Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H\in (0,1)$ called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case $H=1/2$, the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used. Different approaches have been introduced to construct stochastic integrals with respect to fBm:...

Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process ${W_t^H}_{t\in [0,T]}$ with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) $d{X}_t=A{X}_{t}dt+Bd{W}_t^H$ (t∈ [0,T]), $X_0=0$ almost surely, where A is the generator of a $C_0$-semigroup ${S\left(t\right)}_{t\ge 0}$ of bounded linear operators on...

A generalization of the Poisson driven stochastic differential equation is considered. A sufficient condition for asymptotic stability of a discrete time-nonhomogeneous Markov process is proved.

We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

We have seen in a previous article how the theory of “rough paths” allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals...

In this article we prove the pathwise uniqueness for stochastic differential equations in ${ℝ}^d$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha$-stable processes provided that $\alpha \in (1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha \in (\frac{2d}{d+1},2)$. Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C₀-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control problems....

Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac{1}{2}$ we show that a function $\Phi :(0,T)\to ℒ(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...

In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions...

We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes...