Weak solutions of backward stochastic differential equations with continuous generator

We prove the existence of a weak solution to a backward stochastic differential equation (BSDE) Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s in a finite-dimensional space, where $f(t,x,y,z)$ is affine with respect to $z$, and satisfies a sublinear growth condition and a continuity condition This solution takes the form of a triplet $(Y,Z,L)$ of processes defined on an extended probability space and satisfying Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s-(L_T-L_t) where $L$ is a continuous martingale which is orthogonal to any \wien. The solution is constructed on an extended probability space, using Young measures on the space of trajectories. One component of this space is the Skorokhod space D endowed with the topology S of Jakubowski

Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients

We show that, contrarily to what is claimed in some papers, the nontrivial solutions of some stochastic differential equations with almost periodic coefficients are never mean square almost periodic (but they can be almost periodic in distribution)

Existence of weak solutions to stochastic evolution inclusions

We consider the Cauchy problem for a semilinear stochastic differential inclusion in a Hilbert space. The linear operator generates a strongly continuous semigroup and the nonlinear term is multivalued and satisfies a condition which is more heneral than the Lipschitz condition. We prove the existence of a mild solution to this problem. This solution is not "strong" in the probabilistic sense, that is, it is not defined on the underlying probability space, but on a larger one, which provides a "very good extension" in the sense of Jacod and Memin. Actually, we construct this solution as a Young measure, limit of approximated solutions provided by the Euler scheme. The compactness in the space of Young measures of this sequence of approximated solutions is obtained by proving that some measure of noncompactness equals zero

Parametrized Kantorovich-Rubinstein theorem and application to the coupling of random variables

We prove a version for random measures of the celebrated Kantorovich-Rubinstein duality theorem and we give an application to the coupling of random variables which extends and unifies known results.Comment: date de redaction 22 octobre 200

Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application

We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases.Comment: 18 page

Almost periodic solution in distribution for stochastic differential equations with Stepanov almost periodic coefficients

This paper deals with the existence and uniqueness of ($\mu$-pseudo) almost periodic mild solution to some evolution equations with Stepanov ($\mu$-pseudo) almost periodic coefficients, in both determinist and stochastic cases. After revisiting some known concepts and properties of Stepanov ($\mu$-pseudo) almost periodicity in complete metric space, we consider a semilinear stochastic evolution equation on a Hilbert separable space with Stepanov ($\mu$-pseudo) almost periodic coefficients. We show existence and uniqueness of the mild solution which is ($\mu$-pseudo) almost periodic in 2-distribution. We also generalize a result by Andres and Pennequin, according to which there is no purely Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients

Dynamics of a prey-predator system with modified Leslie-Gower and Holling type II schemes incorporating a prey refuge

We study a modified version of a prey-predator system with modified Leslie-Gower and Holling type II functional response studied by M.A. Aziz-Alaoui and M. Daher-Okiye. The modification consists in incorporating a refuge for preys, and substantially complicates the dynamics of the system. We study the local and global dynamics and the existence of cycles. We also investigate conditions for extinction or existence of a stationary distribution, in the case of a stochastic perturbation of the system