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Optimal Stopping for Systems Driven by the Brownian Sheet

Nacira Agram, Bernt Oksendal, Frank Proske, Olena Tymoshenko

Abstract

We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part we derive explicit solutions to two optimal stopping problems for the exponentially discounted Brownian sheet. The first problem consists in determining the optimal two-parameter first hitting point tau = (tau1,tau2) maximizing E[exp(-rho tau1 tau2) h(B(tau1,tau2))], where rho > 0 is a discount factor and h is a reward function. Restricting attention to first hitting points of levels, we obtain a closed-form characterization of the optimal stopping threshold. In particular, for linear rewards h(y)=y the optimal level is y_hat = (2 rho)^(-1/2). The second problem concerns optimal stopping of the integrated discounted Brownian sheet with payoff E[int_0^{tau1} int_0^{tau2} exp(-rho t x) B(t,x) dt dx]. We show that the optimal first hitting level is strictly positive and give an explicit representation of the value function in terms of the exponential integral function. The optimal threshold is characterized as the unique solution of a nonlinear equation derived from a Laplace transform identity for the product tau1 tau2. In the second and main part of the paper we develop a potential theoretic framework for two-parameter optimal stopping problems associated with stochastic partial differential equations driven by the Brownian sheet, proving that the value function is the least superharmonic majorant of the reward and establishing existence of optimal stopping points in the plane.

Optimal Stopping for Systems Driven by the Brownian Sheet

Abstract

We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part we derive explicit solutions to two optimal stopping problems for the exponentially discounted Brownian sheet. The first problem consists in determining the optimal two-parameter first hitting point tau = (tau1,tau2) maximizing E[exp(-rho tau1 tau2) h(B(tau1,tau2))], where rho > 0 is a discount factor and h is a reward function. Restricting attention to first hitting points of levels, we obtain a closed-form characterization of the optimal stopping threshold. In particular, for linear rewards h(y)=y the optimal level is y_hat = (2 rho)^(-1/2). The second problem concerns optimal stopping of the integrated discounted Brownian sheet with payoff E[int_0^{tau1} int_0^{tau2} exp(-rho t x) B(t,x) dt dx]. We show that the optimal first hitting level is strictly positive and give an explicit representation of the value function in terms of the exponential integral function. The optimal threshold is characterized as the unique solution of a nonlinear equation derived from a Laplace transform identity for the product tau1 tau2. In the second and main part of the paper we develop a potential theoretic framework for two-parameter optimal stopping problems associated with stochastic partial differential equations driven by the Brownian sheet, proving that the value function is the least superharmonic majorant of the reward and establishing existence of optimal stopping points in the plane.
Paper Structure (13 sections, 19 theorems, 188 equations, 4 figures)

This paper contains 13 sections, 19 theorems, 188 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Two classical cases with one parameter Brownian motion
  3. Optimal stopping of discounted Brownian motion
  4. Optimal stopping of the integral of the discounted Brownian motion
  5. Exponential of Brownian sheet integrals
  6. The two parameter Itô formula
  7. The First Optimal Stopping Problem: Discounted Brownian sheet
  8. The second optimal stopping problem: The integrated discounted Brownian sheet
  9. Existence of an optimal stopping point in the plane
  10. Supermeanvalued, superharmonic and excessive functions
  11. Least superharmonic majorants
  12. Completing the proof of Theorem \ref{['MainOpt']}
  13. Appendix: A multidimensional two parameter Itô formula

Key Result

Proposition 3.2

Let $Y(z)$ and $G(z)$ be the stochastic exponential fields defined by Y_def and Gz_def respectively. Then $G(z)$ admits the following representation

Figures (4)

  • Figure 1: Geometric interpretation of rectangle integration, the coordinatewise maximum $\zeta\vee\zeta'$, and the mixed partial order.
  • Figure 2: Comparison of first hitting times/points in the one parameter Brownian motion (left) and the two parameter Brownian sheet (right), both with threshold $y = \hat{y}$.
  • Figure 3: Graph of the value function $F(y)$ defined in \ref{['5']}.
  • Figure 4: Geometric comparison of the one and two parameter discount structures. The multiplicative factor $e^{-\rho tx}$ induces a hyperbolic geometry and suppresses large rectangles, leading to a qualitatively different optimal stopping boundary.

Theorems & Definitions (37)

  • Remark 2.2
  • Remark 2.4
  • Proposition 3.2: two parameter Exponential Formula
  • Theorem 3.3: Itô formula
  • Proposition 3.4
  • Remark 3.5
  • Definition 4.1
  • Proposition 4.2
  • Definition 4.3
  • Remark 4.4
  • ...and 27 more