Numerical solution of stochastic backward differential equations.

Summary The first part of this thesis aims at building a probabilistic algorithm for numerically solving stochastic backward differential equations (sdDEs) in the Markovian case, where the equation is associated with a forward process solution of a sdDE. We describe a first algorithm that relies on a double discretization of the equation, in time and in space, and uses simulations of trajectories of the forward process. The discretization in time is an extension of Euler's scheme for eds, where we replace the Brownian motion by a random walk. We then introduce an additional approximation by projecting at each discretization time the forward process on the set of simulated trajectories. This avoids an algorithmic complexity that would be exponential. We show a speed of convergence for this algorithm in dimension 1. We also present a variant of this algorithm, adapted to edsr whose parameters are less regular, by replacing in particular the euler scheme in the discretization of the forward process by the milshtein scheme. This allows us to write an algorithm for the discretization of reflected edsr. In a second part, we analyze the macmillan, and barone-adesi and whaley approximation, used in finance to estimate the price of an American option. By writing the price of the American option as the solution of a certain reflected backward stochastic differential equation, we obtain a general bound for the error of the approximation and show that the approximation converges to the exact price when the volatility of the underlying asset tends to zero. We then propose a second, more elementary, demonstration of this asymptotic result, using the price of a perpetual put.