Applications of the error theory using Dirichlet forms.

Summary This thesis is devoted to the study of applications of the theory of errors by Dirichlet forms. Our work is divided into three parts. The first one analyzes models governed by a stochastic differential equation. After a short technical chapter, an innovative model for order books is proposed. We consider that the bid-ask spread is not a defect, but rather an intrinsic property of the market. Uncertainty is carried by the Brownian motion that drives the asset. We show that the evolution of the spread can be evaluated using closed formulas and we study the impact of the uncertainty of the underlying on derivatives. We then introduce the PBS model for pricing European options. The innovative idea is to distinguish the market volatility from the parameter used by traders to hedge. We assume the former constant, while the latter becomes a subjective and erroneous estimate of the former. We prove that this model predicts a bid-ask spread and a volatility smile. The most interesting properties of this model are the existence of closed formulas for pricing, the impact of the underlying drift and an efficient calibration strategy. The second part focuses on models described by a partial differential equation. The linear and nonlinear cases are analyzed separately. In the first case we show interesting relations between error theory and wavelet theory. In the nonlinear case we study the sensitivity of the solutions using the error theory. Except in the case of an exact solution, there are two possible approaches: one can first discretize the PDE and study the sensitivity of the discretized problem, or demonstrate that the theoretical sensitivities verify PDEs. Both cases are studied, and we prove that the sharp and the bias are solutions of linear PDEs depending on the solution of the original PDE and we propose algorithms to numerically evaluate the sensitivities. Finally, the third part is dedicated to stochastic partial differential equations. Our analysis is divided into two chapters. First, we study the transmission of the uncertainty, present in the initial condition, to the solution of the SPDE. Then, we analyze the impact of a perturbation in the functional terms of the EDPS and in the coefficient of the associated Green's function. In both cases, we prove that the sharp and the bias are solutions of two linear EDPS depending on the solution of the original EDPS.