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Set-Valued Stochastic Differential Equations with Unbounded Coefficients

Atiqah Almuzaini, Jin Ma

Abstract

In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential equations with unbounded coefficients. The space that we will be focusing on are convex, closed sets that are "generated" by a given cone, in the sense that the Hausdorff distance of all elements to the "generating" cone is finite. Such space should in particular include the so-called "upper sets", and has many useful cases in finance, such as the well-known set-valued risk measures, as well as the solvency cone in some super-hedging problems. We shall argue that, for such a special class of unbounded sets, under some conditions, the cancellation law is still valid, eliminating a major obstacle for extending the set-valued analysis to non-compact sets. We shall establish some basic algebraic and topological properties of such spaces, and show that some standard techniques will again be valid in studying the set-valued SDEs with unbounded (drift) coefficients which, to the best of our knowledge, is new.

Set-Valued Stochastic Differential Equations with Unbounded Coefficients

Abstract

In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential equations with unbounded coefficients. The space that we will be focusing on are convex, closed sets that are "generated" by a given cone, in the sense that the Hausdorff distance of all elements to the "generating" cone is finite. Such space should in particular include the so-called "upper sets", and has many useful cases in finance, such as the well-known set-valued risk measures, as well as the solvency cone in some super-hedging problems. We shall argue that, for such a special class of unbounded sets, under some conditions, the cancellation law is still valid, eliminating a major obstacle for extending the set-valued analysis to non-compact sets. We shall establish some basic algebraic and topological properties of such spaces, and show that some standard techniques will again be valid in studying the set-valued SDEs with unbounded (drift) coefficients which, to the best of our knowledge, is new.
Paper Structure (6 sections, 18 theorems, 94 equations)

This paper contains 6 sections, 18 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L_C$-Space and Basic Properties
  4. Set-Valued Lebesgue Integral on $L_C$-Space
  5. Set-Valued SDEs with Unbounded Coefficients
  6. Connections to SDIs and Applications

Key Result

Proposition 2.2

(i) If $F:\mathbb{X}\rightarrow \mathscr{C}(\mathbb{R}^d)$ is measurable, then $F$ admits a measurable selector, i.e. there exists an $\mathcal{\mathscr{M}}/\mathscr{B}(\mathbb{R}^d)$-measurable selector $f:\mathbb{X}\to \mathbb{R}^d$ of $F$; (ii) (Castaing Representation) $F:\mathbb{X}\rightarrow \

Theorems & Definitions (26)

  • Definition 2.1
  • Proposition 2.2: MK
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5: Kisielewicz-Mitchta KisMic16
  • Proposition 2.6: MK
  • Proposition 2.7: SUP
  • Theorem 3.1
  • Example 3.2
  • Proposition 3.3
  • ...and 16 more