Special Issue on Stochastic Differential Equations in a Banach Space, New Methods in Development

Niko Muschelishvili Institute of Computational Mathematics, Technical University of Georgia, Tbilisi, Georgia
• Guest Editor
• Introduction

In developing the stochastic differential equations in an arbitrary separable Banach space the finite dimensional and Hilbert space methods do not give desired results: the first problem is to define the Ito stochastic integral. We defined a generalized Ito stochastic integral as a generalized random element (a linear random function) and the problem of the existence of the stochastic integral reduced to the well-known problem of decomposability of the generalized random element . The second problem is to estimate the stochastic integral using the classical fixed point theorem, which is impossible for an arbitrary Banach space case. We considered the Banach space of generalized random elements, introduced there the stochastic differential equation for generalized random processes (GRP). Here it is possible to use traditional methods to develop the problem of the existence and uniqueness of a solution as a GRP. Afterward, from the main stochastic differential equation in a Banach space we received the equation for GRP and get the solution as a GRP. Thus, we reduced the problem of the existence of the solution, to the problem of decomposability of the GRP. This method we realized in the paper “B. Mamporia. Stochastic differential equation for generalized random processes in a Banach space. Theory of probability and its Applications, 56(4), 602-620, 2012,SIAM. Teoriya Veroyatnostei I ee Primeneniya, 56:4 (2011), 704-725”, where we considered the stochastic differential equation in the case when the Wiener process is one dimensional and the integrand function is Banach space valued (first direction). Our approach gives possibility to further developent the theory in this case (to develop the linear stochastic differential equations and so forth). Later we consider the second direction when the Wiener process is Banach space valued and the integrand predictable function is operator-valued; we developed the problem of the existence and uniqueness of a solution in this case, the Ito formula and will consider the linear stochastic differential equations, the functionals of the Wiener process in a Banach space and so forth. The third direction constitutes the case, when the Wiener process is cylindrical (canonical generalized Wiener process) in Hilbert space and the integrand function is operator-valued from Hilbert space to Banach space. We will develop as well the theory in this case too.

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