In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.

This paper deals with nonlinear filtering problems with delays, i.e., we consider a system (X,Y), which can be represented by means of a system (X,Ŷ), in the sense that Yt = Ŷa(t), where a(t) is a delayed time transformation. We start with X being a Markov process, and then study Markovian systems, not necessarily diffusive, with correlated noises. The interest is focused on the existence of explicit representations of the corresponding filters as functionals depending on the observed trajectory....

We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In the proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.

The paper deals with a filter design for nonlinear continuous stochastic systems with discrete-time measurements. The general recursive solution is given by the Fokker–Planck equation (FPE) and by the Bayesian rule. The stress is laid on the computation of the predictive conditional probability density function from the FPE. The solution of the FPE and its integration into the estimation algorithm is the cornerstone for the whole recursive computation. A new usable numerical scheme for the FPE is...

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.

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...