On Ito's formula of Föllmer and Protter
On least squares estimation in continuous time linear stochastic systems
On Lipschitz Dependence in Systems with Differentiated Inputs.
On logarithmic Sobolev inequalities for normal martingales
On mild solutions of gradient systems in Hilbert spaces
We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.
On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps, with applications to finance
We establish necessary and sufficient conditions of near-optimality for nonlinear systems governed by forward-backward stochastic differential equations with controlled jump processes (FBSDEJs in short). The set of controls under consideration is necessarily convex. The proof of our result is based on Ekeland's variational principle and continuity in some sense of the state and adjoint processes with respect to the control variable. We prove that under an additional hypothesis, the near-maximum...
On Newton's method for stochastic differential equations
On nonperturbative localization with quasi-periodic potential.
On nonuniform dichotomy for stochastic skew-evolution semiflows in Hilbert spaces
In this paper we study a general concept of nonuniform exponential dichotomy in mean square for stochastic skew-evolution semiflows in Hilbert spaces. We obtain a variant for the stochastic case of some well-known results, of the deterministic case, due to R. Datko: Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3(1972), 428–445. Our approach is based on the extension of some techniques used in the deterministic case for the study of asymptotic behavior...
On One Estimation Problem
On one estimation problem.
On one-dimensional diffusion processes living in a bounded space interval
We prove that under some assumptions a one-dimensional Itô equation has a strong solution concentrated on a finite spatial interval, and the pathwise uniqueness holds.
On pathwise uniqueness and expansion of filtrations
On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes
We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order with drift and diffusion coefficients , . When , we investigate pathwise uniqueness for this equation. When , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether or and on whether the driving stable process is symmetric or not. Our assumptions...
On Poisson-Dirichlet problems with polynomial data
In this note we provide a probabilistic proof that Poisson and/or Dirichlet problems in an ellipsoid in Rd, that have polynomial data, also have polynomial solutions. Our proofs use basic stochastic calculus. The existing proofs are based on famous lemma by E. Fisher which we do not use, and present a simple martingale proof of it as well.
On processes with conditional independent increments and stable convergence in law
On random boundary value problems for ordinary differential equations
On random evolutions induced by countable state space Markov chains
On random orthogonal polynomials.
On regular points in Burgers turbulence with stable noise initial data