arXiv subjects
Publications and source records attributed to Paul Raynaud De Fitte.
This paper deals with the existence, the uniqueness and an approximation scheme of the solution to sweeping processes perturbed by a continuous signal of finite $p$-variation with $p\in [1,3[$. It covers pathwise stochastic noises directed by a fractional Brownian motion of Hurst parameter greater than $1/3$.
We prove the existence of a weak solution to a backward stochastic differential equation (BSDE) $$ Y_t=ξ+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s$$ 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 $$ Y_t=ξ+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s-(L_T-L_t)$$ 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.
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).
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.
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.