Jean Daniel Mukam

Antoine Tambue, Jean Daniel Mukam

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to substain the theoretical results are provided.

Jean Daniel Mukam, Antoine Tambue

In this paper, we investigate a numerical approximation of a general second order semilinear parabolic non-autonomous stochastic partial differential equation (SPDE) driven by additive noise. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature while the non-autonomous case is not yet well understood. We discretize the non-autonomous SPDE in space by the finite element method and in time by the Magnus-type integrator. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. Appropriate assumptions on the drift term and the noise allow to achieve optimal convergence order in time greater than $1/2$, without any logarithmic reduction of convergence order in time. In particular, for trace class noise, we achieve optimal convergence orders $\mathcal{O}\left(h^{2-ε}+Δt\right)$, where $ε$ is a positive number small enough. Numerical simulations are provided to illustrate our theoretical results.

Antoine Tambue, Jean Daniel Mukam

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order $\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+Δt^{1/2}\right)$. Numerical simulations to illustrate our theoretical finding are provided.

Antoine Tambue, Jean Daniel Mukam

This paper aims to investigate a full numerical approximation of non-autonomous semilnear parabolic partial differential equations (PDEs) with nonsmooth initial data. Our main interest is on such PDEs where the nonlinear part is stronger than the linear part, also called reactive dominated transport equations. For such equations, many classical numerical methods lose their stability properties. We perform the space and time discretizations respectively by the finite element method and an exponential integrator. We obtain a novel explicit, stable and efficient scheme for such problems called Magnus-Rosenbrock method. We prove the convergence of the fully discrete scheme toward the exact solution. The result shows how the convergence orders in both space and time depend on the regularity of the initial data. In particular, when the initial data belongs to the domain of the family of the linear operator, we achieve convergence orders $\mathcal{O}\left(h^{2}+Δt^{2-ε}\right)$, for an arbitrarily small $ε>0$. Numerical simulations to illustrate our theoretical result are provided.

Antoine Tambue, Jean Daniel Mukam

Under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution, while Euler implicit method converges but requires much computational efforts. Tamed scheme was first introduced in [2] to overcome this failure of the standard explicit method. This technique is extended to SDEs driven by Poisson jump in [3] where several schemes were analyzed. In this work, we investigate their nonlinear stability under non-global Lipschitz and their linear stability. Numerical simulations to sustain the theoretical results are also provided.

Jean Daniel Mukam

In this work we consider a stochastic differential equation (SDEs) with jump. We prove the existence and the uniqueness of solution of this equation in the strong sense under global Lipschitz condition. Generally, exact solutions of SDEs are unknowns. The challenge is to approach them numerically. There exist several numerical techniques. In this thesis, we present the compensated stochastic theta method (CSTM) which is already developed in the literature. We prove that under global Lipschitz condition, the CSTM converges strongly with standard order 0.5. We also investigated the stability behaviour of both CSTM and stochastic theta method (STM). Inspired by the tamed Euler scheme developed in [8], we propose a new scheme for SDEs with jumps called compensated tamed Euler scheme. We prove that under non-global Lipschitz condition the compensated tamed Euler scheme converges strongly with standard order 0.5. Inspired by [11], we propose the semi-tamed Euler for SDEs with jumps under non-global Lipschitz condition and prove its strong convergence of order 0.5. This latter result is helpful to prove the strong convergence of the tamed Euler scheme. We analyse the stability behaviours of both tamed and semi-tamed Euler scheme We present also some numerical experiments to illustrate our theoretical results.