling, which is explicit in terms of quantile functions of the marginals. The integral M_1(μ,ν) of |x-y| against each of these couplings is increased by twice the Wasserstein distance W_1(μ,ν) between μ and ν. We show a similar inequality when |x-y| and W_1 are respectively replaced by |x-y|^ρ and the product of W_ρ by the centered moment of order ρ of the second marginal raised to the exponent ρ-1, for any ρ∈[1,+∞[. We then study the generalization of this new stability inequality to the higher dimension. Finally, we establish a strong connection between our new family of martingale couplings and the projection of a coupling between two comparable given marginals in convex order onto the set of martingale couplings between these same marginals. This last projection is taken with respect to the adapted Wasserstein distance, which majors the usual Wasserstein distance and thus induces a finer topology better suited for financial modeling, since it takes into account the temporal structure of martingales. In the second part, we prove that any martingale coupling whose marginals are approximated by comparable probability measures in convex order can itself be approximated by martingale couplings in the sense of the adapted Wasserstein distance. We then discuss various applications of this result. In particular, we strengthen a stability result for the weak optimal transport problem and establish a stability result for the weak optimal martingale transport problem. We derive stability with respect to the marginals of the over-replication price of VIX futures contracts.

This thesis is devoted to the theoretical and numerical study of the weak error of time and particle discretization of nonlinear Stochastic Differential Equations in the McKean sense. In the first part, we analyze the weak convergence speed of the time discretization of standard SDEs. More specifically, we study the convergence in total variation of the Euler-Maruyama scheme applied to d-dimensional DEs with a measurable drift coefficient and additive noise. We obtain, assuming that the drift coefficient is bounded, a weak order of convergence 1/2. By adding more regularity on the drift, namely a spatial divergence in the sense of L[rho]-space-uniform distributions in time for some [rho] greater than or equal to d, we reach a convergence order equal to 1 (within a logarithmic factor) at terminal time. In dimension 1, this result is preserved when the spatial derivative of the drift is a measure in space with a total mass bounded uniformly in time. In the second part of the thesis, we analyze the weak discretization error in both time and particles of two classes of nonlinear DHSs in the McKean sense. The first class consists of multi-dimensional SDEs with regular drift and diffusion coefficients in which the law dependence occurs through moments. The second class consists of one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where the law dependence occurs through the distribution function. We approximate the EDS by the Euler-Maruyama schemes of the associated particle systems and we obtain for both classes a weak order of convergence equal to 1 in time and in particles. In the second class, we also prove a result of chaos propagation of optimal order 1/2 in particles and a strong order of convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by numerical simulations.

In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivatives of various orders. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. As a consequence of this result, we also analyse the convergence of fluctuation between the empirical measure of particles in an interacting particle system and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear.

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We obtain Gaussian-type concentration. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

For µ and ν two probability measures on the real line such that µ is smaller than ν in the convex order, this property is in general not preserved at the level of the empirical measures µI = 1 I I i=1 δX i and νJ = 1 J J j=1 δY j , where (Xi) 1≤i≤I (resp. (Yj) 1≤j≤J) are independent and identically distributed according to µ (resp. ν). We investigate modifications of µI (resp. νJ) smaller than νJ (resp. greater than µI) in the convex order and weakly converging to µ (resp. ν) as I, J → ∞. According to Kertz and Rösler (1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For µ and ν in this set, this enables us to define a probability measure µ ∨ ν (resp. µ ∧ ν) greater than µ (resp. smaller than ν) in the convex order. We give efficient algorithms permitting to compute µ ∨ ν and µ ∧ ν (and therefore µI ∨ νJ and µI ∧ νJ) when µ and ν have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

By drawing a parallel between metadynamics and self interacting models for polymers, we study the longtime convergence of the original metadynamics algorithm in the adiabatic setting, namely when the dynamics along the collective variables decouples from the dynamics along the other degrees of freedom. We also discuss the bias which is introduced when the adiabatic assumption does not holds.

In this paper, we analyse the rate of convergence of a system of $N$ interacting particles with mean-field rank based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov to check trajectorial propagation of chaos with optimal rate $N^{-1/2}$ to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy to check convergence in $L^1(\mathbb{R})$ with rate ${\mathcal O}(\frac{1}{\sqrt N} + h)$ of the empirical cumulative distribution function of the Euler discretization with step $h$ of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves in ${\mathcal O}(\frac{1}{N} + h)$. We provide numerical results which confirm our theoretical estimates.

This thesis contains two parts. In the first part, we prove two limit theorems of optimal quantization. The first limit theorem is the characterization of the convergence under the Wasserstein distance of a sequence of probability measures by the simple convergence of the quantization error functions. These results are established in Rd and also in a separable Hilbert space. The second limit theorem shows the speed of convergence of the optimal grids and the quantization performance for a sequence of probability measures which converge under the Wasserstein distance, in particular the empirical measure. The second part of this thesis focuses on the approximation and simulation of the McKean-Vlasov equation. We start this part by proving, by Feyel's method (see Bouleau (1988) [Section 7]), the existence and uniqueness of a strong solution of the McKean-Vlasov equation dXt = b(t, Xt, μt)dt + σ(t, Xt, μt)dBt under the condition that the coefficient functions b and σ are lipschitzian. Then, the convergence speed of the theoretical Euler scheme of the McKean-Vlasov equation is established and also the convex order functional results for the McKean-Vlasov equations with b(t,x,μ) = αx+β, α,β ∈ R. In the last chapter, the error of the particle method, several quantization-based schemes and a hybrid particle-quantization scheme are analyzed. At the end, two example simulations are illustrated: the Burgers equation (Bossy and Talay (1997)) in dimension 1 and the FitzHugh-Nagumo neural network (Baladron et al. (2012)) in dimension 3.

We are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order. We propose to alternate transitions according to a martingale Markov kernel mapping a probability measure in the sequence to the next and dual quantization steps. In the case of ARCH models and in particular of the Euler scheme of a driftless Brownian diffusion, the noise has to be truncated to enable the dual quantization step. We analyze the error between the original ARCH model and its approximation with truncated noise and exhibit conditions under which the latter is dominated by the former in the convex order at the level of sample-paths. Last, we analyse the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.

In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu$ and $\nu$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $\mu$ and $\nu$. It contains the inverse transform martingale coupling which is explicit in terms of the associated cumulative distribution functions. The integral of $\vert x-y\vert$ with respect to each of these couplings is smaller than twice the $W_1$ distance between $\mu$ and $\nu$. When $\mu$ and $\nu$ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.

In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes.

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W22(μ,ν) between two probability measures μ and ν with finite second order moments on Rd is the composition of a martingale coupling with an optimal transport map T. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between μ and T#μ. Next, we prove that σ↦W22(σ,ν) is differentiable at μ in the Lions~\cite{Lions} and the geometric senses iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. Besides, we give a self-contained proof that mere Fréchet differentiability of a law invariant function F on L2(Ω,P.Rd) is enough for the Fréchet differential at X to be a measurable function of X.

In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(1/N + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

For µ and ν two probability measures on the real line such that µ is smaller than ν in the convex order, this property is in general not preserved at the level of the empirical measures µI = 1 I I i=1 δX i and νJ = 1 J J j=1 δY j , where (Xi) 1≤i≤I (resp. (Yj) 1≤j≤J) are independent and identically distributed according to µ (resp. ν). We investigate modifications of µI (resp. νJ) smaller than νJ (resp. greater than µI) in the convex order and weakly converging to µ (resp. ν) as I, J → ∞. According to Kertz and Rösler (1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For µ and ν in this set, this enables us to define a probability measure µ ∨ ν (resp. µ ∧ ν) greater than µ (resp. smaller than ν) in the convex order. We give efficient algorithms permitting to compute µ ∨ ν and µ ∧ ν (and therefore µI ∨ νJ and µI ∧ νJ) when µ and ν have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

n Gerbi et al. (2016) we proved strong convergence with order 1/2 of the Ninomiya–Victoir schemeXN V,ηwith time stepT/Nto the solutionXof the limiting SDE. In this paper we check that thenormalized error defined by√N(X−XN V,η)converges to an affine SDE with source terms involvingthe Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher randomvariablesη. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the centrallimit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel MonteCarlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes.This suggests that the rate of convergence is greater than 1/2 in this case and we actually prove strongconvergence with order 1 and study the limit of the normalized errorN(X−XN V,η). The limiting SDEinvolves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. Whenall the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoirscheme coincides with the solution to the SDE on the discretization grid.

No summary available.

We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is an adaptive importance technique useful to sample multimodal target distributions. The importance function is based on the weights of disjoint sets which form a partition of the space. In the context of computational statistical physics, the logarithm of these weights is, up to a multiplicative constant, the free energy, and the discrete valued function defining the partition is called the reaction coordinate. The algorithm is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets and (ii) the target distribution is biased using only a fraction of the free energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. The algorithm can also be seen as a generalization of well-tempered metadynamics. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.

In this thesis, we study several financial mathematics problems related to the valuation of derivatives. Through different asymptotic approaches, we develop methods to compute accurate approximations of the price of certain types of options in cases where no explicit formula exists.In the first chapter, we focus on the valuation of options whose payoff depends on the trajectory of the underlying by Monte Carlo methods, when the underlying is modeled by an affine process with stochastic volatility. We prove a principle of large trajectory deviations in long time, which we use to compute, using Varadhan's lemma, an asymptotically optimal change of measure, allowing to significantly reduce the variance of the Monte-Carlo estimator of option prices.The second chapter considers the valuation by Monte-Carlo methods of options depending on multiple underlyings, such as basket options, in Wishart's stochastic volatility model, which generalizes the Heston model. Following the same approach as in the previous chapter, we prove that the process vérifie a principle of large deviations in long time, which we use to significantly reduce the variance of the Monte Carlo estimator of option prices, through an asymptotically optimal change of measure. In parallel, we use the principle of large deviations to characterize the long-time behavior of the Black-Scholes implied volatility of basket options.In the third chapter, we study the valuation of realized variance options, when the spot volatility is modeled by a constant volatility diffusion process. We use recent asymptotic results on the densities of hypo-elliptic diffusions to compute an expansion of the realized variance density, which we integrate to obtain the expansion of the option price, and then their Black-Scholes implied volatility.The final chapter is devoted to the valuation of interest rate derivatives in the Lévy model of the Libor market, which generalizes the classical Libor market model (log-normal) by adding jumps. By writing the former as a perturbation of the latter and using the Feynman-Kac representation, we explicitly compute the asymptotic expansion of the price of interest rate derivatives, in particular, caplets and swaptions.

This thesis is devoted to the theoretical and numerical study of two nonlinear problems in the McKean sense in finance. In the first part, we address the problem of calibrating a model with local and stochastic volatility to take into account the prices of European vanilla options observed on the market. This problem results in the study of a nonlinear stochastic differential equation (SDE) in the McKean sense because of the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor with respect to the SDE solution. We obtain the existence of the process in the particular case where the stochastic volatility factor is a jump process with a finite number of states. We also obtain the weak convergence at order 1 of the time discretization of the nonlinear DHS in the McKean sense for general stochastic volatility factors. In the industry, the calibration is efficiently performed using a regularization of the conditional expectation by a Nadaraya-Watson type kernel estimator, as proposed by Guyon and Henry-Labordère in [JGPHL]. We also propose a half-time numerical scheme and study the associated particle system that we compare to the algorithm proposed by [JGPHL]. In the second part of the thesis, we focus on a problem of contract valuation with margin calls, a problem that appeared with the application of new regulations since the financial crisis of 2008. This problem can be modeled by an anticipatory stochastic differential equation (SDE) with dependence on the law of the solution in the generator. We show that this equation is well-posed and propose an approximation of its solution using standard linear SRDEs when the liquidation time of the option in case of default is small. Finally, we show that the computation of the solutions of these standard EDSRs can be improved using the multilevel Monte Carlo method introduced by Giles in [G].

In this thesis we study various aspects of martingale optimal transport in dimension greater than one, from duality to local structure, and finally propose methods of numerical approximation.We first prove the existence of irreducible components intrinsic to martingale transports between two given measures, and the canonicity of these components. We then prove a duality result for the optimal martingale transport in any dimension, the point by point duality is no longer true but a form of quasi-safe duality is proved. This duality allows us to prove the possibility of decomposing the quasi-safe optimal transport into a series of subproblems of point by point optimal transports on each irreducible component. We finally use this duality to prove a martingale monotonicity principle, analogous to the famous monotonicity principle of classical optimal transport. We then study the local structure of optimal transports, deduced from differential considerations. We obtain a characterization of this structure using real algebraic geometry tools. We deduce the structure of martingale optimal transports in the case of Euclidean norm power costs, thus solving a conjecture dating back to 2015. Finally, we compare existing numerical methods and propose a new method that is shown to be more efficient and to deal with an intrinsic problem of the martingale constraint that is the convex order defect. We also give techniques to handle the numerical problems in practice.

We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter λ, and admitting a unique invariant measure for any value of λ around λ = 0. Our aim is to compute the derivative with respect to λ of averages with respect to the invariant measure, at λ = 0. We analyze a numerical method which consists in simulating the process at λ = 0 together with its derivative with respect to λ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to λ of the mean of an observable through Monte Carlo simulations.

No summary available.

The Keller-Segel partial differential equation is a two-dimensional model for chemotaxis. When the total mass of the initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $\chi$ of bacteria to the chemo-attractant is larger than $8\pi$. We investigate its approximation by a system of $N$ two-dimensional Brownian particles interacting through a singular attractive kernel in the drift term. In the very subcritical case $\chi<2\pi$, the diffusion strongly dominates this singular drift: we obtain existence for the particle system and prove that its flow of empirical measures converges, as $N\to\infty$ and up to extraction of a subsequence, to a weak solution of the Keller-Segel equation. We also show that for any $N\ge 2$ and any value of $\chi>0$, pairs of particles do collide with positive probability: the singularity of the drift is indeed visited. Nevertheless, when $\chi<2\pi N$, it is possible to control the drift and obtain existence of the particle system until the first time when at least three particles collide. We check that this time is a.s. infinite, so that global existence holds for the particle system, if and only if $\chi\leq 8\pi(N-2)/(N-1)$. Finally, we remark that in the system with $N=2$ particles, the difference between the two positions provides a natural two-dimensional generalization of Bessel processes, which we study in details.

No summary available.

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We obtain Gaussian-type concentration. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

Motivated by the approximation of Martingale Optimal Transport problems, we study sampling methods preserving the convex order for two probability measures $\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When $(X_i)_{1\le i\le I}$ (resp.

In this paper, we summarize the results about the strong convergence rate of the Ninomiya-Victoir scheme and the stable convergence in law of its normalized error that we obtained in previous papers. We then recall the properties of the multilevel Monte Carlo estimators involving this scheme that we introduced and studied before. Last, we are interested in the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme. We prove that this error converges with strong order 2 when an explicit Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order 5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the same order of strong convergence. Moreover, the properties of our multilevel Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.

This thesis contains two contributions to the space-time homogenization of the first order Hamilton-Jacobi equations. These equations are related to the modelling of road traffic. Finally, some results of homogenization in a nearly periodic environment are presented. The first chapter is devoted to the homogenization of an infinite system of coupled differential equations with delay time. This system is derived from a microscopic model of simple road traffic. The drivers follow each other on an infinite straight road and their reaction time is taken into account. The speed of each driver is assumed to be a function of the distance to the preceding driver: we speak of a "follow-the-leader" model. Thanks to a strict comparison principle, we show the convergence to a macroscopic model for reaction times lower than a critical value. In a second step, we show a counterexample to the homogenization for a reaction time higher than this critical value, for particular initial conditions. For this purpose, we perturb the stationary solution in which the vehicles are all equidistant at the initial times. The second chapter deals with the homogenization of a Hamilton-Jacobi equation whose Hamiltonian is spatially discontinuous. The associated traffic model is a straight road with an infinite number of traffic lights. The traffic lights are assumed to be identical, equally spaced and the time delay between two successive lights is assumed to be constant. We study the large-scale influence of this phasing on the traffic. We distinguish between the free road portion, which will be represented by a macroscopic model, and the traffic lights, which will be modeled by time-dependent flow limiters. The theoretical framework is the one by C. Imbert and R. Monneau (2017) for Hamilton-Jacobi equations on networks. The study consists in the theoretical homogenization, where the effective Hamiltonian depends on the phasing, and then the obtaining of qualitative properties of this Hamiltonian with the help of observations via numerical simulations. The third chapter presents results of homogenization in an almost periodic environment. First, we study an evolution problem with a stationary Hamiltonian, almost periodical in space. Using almost periodical arguments, we carry out in a second time a new proof of the homogenization result of the second chapter. The Hamiltonian is then periodic in time and almost periodic in space. We also have some open questions, especially in the case where the Hamiltonian is almost periodic in time-space, and in the case of a traffic model where the traffic lights are quite close, with therefore a microscopic model between the lights.

This thesis deals with various control and optimization problems for which only approximate solutions exist to date. On the one hand, we are interested in techniques to reduce or eliminate approximations in order to obtain more precise or even exact solutions. On the other hand, we develop new approximation methods to deal more quickly with larger scale problems. We study numerical methods for simulating stochastic differential equations and for improving expectation calculations. We implement quantization-type techniques for the construction of control variables and the stochastic gradient method for solving stochastic control problems. We are also interested in clustering methods related to quantization, as well as in information compression by neural networks. The problems studied are not only motivated by financial issues, such as stochastic control for option hedging in incomplete markets, but also by the processing of large media databases commonly referred to as Big data in Chapter 5. Theoretically, we propose different majorizations of the convergence of numerical methods on the one hand for the search of an optimal hedging strategy in incomplete market in chapter 3, on the other hand for the extension of the Beskos-Roberts technique of differential equation simulation in chapter 4. We present an original use of the Karhunen-Loève decomposition for a variance reduction of the expectation estimator in chapter 2.

Monte-Carlo methods are widely used by the financial industry to price derivatives, estimate risks, or to calibrate/estimate models. They can also be used to handle big data, in machine learning, to perform online optimization, to study the propagation of uncertainty in fluid mechanics or geophysics. Under the same label Monte-Carlo, one actually finds very different techniques and communities that evolve in different directions. The thematic cycle that we organized from october 2015 to July 2016 aimed at confronting the different viewpoints of these communities and at contributing to a general thinking on how these techniques can be used by the financial industry and the economic world in general. It benefited from the financial support of the Louis Bachelier Institute, the Chaire Risques Financiers, the Chaire Finance et D´eveloppement durable, the Chaire Economie des nou- ´ velles donn´ees, the Chaire March´es en mutation, the ANR program ISOTACE ANR-12-MONU-0013 and the Institut Henri Poincar´e. Three topics were covered by academic lectures followed by a one-day workshop: propagation of uncertainty, particle methods for the management of risks, stochastic algorithms and big data. We thank Areski Cousin, Virginie Ehrlacher, Romuald Elie, Gersende Fort, St´ephane Gaiffas and Gilles Pag`es for having coordinated these workshops. The cycle was concluded by a one week closing conference with twelve plenary talks and sixteen minisymposia: see the website https://montecarlo16.sciencesconf.org Of course the six papers in these proceedings cannot account for all the topics addressed during the cycle. But they give qualitative spotlights on some of the active fields of research on stochastic methods in finance. We thank their authors for these valuable contributions.

This paper is devoted to the Paveri-Fontana model and its computation. The master equation of this model has no analytic solution in nonequilibrium case. We develop a stochastic approach to approximate this evolution equation. First, we give a probabilistic interpretation of the equation as a nonlinear Fokker- Planck equation. Replacing the nonlinearity by interaction, we deduce how to approximate its solution thanks to an algorithm based on a fictitious jump simulation of the interacting particle system. This algorithm is improved to obtain a linear complexity regarding the number of particles. Finally, the numerical method is illustrated on one traffic flow scenario and compared with a finite differences deterministic method.

This article is dedicated to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions, or more generally nonconstant monotonic bounded functions, as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multitype version of the sticky particle dynamics and obtain existence of global weak solutions by compactness. We then derive a $L^p$ stability estimate on the particle system uniform in the number of particles. This allows to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all orders, which encompasses the classical $L^1$ estimate and generalises to diagonal systems the results by Bolley, Brenier and Loeper [J. Hyperbolic Differ. Equ., 2005] in the scalar case. Our results are obtained without any smallness assumption on the variation of the data, and only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic.

Brenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bres-san [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the L 1 error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time t is bounded from above by a term of order (1 + t)/n, where n denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using It\^o's calculus and ideas of Arnold, Carlen and Ju \cite{Arnoldcarlenju}, we obtain moreover a new Bakry Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is non-intrisic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide examples where the classic Bakry Emery criterion fails, but our non-intrisic criterion applies without modifying the law of the diffusion process.

By Gyongy's theorem, a local and stochastic volatility model is calibrated to the market prices of all call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labord\`ere (2011), provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to Figalli (2008), we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called Regime Switching Local Volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. We then deduce existence of the calibrated model by extending the results in Figalli (2008).

We deal with the problem of outsourcing the debt for a big investment, according two situations: either the firm outsources both the investment (and the associated debt) and the exploitation to a private consortium, or the firm supports the debt and the investment but outsources the exploitation. We prove the existence of Stackelberg and Nash equilibria between the firm and the private consortium, in both situations. We compare the benefits of these contracts. We conclude with a study of what happens in case of incomplete information, in the sense that the risk aversion coefficient of each partner may be unknown by the other partner.

This thesis is devoted to the study of the strong convergence properties of the Ninomiya and Victoir scheme. The authors of this scheme propose to approximate the solution of a stochastic differential equation (SDE), denoted $X$, by solving $d+1$ ordinary differential equations (ODE) on each time step, where $d$ is the dimension of the Brownian motion. The aim of this study is to analyze the use of this scheme in a multi-step Monte-Carlo method. Indeed, the optimal complexity of this method is directed by the order of convergence towards $0$ of the variance between the schemes used on the coarse and on the fine grid. This order of convergence is itself related to the strong order of convergence between the two schemes. We then show in chapter $2$, that the strong order of the Ninomiya-Victor scheme, denoted $X^{NV,eta}$ and of time step $T/N$, is $1/2$. Recently, Giles and Szpruch proposed a multi-step Monte Carlo estimator realizing $Oleft(epsilon^{-2}right)$ complexity using a modified Milstein scheme. In the same spirit, we propose a modified Ninomiya-Victoir scheme that can be coupled at high order $1$ with the Giles and Szpruch scheme at the last level of a multi-step Monte Carlo method. This idea is inspired by Debrabant and Rossler. These authors suggest using a high low order scheme at the finest discretization level. Since the optimal number of discretization levels of a multi-step Monte Carlo method is directed by the low error of the scheme used on the fine grid of the last discretization level, this technique allows to accelerate the convergence of the multi-step Monte Carlo method by obtaining a high low order approximation. The use of the $1$ coupling with the Giles-Szpruch scheme allows us to keep a multi-step Monte-Carlo estimator realizing an optimal complexity $Oleft( epsilon^{-2} right)$ while taking advantage of the $2$ low order error of the Ninomiya-Victoir scheme. In the third chapter, we are interested in the renormalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$. We show the stable law convergence to the solution of an affine SDE, whose source term is formed by the Lie brackets between the Brownian vector fields. Thus, when at least two Brownian vector fields do not commute, the limit is non-trivial. This ensures that the strong order $1/2$ is optimal. On the other hand, this result can be seen as a first step towards proving a central limit theorem for multi-step Monte-Carlo estimators. To do so, we need to analyze the stable law error of the scheme between two successive discretization levels. Ben Alaya and Kebaier proved such a result for the Euler scheme. When the Brownian vector fields commute, the limit process is zero. We show that in this particular case, the strong order is $1$. In chapter 4, we study the convergence to a stable law of the renormalized error $Nleft(X - X^{NV}right)$ where $X^{NV}$ is the Ninomiya-Victor scheme when the Brownian vector fields commute. We demonstrate the convergence of the renormalized error process to the solution of an affine SDE. When the dritf vector field does not commute with at least one of the Brownian vector fields, the strong convergence speed obtained previously is optimal.

No summary available.

In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity ${\mathcal O}(\epsilon^{-2})$ for the precision $\epsilon$. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order 1 to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.

In this paper, we are interested in the strong convergence properties of the Ninomiya–Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity O(ϵ−2) for the precision ϵ. In the same spirit, we propose a modified Ninomiya–Victoir scheme, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya–Victoir scheme permits to reduce the number of discretisation levels.

In a previous work, we proved strong convergence with order 1 of the Ninomiya-Victoir scheme $X^{\rm NV}$ with time step $T/N$ to the solution $X$ of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process $N(X−X^{\rm NV})$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually 1 when the Brownian vector fields commute, but at least one of them does not commute with the drift vector field. When all the vector fields commute the limit vanishes. Our result is consistent with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.

This thesis studies risk management and hedging using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) as risk measures. The first part proposes a price evolution model that we confront with real data from the Paris stock exchange (Euronext PARIS). Our model takes into account the probabilities of occurrence of extreme losses and the regime changes observed in the data. Our approach consists in detecting the different periods of each regime by constructing a hidden Markov chain and estimating the tail of each regime distribution by power laws. We show empirically that the latter are more suitable than normal and stable distributions. The VaR estimation is validated by several backtests and compared to the results of other classical models on a base of 56 stock assets. In the second part, we assume that stock prices are modeled by exponential Lévy processes. First, we develop a numerical method for computing the cumulative VaR and CVaR. This problem is solved using the formalization of Rockafellar and Uryasev, which we evaluate numerically by Fourier inversion. In a second step, we focus on minimizing the hedging risk of European options, under a budget constraint on the initial capital. By measuring this risk by the CVaR, we establish an equivalence between this problem and a Neyman-Pearson type problem, for which we propose a numerical approximation based on the relaxation of the constraint.

In this thesis, we propose a progressive (forward) approach for the probabilistic representation of nonlinear and nonconservative Partial Differential Equations (PDEs), allowing to develop a particle-based algorithm to numerically estimate their solutions. The Nonlinear Stochastic Differential Equations of McKean type (NLSDE) studied in the literature constitute a microscopic formulation of a phenomenon modeled macroscopically by a conservative PDE. A solution of such a NLSDE is the data of a couple $(Y,u)$ where $Y$ is a solution of a stochastic differential equation (SDE) whose coefficients depend on $u$ and $t$ such that $u(t,cdot)$ is the density of $Y_t$. The main contribution of this thesis is to consider nonconservative PDEs, i.e. conservative PDEs perturbed by a nonlinear term of the form $Lambda(u,nabla u)u$. This implies that a pair $(Y,u)$ will be a solution of the associated probabilistic representation if $Y$ is still a stochastic process and the relation between $Y$ and the function $u$ will then be more complex. Given the law of $Y$, the existence and uniqueness of $u$ are proved by a fixed point argument via an original Feynmann-Kac formulation.

This volume presents elementary random models (discrete and continuous time Markov chains, extreme value laws) and some of their current applications: optimization algorithms, supply management, queue sizing, reliability and design of structures. More recent problems are also addressed: search for exceptional sequences and homogeneous zones of DNA, estimation of the mutation rate of DNA, coagulation phenomena of polymer molecules or aerosols.This volume is intended for a very wide audience of students and teachers. The prerequisite for its reading is the mastery of the contents of an introductory course in probability (4th cover).

No summary available.

We consider the random walk Metropolis algorithm on $\R^n$ with Gaussian proposals, and when the target probability is the $n$-fold product of a one dimensional law. In the limit $n \to \infty$, it is well-known that, when the variance of the proposal scales inversely proportional to the dimension $n$ whereas time is accelerated by the factor $n$, a diffusive limit is obtained for each component of the Markov chain if this chain starts at equilibrium. This paper extends this result when the initial distribution is not the target probability measure. Remarking that the interaction between the components of the chain due to the common acceptance/rejection of the proposed moves is of mean-field type, we obtain a propagation of chaos result under the same scaling as in the stationary case. This proves that, in terms of the dimension $n$, the same scaling holds for the transient phase of the Metropolis-Hastings algorithm as near stationarity. The diffusive and mean-field limit of each component is a nonlinear diffusion process in the sense of McKean. This opens the route to new investigations of the optimal choice for the variance of the proposal distribution in order to accelerate convergence to equilibrium.

We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.

This thesis deals with the development of Monte Carlo methods for computing Feynman-Kac representations involving operators in divergence form with a piecewise constant diffusion coefficient. The proposed methods are variants of the walk on spheres within zones with a constant diffusion coefficient and stochastic finite difference techniques to deal with interface conditions as well as boundary conditions of different types. By combining these two techniques, we obtain random walks whose score computed along the path provides a biased estimator of the solution of the partial differential equation considered. We show that the global bias of our algorithm is in general of order two with respect to the finite difference step. These methods are then applied to the direct problem related to electrical impedance tomography for tumor detection. A variance reduction technique is also proposed in this framework. Finally, the inverse problem of tumor detection from surface measurements is addressed using two stochastic algorithms based on a parametric representation of the tumor or tumors as one or more spheres. Numerous numerical tests are proposed and show convincing results in the localization of tumors.

The Self-Healing Umbrella Sampling (SHUS) algorithm is an adaptive biasing algorithm which has been proposed to efficiently sample a multimodal probability measure. We show that this method can be seen as a variant of the well-known Wang-Landau algorithm. Adapting results on the convergence of the Wang-Landau algorithm, we prove the convergence of the SHUS algorithm. We also compare the two methods in terms of efficiency. We finally propose a modification of the SHUS algorithm in order to increase its efficiency, and exhibit some similarities of SHUS with the well-tempered metadynamics method.

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and Hölder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C \left(1+\mathbf{1}_{\gamma=1} \sqrt{\ln(N)}\right)N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio, Gigli and Savaré to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and Holder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C \left(1+\mathbf{1}_{\gamma=1} \sqrt{\ln(N)}\right)N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio, Gigli and Savare to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

We consider the Random Walk Metropolis algorithm on $\R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known (see Roberts et al. (1997))) that, in the limit $n \to \infty$, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain.

We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.

We study a mean-field version of rank-based models of equity markets such as the Atlas model introduced by Fernholz in the framework of Stochastic Portfolio Theory. We obtain an asymptotic description of the market when the number of companies grows to infinity. Then, we discuss the long-term capital distribution. We recover the Pareto-like shape of capital distribution curves usually derived from empirical studies, and provide a new description of the phase transition phenomenon observed by Chatterjee and Pal. Finally, we address the performance of simple portfolio rules and highlight the influence of the volatility structure on the growth of portfolios.

This thesis consists of three essentially independent parts, each of which is devoted to the study of a system of particles, following a deterministic or random dynamics, and inside which the interactions are done only at the collisions between the particles.Part I proposes a numerical and theoretical study of the non-equilibrium stationary states of the Complete Exchange Model, introduced in physics to understand the transport of heat in some porous materials.Part II is devoted to a system of Brownian particles evolving on the real straight line and interacting through their rank. The limiting behavior of this system, in long time and with a large number of particles, is described, then the results are applied to the study of a financial market model called mean field Atlas model.Part III introduces a multitype version of the particle system studied in the previous part, which allows to approach parabolic systems of nonlinear partial differential equations. The small noise limit of this system is called multitype sticky particle dynamics and approaches this time hyperbolic systems. A detailed study of this dynamics gives stability estimates in Wasserstein distance on the solutions of these systems.

We introduce order-based diffusion processes as the solutions to multidimensional stochastic differential equations, with drift coefficient depending only on the ordering of the coordinates of the process and diffusion matrix proportional to the identity. These processes describe the evolution of a system of Brownian particles moving on the real line with piecewise constant drifts, and are the natural generalization of the rank-based diffusion processes introduced in stochastic portfolio theory or in the probabilistic interpretation of nonlinear evolution equations. Owing to the discontinuity of the drift coefficient, the corresponding ordinary differential equations are ill-posed. Therefore, the small noise limit of order-based diffusion processes is not covered by the classical Freidlin-Wentzell theory. The description of this limit is the purpose of this article. We first give a complete analysis of the two-particle case. Despite its apparent simplicity, the small noise limit of such a system already exhibits various behaviours. In particular, depending on the drift coefficient, the particles can either stick into a cluster, the velocity of which is determined by elementary computations, or drift away from each other at constant velocity, in a random ordering. The persistence of randomness in the zero noise limit is of the very same nature as in the pioneering works by Veretennikov (Mat. Zametki, 1983) and Bafico and Baldi (Stochastics, 1981) concerning the so-called Peano phenomenon. In the case of rank-based processes, we use a simple convexity argument to prove that the small noise limit is described by the sticky particle dynamics introduced by Brenier and Grenier (SIAM J. Numer. Anal., 1998), where particles travel at constant velocity between collisions, at which they stick together. In the general case of order-based processes, we give a sufficient condition on the drift for all the particles to aggregate into a single cluster, and compute the velocity of this cluster. Our argument consists in turning the study of the small noise limit into the study of the long time behaviour of a suitably rescaled process, and then exhibiting a Lyapunov functional for this rescaled process.

In this paper, we are interested in the long-time behaviour of stochastic systems of n interacting vortices: the position in R2 of each vortex evolves according to a Brownian motion and a drift summing the influences of the other vortices computed through the Biot and Savart kernel and multiplied by their respective vorticities. For fixed n, we perform the rescalings of time and space used successfully by Gallay and Wayne [5] to study the long-time behaviour of the vorticity formulation of the two dimensional incompressible Navier-Stokes equation, which is the limit as n → ∞ of the weighted empirical measure of the system under mean-field interaction. When all the vorticities share the same sign, the 2n-dimensional process of the rescaled positions of the vortices is shown to converge exponentially fast as time goes to infinity to some invariant measure which turns out to be Gaussian if all the vorticities are equal. In the particular case n = 2 of two vortices, we prove exponential convergence in law of the 4-dimensional process to an explicit random variable, whatever the choice of the two vorticities. We show that this limit law is not Gaussian when the two vorticities are not equal.

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with $N$ steps is smaller than $O(N^{-2/3+\varepsilon})$ where $\varepsilon$ is an arbitrary positive constant. This rate is intermediate between the strong error estimation in $O(N^{-1/2})$ obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation $O(N^{-1})$ obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time $T$. We also check that the supremum over $t\in[0,T]$ of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time $t$ and the Euler scheme at time $t$ behaves like $O(\sqrt{\log(N)}N^{-1})$.

We are interested in the Wasserstein distance between two probability measures on $\R^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. It turns out that for cost functions $c(x,y)$ equal to the $p$-th power of the $L^q$ norm of $x-y$ in $\R^n$, this coupling is optimal only when $p=q$ i.e. when $c(x,y)$ may be decomposed as the sum of coordinate-wise costs.

In the present paper, we first deal with the discretization of stochastic dierential equa- tions. We elaborate on the analysis of the weak error of the Euler scheme by Talay and Tubaro (31) to contruct schemes with quicker weak rate of convergence for SDEs corresponding to an infinitesimal generator with smooth coecients. We also extend this analysis to the case of a discontinuous drift coecient. In a second part, we present two applications of stochastic gradient algorithms in finance.

No summary available.

Stochastic volatility models can be seen as a particular family of two-dimensional stochastic differential equations (SDE) in which the volatility process follows an autonomous one-dimensional SDE. We take advantage of this structure to propose an efficient discretization scheme with order two of weak convergence. We prove that the order two holds for the asset price and not only for the log-asset as usually found in the literature. Numerical experiments confirm our theoretical result and we show the superiority of our scheme compared to the Euler scheme, with or without Romberg extrapolation.

No summary available.

We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all times. We prove the uniqueness of such a solution and we deduce the existence from a propagation of chaos result on a system of scalar diffusion processes, the interactions of which only depend on their ranking. We then investigate the long time behaviour of the solution. Using a probabilistic argument and under weak assumptions, we show that the flow of the Wasserstein distance between two solutions is contractive. Under more stringent conditions ensuring the regularity of the probabilistic solutions, we finally derive an explicit formula for the time derivative of the flow and we deduce the convergence of solutions to equilibrium.

In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, - a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a, 2008b].

In this thesis, we treat two stochastic optimal control problems. Each problem corresponds to a part of this paper. The first problem is very specific, it is the valuation of American put contracts in the presence of discrete dividends (Part I). The second one is more general, since it is about proving the existence of a dynamic programming principle under probability constraints in a discrete time framework (Part II). Although the two problems are quite distinct, the dynamic programming principle is at the heart of both problems. The relation between the valuation of an American Put and a free boundary problem has been proved by McKean. The frontier of this problem has a clear economic meaning since it corresponds at any moment to the upper bound of the set of asset prices for which it is preferable to exercise one's right to sell immediately. The shape of this frontier in the presence of discrete dividends has not been solved to our knowledge. Under the assumption that the dividend is a deterministic function of the asset price at the time preceding its payment, we study how the frontier is modified. In the vicinity of the dividend dates, and in the model of Chapter 3, we know how to qualify the monotonicity of the frontier, and in some cases quantify its local behavior. In Chapter 3, we show that the smooth-fit property is satisfied at all dates except the dividend dates. In both Chapters 3 and 4, we give conditions to guarantee the continuity of this frontier outside the dividend dates. Part II is originally motivated by the optimal management of the production of a hydro-electric plant with a constraint in probability on the water level of the dam at certain dates. Using Balder's work on Young's relaxation of optimal control problems, we focus more specifically on solving them by dynamic programming. In Chapter 5, we extend the results of Evstigneev to the framework of Young's measurements. We then establish that it is possible to solve by dynamic programming some problems with constraints in conditional expectations. Thanks to the work of Bouchard, Elie, Soner and Touzi on stochastic target problems with controlled loss, we show in Chapter 6 that a problem with expectation constraints can be reduced to a problem with conditional expectation constraints. As a special case, we prove that the initial dam management problem can be solved by dynamic programming.

Vanilla calibration is a major problem in finance. We try to solve it for three classes of models: local and stochastic volatility models, the so-called "local correlation" model and a hybrid model of local volatility with stochastic rates. From a mathematical point of view, the calibration equation is a particularly complex nonlinear and integro-differential equation. In a first part, we prove the existence of solutions for this equation, as well as for its adjoint (simpler to solve). These results are based on fixed point methods in Hölder spaces and require classical theorems related to parabolic partial differential equations, as well as some a priori estimates in short time. The second part deals with the application of these existence results to the three financial models mentioned above. We also present the numerical results obtained by solving the edp. The calibration by this method is quite satisfactory. Finally, we focus on the algorithm used for the numerical solution: a predictor-corrector ADI scheme, which is modified to take into account the nonlinear character of the equation. We also describe an instability phenomenon of the edp solution that we try to explain from a theoretical point of view thanks to the so-called "Hadamard instability".

This work presents some results on interacting particle systems for the probabilistic interpretation of partial differential equations, with applications to questions of molecular dynamics and quantum chemistry. In particular, a particle method is presented to analyze the adaptive biasing force process, used in molecular dynamics for the calculation of free energy differences. The sensitivity of stochastic dynamics with respect to a parameter is also studied, in view of the calculation of forces in the Born-Oppenheimer approximation to search for the fundamental quantum state of molecules. Finally, we present a numerical scheme based on a system of particles to solve scalar conservation laws, with an anomalous diffusion term resulting in jump dynamics on the particles.

The first part of this thesis is devoted to numerical methods for the simulation of random processes defined by stochastic differential equations (SDE). We start by studying the algorithm of Beskos et al [13] which allows us to simulate exactly the trajectories of a process that is a solution of a SDE in dimension 1. We propose an extension of this algorithm for the exact computation of expectations and we study the application of these ideas to the pricing of Asian options in the Black & Scholes model. We then turn our attention to numerical schemes. In the second chapter, we propose two discretization schemes for a family of stochastic volatility models and study their convergence properties. The first scheme is adapted to the pricing of path-dependent options and the second to vanilla options. We also study the special case where the process driving the volatility is an Ornstein-Uhlenbeck process and we exhibit a discretization scheme that has better convergence properties. Finally, in the third chapter, we discuss the weak trajectory convergence of the Euler scheme. We provide a first answer by controlling the Wasserstein distance between the marginals of the solution process and the Euler scheme, uniformly in time. The second part of the thesis deals with the modeling of dependence in finance and this through two distinct problems: the joint modeling between a stock index and its component stocks and the management of default risk in credit portfolios. In the fourth chapter, we propose an original modeling framework in which the volatilities of the index and its components are linked. We obtain a simplified model when the index size is large, in which the index follows a local volatility model and the individual stocks follow a stochastic volatility model composed of an intrinsic part and a common part driven by the index. We study the calibration of these models and show that it is possible to calibrate to observed market option prices for both the index and the stocks, which is a considerable advantage. Finally, in the last chapter of the thesis, we develop an intensity model that allows us to simultaneously and consistently model all rating transitions that occur in a large credit portfolio. In order to generate higher levels of dependence, we introduce the dynamic frailty model in which an unobservable dynamic variable acts multiplicatively on the intensities of transitions. Our approach is purely historical and we study maximum likelihood estimation of the parameters of our models based on data of past rating transitions.

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This work focuses on the mathematical analysis of multi-scale models for the simulation of polymeric fluids. These models couple, at the microscopic level, a molecular description of the evolution of polymer chains (in the form of a stochastic differential equation) with, at the macroscopic level, the conservation of mass and momentum equations for the solvent (in the form of partial differential equations). Chapter 1 introduces the models and gives the main results obtained. In chapters 2, 4, 5 and 7 we show in which sense the equations are well posed for various polymer models, considering either homogeneous flows or plane sheared flows. In chapters 2, 3, 6 and 7, we analyze and prove the convergence of numerical methods for these models. Finally, chapter 8 deals with the long time behavior of the system. A second part of this paper consists of three chapters dealing with work in magnetohydrodynamics (MHD), in collaboration with industry. Chapter 9 is an introduction to the problem and to the numerical methods used. Chapter 10 describes a new test case in MHD. Finally, chapter 11 gives an analysis of the stability of the numerical scheme used.

The implied volatility smile observed in the options markets reflects the inadequacy of the Black-Scholes model. With the need to develop a more satisfactory financial asset model, comes the need to calibrate it, which is the subject of this thesis. Calibration by relative entropy minimization has been recently proposed in the framework of the Monte-Carlo method. The convergence and stability of this method have been studied and it has been extended to more general criteria than relative entropy. For the absence of arbitrage opportunity to exist, the discounted underlying must be a martingale. The consideration of this necessity is absolved from the perspective of a moments problem. In the second part, we considered a simple model of the crash phenomenon by introducing in particular jumps in the volatility of the underlying. We computed the quadratic risk and performed an approximate development of the smile which constitutes a tool for the calibration. Finally, in the third part, we use the relative entropy to calibrate the intensity of jumps in a diffusion model with jumps and local volatility. The stability of the method is proved using optimal control techniques and the implicit function theorem.

The approach adopted in this thesis is the following. Considering a nonlinear evolution equation, we try to associate a probability p on a space of trajectories such that : - either the marginals in time of p have densities with respect to the lebesgue measure in space which are weak solution of the evolution equation - or, in the case of dimension one of space, the distribution functions of the marginals are weak solution of the equation Once the probability p is obtained as the unique solution of a nonlinear martingale problem, we construct a system of n probabilistically interacting particles whose empirical measure converges to p when n tends to infinity. Such a convergence result, called chaos propagation, allows us to approach the solutions of the evolution equation by simulating the system of particles. For the most part, we treat parabolic type equations such as the viscous burgers equation and the porous media equation in this program. Under p, the canonical process on the space of continuous trajectories is then a nonlinear diffusion. We are also interested in a kinetic equation related to scalar conservation laws and work for that on a space of trajectories with jumps. Finally, we show on examples that it is not necessary to restrict ourselves to the natural case where the initial condition of the evolution equation is a probability or a probabilistic distribution function. It is possible to adapt the approach to take into account bounded signed measures or their distribution functions.

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