Set-Valued Stochastic Differential Equations with Unbounded Coefficients and Applications

Abstract

In this dissertation, we focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets. By establishing a new theoretical framework on such sets, which is beyond the existing theory of set-valued analysis, we shall study the set-valued SDEs (SV-SDEs) with unbounded coefficients, and their applications in the super-hedging problem of a continuous-time model with transac- tion costs in finance. The space that we will be focusing on is convex, closed sets that are “generated” by a given cone with certain constraints. We shall argue that, for such a special class of unbounded sets, the cancellation law could still be valid, and many algebraic and topological properties of the existing theory of set-valued analysis on compact sets and standard techniques for studying SV-SDEs can be ex- tended to the case with unbounded (drift) coefficients. In the super-hedging problem of discrete-time models with transaction costs, the set of self-financing portfolios are often described by the (unbounded) “solvency cone”. Our study of unbounded sets is therefore crucial in extending the theory to the continuous-time model. In the model with transaction costs and vector-valued contingent claims, the set of super-hedging positions is inherently a closed convex unbounded set. We shall argue that the (dy- namic) super-hedging set can be expressed as set-valued integrals of the solvency cones, and define a set-valued dynamic risk measure. Finally, after some refinement, we show that the dynamic super-hedging sets satisfy a recursive relation which can be considered as a geometric dynamic programming principle (DPP).