In this paper, we present a result on relaxability of partially observed control problems for infinite dimensional stochastic systems in a Hilbert space. This is motivated by the fact that measure valued controls, also known as relaxed controls, are difficult to construct practically and so one must inquire if it is possible to approximate the solutions corresponding to measure valued controls by those corresponding to ordinary controls. Our main result is the relaxation theorem which states that...

It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.

Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_{\varphi }$ of Wiener measure and for a partition $0=t_0<t_1<\cdots <t_n<t_{n+1}=T$ of $[0,T]$, let $X_{n}C[0,T]\to {ℝ}^{n+1}$ and $X_{n+1}C[0,T]\to {ℝ}^{n+2}$ be given by $X_n\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_n\right))$ and $X_{n+1}\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_{n+1}\right))$, respectively. In this paper, using a simple formula for the conditional $w_{\varphi }$-integral of functions on $C[0,T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_{\varphi }$-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_{\varphi }$-integral...

A stochastic integral equation corresponding to a probability space $(\Omega ,{\Sigma }_{\omega },P_{\omega })$ is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable $u\left(·\right):{ℝ}^1\to {ℝ}^m$. One constructs stochastic processes ${\eta }^{\left(1\right)}\left(t\right)$, ${\eta }^{\left(2\right)}\left(t\right)$ connected with a Markov chain and with the space $(\Omega ,{\Sigma }_{\omega },P_{\omega })$. The expected values of ${\eta }^{\left(i\right)}\left(t\right)$ (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative $x_u^{\text{'}}\left(t\right)$.