Dans cet article, nous étudions les résultats de grandes déviations associés au couple $(X^{\epsilon },{\nu }^{\epsilon })$, solution de l’E.D.S. interprétée au sens d’Itô :$$d{X}_t^{\epsilon }=\sqrt{\epsilon }{\sigma }_{{\nu }^{\epsilon }\left(t\right)}\left(X_t^{\epsilon }\right)d{W}_t+b_{{\nu }^{\epsilon }\left(t\right)}\left(X_t^{\epsilon }\right)dt;X_0^{\epsilon }=x\in {ℝ}^d$$avec des conditions assez générales sur les coefficients et dans les deux cas suivants :Premier cas : ${\nu }^{\epsilon }$ est indépendant du mouvement brownien $W$ et satisfait à un principe de grandes déviations ;Deuxième cas : ${\nu }^{\epsilon }$ est un processus markovien avec un nombre fini d’états $\{1,...,n\}$ vérifiant$$\mathbb{P}\{{\nu }^{\epsilon }(t+\Delta )=j/{\nu }^{\epsilon }\left(t\right)=i,X^{\epsilon }\left(t\right)=x\}=d_{ij}\left(x\right)\Delta +o\left(\Delta \right)$$uniformément dans ${ℝ}^d$ pourvu que $\Delta \downarrow 0,1\le i,j\le n,i\ne j$.Ces résultats sont des extensions de ceux de Bezuidenhout...

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit distribution from a given well for a single process can be approximated by...

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice ${\mathbb{Z}}^d$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with...

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice ${\mathbb{Z}}^d$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with...

We define approximation schemes for generalized backward stochastic differential systems, considered in the Markovian framework. More precisely, we propose a mixed approximation scheme for the following backward stochastic variational inequality: $$d{Y}_t+F(t,X_t,Y_t,Z_t)dt\in \partial \phi \left(Y_t\right)dt+Z_{t}d{W}_t,$$ where ∂φ is the subdifferential operator of a convex lower semicontinuous function φ and (X t)t∈[0;T] is the unique solution of a forward stochastic differential equation. We use an Euler type scheme for the system of decoupled forward-backward variational...

This paper deals with the construction of numerical solution of the Black-Scholes (B-S) type equation modeling option pricing with variable yield discrete dividend payment at time $t_d$. Firstly the shifted delta generalized function $\delta (t-t_d)$ appearing in the B-S equation is approximated by an appropriate sequence of nice ordinary functions. Then a semidiscretization technique applied on the underlying asset is used to construct a numerical solution. The limit of this numerical solution is independent of the...

We study a new class of Markov type semigroups (not strongly continuous in general) in the space of all real, uniformly continuous and bounded functions on a separable metric space E. Our results allow us to characterize the generators of Markov transition semigroups in infinite dimensions such as the heat and the Ornstein-Uhlenbeck semigroups.