arXiv subjects

Salah-Eldin A. Mohammed

Publications and source records attributed to Salah-Eldin A. Mohammed.

4 recordsLinked to original sources

arXiv2015

Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives

In this article we consider a stochastic optimal control problem where the dynamics of the state process, $X(t)$, is a controlled stochastic differential equation with jumps, delay and \emph{noisy memory}. The term noisy memory is, to the best of our knowledge, new. By this we mean that the dynamics of $X(t)$ depend on $\int_{t-δ}^t X(s) dB(s)$ (where $B(t)$ is a Brownian motion). Hence, the dependence is noisy because of the Brownian motion, and it involves memory due to the influence from the previous values of the state process. We derive necessary and sufficient maximum principles for this stochastic control problem in two different ways, resulting in two sets of maximum principles. The first set of maximum principles is derived using Malliavin calculus techniques, while the second set comes from reduction to a discrete delay optimal control problem, and application of previously known results by Øksendal, Sulem and Zhang. The maximum principles also apply to the case where the controller only has partial information, in the sense that the admissible controls are adapted to a sub-$σ$-algebra of the natural filtration.

math.OC↗

arXiv2015

Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms \[\mathbb{R}^d\ni x\quad\longmapsto\quadϕ_{s,t}(x)\in \mathbb{R}^d,\qquad s,t\in\mathbb{R}\] for a stochastic differential equation (SDE) of the form \[dX_t=b(t,X_t)\,dt+dB_t,\qquad s,t\in\mathbb{R},X_s=x\in\mathbb{R}^d.\] The above SDE is driven by a bounded measurable drift coefficient $b:\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^d$ and a $d$-dimensional Brownian motion $B$. More specifically, we show that the stochastic flow $ϕ_{s,t}(\cdot)$ of the SDE lives in the space $L^2(Ω;W^{1,p}(\mathbb{R}^d,w))$ for all $s,t$ and all $p\in (1,\infty)$, where $W^{1,p}(\mathbb{R}^d,w)$ denotes a weighted Sobolev space with weight $w$ possessing a $p$th moment with respect to Lebesgue measure on $\mathbb {R}^d$. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant "culture" in these dynamical systems is that the flow "inherits" its spatial regularity from that of the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation \[\cases{\displaystyle d_tu(t,x)+\bigl(b(t,x)\cdot Du(t,x)\bigr)\,dt+\sum_{i=1}^de_i\cdot Du(t,x)\circ dB_t^i=0,\cr u(0,x)=u_0(x),}\] where $b$ is bounded and measurable, $u_0$ is $C_b^1$ and $\{e_i\}_{i=1}^d$ a basis for $\mathbb{R}^d$. It is well known that the deterministic counterpart of the above equation does not in general have a solution.

math.PR↗

arXiv2005

The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations II: Existence of stable and unstable manifolds

This article is a sequel to [M.Z.Z.1] aimed at completing the characterization of the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Stationary solution are viewed as random points in the infinite-dimensional state space, and the characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see's and spde's (Theorems 4.1-4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments (Theorem 2.1).

math.PR↗

arXiv1998

The Stable Manifold Theorem for Stochastic Differential Equations

We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory.

math.PR↗