Related papers: Deterministic and Stochastic Differential Equation…

We consider stochastic PDEs \[dY_t = L(Y_t)\, dt + A(Y_t).\, dB_t, t > 0\] and associated PDEs \[du_t = L u_t\, dt, t > 0\] with regular initial conditions. Here, $L$ and $A$ are certain partial differential operators involving…

We here consider optimal control problems governed by nonlinear stochastic equations on a Hilbert space H with nonconvex payoff, which is rewritten as a deterministic optimal control problem governed by a Kolmogorov equation in H. We prove…

We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial…

We prove an existence and uniqueness result for the infinitely delayed stochastic evolution equation $$dU(t) = &\big(AU(t) + F(t,U_t)\big) dt + B(t,U_t)dW_H(t), t\in[0,T_0]$$ where $A$ is the generator of an analytic semigroup on a UMD…

The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov…

We study the following backward stochastic differential equation on finite time horizon driven by an integer-valued random measure $\mu$ on $\mathbb R_+\times E$, where $E$ is a Lusin space, with compensator $\nu(dt,dx)=dA_t\,\phi_t(dx)$:…

This paper studies the existence and uniqueness of solution of It\^o type stochastic differential equation $dx(t)=b(t, x(t), \om)dt+\si(t,x(t), \om) d B(t)$, where $B(t)$ is a fractional Brownian motion of Hurst parameter $H>1/2$ and…

This paper is concerned with the solvability of some abstract differential equation of type $$\dot u(t) + Au(t) + Bu(t) \ni f(t), t \in (0,T], u(0) = 0,$$ where $A$ is a linear selfadjoint operator and $B$ is a nonlinear(possibly…

We present a condition for a stochastic differential equation dX_{t}={\mu}(t,X_{t})dt+{\sigma}(t,X_{t})dB_{t} to have a unique functional solution of the form Z(t,B_{t}). The condition expresses a relation between {\mu} and {\sigma}. A…

This paper continues the study of [11, 13] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a…

These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in…

In this article, we show that if $A$ is a maximal monotone operator on a Hilbert space $H$ with $0$ in the range $\textrm{Rg}(A)$ of $A$, then for every $0<s<1$, the Dirichlet problem associated with the Bessel-type equation $$…

In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],…

We consider a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone follower problems and find applications in spatial models of production and climate transition. Let…

Let $A$ be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space $H$. The inverse problems of determining the right-hand side of the equation and the function $\phi$ in the non-local boundary value problem…

Given a real and separable Hilbert space H we consider the measure-valued equation \begin{equation*} \int_H\phi(x)\mu_t(dx)- \int_H\phi(x)\mu(dx)= \int_0^t(\int_HK_0\phi(x)\mu_s(dx))ds, \end{equation*} where K_0 is the Kolmogorov…

We show uniqueness in law for the critical SPDE $$ dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, $$ where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having…

Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following…

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…

This paper is concerned with the sensitivity analysis of a class of parameterized fixed-point problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that, if the operators in the considered fixed-point…