GIORGI Daphne

This article discusses MLMC estimators with and without weights, applied to nested expectations of the form E [f (E [F (Y, Z)|Y ])]. More precisely, we are interested on the assumptions needed to comply with the MLMC framework, depending on whether the payoff function f is smooth or not. A new result to our knowledge is given when f is not smooth in the development of the weak error at an order higher than 1, which is needed for a successful use of MLMC estimators with weights.

This article discusses MLMC estimators with and without weights, applied to nested expectations of the form E [f (E [F (Y, Z)|Y ])]. More precisely, we are interested on the assumptions needed to comply with the MLMC framework, depending on whether the payoff function f is smooth or not. A new result to our knowledge is given when f is not smooth in the development of the weak error at an order higher than 1, which is needed for a successful use of MLMC estimators with weights.

In this work, we will focus on the Multilevel Monte Carlo estimators. These estimators will appear in their standard form, with weights and in their randomized form. We will recall the previous existing results concerning these estimators, in terms of minimization of the simulation cost. We will then show a strong law of large numbers and a central limit theorem.After that, we will focus on two application frameworks.The first one is the diffusions framework with antithetic discretization schemes, where we will extend the Multilevel estimators to Multilevel estimators with weights, and the second is the nested framework, where we will analyze the weak and the strong error assumptions. We will conclude by implementing the randomized form of the Multilevel estimators, comparing this to the Multilevel estimators with and without weights.

We aim at analyzing in terms of a.s. convergence and weak rate the performances of the Multilevel Monte Carlo estimator (MLMC) introduced in [Gil08] and of its weighted version, the Multilevel Richardson Romberg estimator (ML2R), introduced in [LP14]. These two estimators permit to compute a very accurate approximation of $I_0 = \mathbb{E}[Y_0]$ by a Monte Carlo type estimator when the (non-degenerate) random variable $Y_0 \in L^2(\mathbb{P})$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations $(Y_h)_{h \in \mathcal{H}}$ is available. We will carry out these investigations in an abstract framework before applying our results, mainly a Strong Law of Large Numbers and a Central Limit Theorem, to some typical fields of applications: discretization schemes of diffusions and nested Monte Carlo.

In this work, we are interested in Multilevel Monte Carlo estimators. These estimators will appear in their standard form, with weights and in a randomized form. We will recall their definitions and the existing results concerning these estimators in terms of simulation cost minimization. We will then show a strong law of large numbers and a central limit theorem. After that we will study two application frameworks. The first one is that of diffusions with antithetic discretization schemes, where we will extend the Multilevel estimators to Multilevel estimators with weights. The second is the nested framework, where we will focus on strong and weak error assumptions. We will conclude with the implementation of the randomized form of Multilevel estimators, comparing it to Multilevel estimators with and without weights.

The original Bocop package implements a local optimization method. The optimal control problem is approximated by a finite dimensional optimization problem (NLP) using a time discretization (the direct transcription approach). The NLP problem is solved by the well known software Ipopt, using sparse exact derivatives computed by Adol-C. The second package BocopHJB implements a global optimization method. Similarly to the Dynamic Programming approach, the optimal control problem is solved in two steps. First we solve the Hamilton-Jacobi-Bellman equation satisfied by the value fonction of the problem. Then we simulate the optimal trajectory from any chosen initial condition. The computational effort is essentially taken by the first step, whose result, the value fonction, can be stored for subsequent trajectory simulations.