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On the continuity of optimal stopping surfaces for jump-diffusions

Cheng Cai, Tiziano De Angelis, Jan Palczewski

Abstract

We show that optimal stopping surfaces $(t,y)\mapsto x_*(t,y)$ arising from time-inhomogeneous optimal stopping problems on two-dimensional jump-diffusions $(X,Y)$ are continuous (jointly in time and space) under mild monotonicity and regularity assumptions of local nature.

On the continuity of optimal stopping surfaces for jump-diffusions

Abstract

We show that optimal stopping surfaces arising from time-inhomogeneous optimal stopping problems on two-dimensional jump-diffusions are continuous (jointly in time and space) under mild monotonicity and regularity assumptions of local nature.

Paper Structure

This paper contains 7 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Some comments on the operator ${\mathcal{A}}$
  4. Continuity of the stopping boundary: the continuous case
  5. Continuity of the stopping boundary: the general case
  6. Sufficient conditions for jump diffusions
  7. Concluding remarks

Key Result

Lemma 3.3

Let ${\mathcal{U}}\subset[0,T)\times{\mathcal{O}}$ be defined as in eq:cU. Assume that are Hölder continuous in ${\mathcal{U}}$, $\beta_1,\beta_2>0$ in $\overline{{\mathcal{U}}}$ and $\rho\in(-1,1)$. If $v$ is continuous then $v\in C^{1,2}({\mathcal{C}}\cap\mathcal{U})$ and it solves the boundary value problem in eq:PDE.

Theorems & Definitions (17)

  • Remark 2.1: Gain function
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Theorem 4.2
  • ...and 7 more