The Uniqueness of Random Solution to Itô Stochastic Integral Equation

Abstract

The integral equation of Itô integral type is considered, for the deterministic and stochastic cases. We show that under some assumptions the equations have solutions belonging to the space of continuous functions. In this study, we attempt to apply the theoretical techniques of probabilistic functional analysis to examine of existence and Uniqueness of a Random Solution to Itô Stochastic Integral Equation. Also, we shall present another type of stochastic integral equation that has been of considerable importance to applied mathematicians and engineers, involving the Itô or Itô-Doob form of stochastic integrals. We present the extension when W = (W¹, ... ,W^d) is a multidimensional Brownian motion and we replaced the minimal augmented filtration generated by \(W, \mathbb{F}^w\)={\({\mathcal{F}^w_t}\)}\(_{t\epsilon\lceil0,T\rceil}\) by a largest one \(\mathbb{H}=\){\({\mathcal{H}^w_t}\)}\(_{t\epsilon\lceil0,T\rceil}\) (satisfying the usual conditions)  such that the Brownian motion W is a martingale respect to \(\mathbb{H}\) ; i.e., the following property is satisfied \(\mathbb{E}\left[W_t - W_s : \mathcal{H}_s\right]\) = 0 and We have constructed the Itô integral for processes in \(h \epsilon L^0_{\alpha,_T}\) by localizing \(h\) in \(L^2_{\alpha,_T}\). i.e. by considering \(h^n\) = \(h1\)\(_{[{0,T_n}]}\).