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Optimal stopping of an Ornstein-Uhlenbeck bridge

Abel Azze, Bernardo D'Auria, Eduardo García-Portugués

Abstract

We make a rigorous analysis of the existence and characterization of the free boundary related to the optimal stopping problem that maximizes the mean of an Ornstein--Uhlenbeck bridge. The result includes the Brownian bridge problem as a limit case. The methodology hereby presented relies on a time-space transformation that casts the original problem into a more tractable one with an infinite horizon and a Brownian motion underneath. We comment on two different numerical algorithms to compute the free-boundary equation and discuss illustrative cases that shed light on the boundary's shape. In particular, the free boundary generally does not share the monotonicity of the Brownian bridge case.

Optimal stopping of an Ornstein-Uhlenbeck bridge

Abstract

We make a rigorous analysis of the existence and characterization of the free boundary related to the optimal stopping problem that maximizes the mean of an Ornstein--Uhlenbeck bridge. The result includes the Brownian bridge problem as a limit case. The methodology hereby presented relies on a time-space transformation that casts the original problem into a more tractable one with an infinite horizon and a Brownian motion underneath. We comment on two different numerical algorithms to compute the free-boundary equation and discuss illustrative cases that shed light on the boundary's shape. In particular, the free boundary generally does not share the monotonicity of the Brownian bridge case.

Paper Structure

This paper contains 7 sections, 9 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Reformulation of the problem
  4. Solution of the reformulated problem: a direct approach
  5. Solution of the original problem and some extensions
  6. Numerical results
  7. Conclusions

Key Result

Proposition 1

Consider the time change $\upsilon:[0,1]\rightarrow \mathbb{R}$ such that $\upsilon(t) := \psi(t)e^{-\alpha}/\kappa(1)$, and the space change $\eta:\mathbb{R}\rightarrow \mathbb{R}$ with $\eta(x) := x/(\gamma\sqrt{\kappa(1)e^{\alpha}})$. Take $(t, x)\in[0, 1)\times\mathbb{R}$ and set $s = \upsilon(t

Figures (3)

  • Figure 1: Optimal stopping boundary estimation for different values of $\alpha$. The boundary is pulled towards $0$ with a strength that increases as both $|\alpha|$ (values of $\alpha$ with equal absolute values yield the same boundary) and the residual time to the horizon $1 - t$ increases. As $\alpha \rightarrow 0$, the boundary estimation is shown to converge towards the OSB of a BB (dashed line), which is known to be $z + L\sqrt{1 - t}$, for $L\approx 0.8399$.
  • Figure 2: Optimal stopping boundary estimation for different values of $\gamma$. The boundary exhibits an increasing proportional relationship with respect to $\gamma$.
  • Figure 3: Optimal stopping boundary estimation for different values of $z$ and $N$. We display $t\mapsto\beta(t) - z$ to allow a clearer comparison across the different values of $z$. As $N$ increases the boundary estimation is seen to converge.

Theorems & Definitions (23)

  • Proposition 1: Time-space equivalence
  • proof
  • Proposition 2: Existence and shape of the optimal stopping boundary
  • proof
  • Proposition 3: Local Lipschitz continuity of the value function
  • proof
  • Proposition 4: Higher smoothness of the value function and the free-boundary problem
  • proof
  • Proposition 5: Partial derivatives of the value function
  • proof
  • ...and 13 more