On the discretization and small-noise behavior of one-dimensional DHS with singular derivative coefficients.

Summary The first part of this thesis deals with the approximation of solutions of one-dimensional stochastic differential equations with non-Lipschitzian coefficients. Our attention is focused on two classes of equations widely used in finance. We first consider a generalization of the Cox-Ingersoll-Ross and Hull & White models . the drift coefficient is boundedly derivative, while the diffusion coefficient is of the type σ (x) = xα, with ½ ≤ α < 1. We then consider the SDE verified by a Bessel process . the drift coefficient is of type C on x, with C > 0 and thus has a singularity in zero. We place ourselves under assumptions that ensure the existence and uniqueness of solutions with strictly positive trajectories almost surely and propose discretization schemes that preserve the positivity of the approximated processes. On the one hand, we obtain the weak convergence speed of the schemes for a class of regular test functions and, on the other hand, we analyze by a time change method the strong convergence speed of the scheme in the case where the diffusion coefficient is of the type σ (x) = xα. The second part of the thesis addresses the problem of the asymptotic behavior of the solution of a parabolic partial differential equation (PDE) with a discontinuous first order coefficient when the viscosity tends to zero. We show that under an assumption of monotonicity on the first order coefficient, the solution converges weakly to the "measure solution" of the associated transport equation.