On the existence of the optimal control for stochastic functional differential equations subject to external disturbances (Q6570651)

The authors consider a probability space \((\Omega ,F,P)\) with a flow of \( \sigma \)-algebras \(\{F_{t}\), \(t\geq 0\}\), the space \(D\) of right-continuous functions with left-hand boundaries, with values from \(\mathbb{R}\) and with a uniform metric, and the stochastic functional differential equations: \( dx_{i}(t)=a_{i}(t,x^{t})dt+b(t,x^{t})dw(t)+\int_{V}c(t,x^{t},v)\widetilde{ \nu }(dt,dv)\), \(i=1,2\), where \(\{x^{t}=x(t+\theta )\), \(\theta \in (-\infty ,0]\}\), \(a_{i}\) are two continuous functionals \(\mathbb{R}_{+}\times D\rightarrow \mathbb{R}\), satisfying \(a_{1}(t,\varphi )\leq a_{2}(t,\varphi )\leq 0\), \(t\geq 0\), \(\varphi \in D\), \(b:\mathbb{R}_{+}\times D\rightarrow \mathbb{R}\) and \(c:\mathbb{R}_{+}\times D\times V\rightarrow \mathbb{R}\) are continuous functionals which satisfy: \(\left\vert b(t,\varphi )-b(t,\psi )\right\vert +\int_{V}\left\vert c(t,\varphi ,u)-c(t,\psi ,u)\right\vert \Pi (du)\leq \rho \left\Vert \varphi -\psi \right\Vert )\), \(t\geq 0\), for some strictly increasing function \(\rho :\mathbb{R}_{+}\rightarrow \mathbb{R}\) such that \(\rho (0)=0\), \(\int_{0}^{\infty }\rho ^{-2}(x)dx=\infty \), \(x_{i}\) are random processes which are continuous in \(t\) and \(\{F_{t}\}\)-measurable in \(\omega \), \(w\) is a standard Wiener process \(w(t):[0,\infty )\times \Omega \subset \mathbb{R}\) such that \(w(0)=0\) (mod \(P\)), \(\widetilde{\nu }\) is a centered Poisson measure \(\widetilde{\nu }(t,A)=\nu (t,A)-\Pi (A)t\), \( A\in V\subset \mathbb{R}\setminus \{0\}\), where \(\{w(t)\}\) and \(\{\widetilde{ \nu }(t,A)\}\) are independent of each other.\ For the main result, the authors also assume the existence of random processes \(\alpha _{i}\) measurable with respect to \(\{F_{t}\), \(t\geq 0\}\)such that: \( x_{i}(t)-x_{i}(0)=\int_{0}^{t}\alpha _{i}(s)ds+\int_{0}^{t}b(s,x_{i}^{s})dw(s)+\int_{0}^{t} \int_{V}c(t,x_{i}^{s},v)\widetilde{\nu }(dt,dv)\), \(x_{1}(\theta )\leq x_{2}(\theta )\), \(\theta \in (-\infty ,0]\), \(\alpha _{i}(t)\leq a_{i}(t,x_{i}^{t})\), \(t\geq 0\), with probability one.\ The main result proves that \(x_{1}(\theta )\leq x_{2}(\theta )\), \(t\geq 0\), with probability one. From this result, the authors deduce the existence of a stochastic control for a class of stochastic functional differential equations subject to external disturbances taken as random processes.