Financial models and well-posedness properties for symmetric set-valued stochastic differential equations with relaxed Lipschitz condition

Abstract

In this paper, stochastic differential equations are considered in the context of set-valued analysis with solutions that are set-valued stochastic processes. The equations were proposed in the so-called symmetrical form. A variety of set-valued stochastic differential equations that extend well-known single-valued models in financial mathematics are presented. The misconception that the solution of a single-valued equation, starting from a point within the initial value of the set-valued equation, will always remain within the solution of the set-valued equation (i.e., it is a selection) is refuted. Then, the symmetric set-valued differential equation in general form is studied. It is assumed that the coefficients of equation satisfy a very general condition, including that of the Lipschitz type, with a function that appears with a certain integral inequality. The result obtained is that there is a unique solution to the equation considered. It is also shown that the solution is stable with respect to small changes in the equation data. The implications for symmetric set-valued random differential equations and deterministic symmetric set-valued differential equations are also stated.