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On the optimal stopping of Gauss-Markov bridges with random pinning points

Abel Azze, Bernardo D'Auria

TL;DR

The paper advances the theory of optimal stopping for Gauss–Markov processes conditioned on a prescribed terminal distribution by mapping the problem to a randomized Brownian bridge through a time–space transformation. It proves the optimal stopping rule is the first passage into the stopping region and establishes Lipschitz regularity of the value function via coupling of terminal pins, along with likelihood-ratio based comparison results and boundary bounds under bounded-support priors. It provides structural results for the stopping boundary, identifying conditions under which it is a graph of a function, including Gaussian and strongly log-concave priors, and presents explicit integral equations characterizing the boundary in Gaussian and Dirac-delta prior cases. Numerical experiments illustrate diverse boundary geometries and demonstrate the practical computation of OS boundaries, supported by open-source code for reproducibility.

Abstract

We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random endpoint with a time-dependent payoff. We prove that the optimal rule is the first entry into the stopping region, and establish that the value function is Lipschitz continuous on compacts via a coupling of terminal pinning points across different initial conditions. A comparison theorems then order value functions according to likelihood-ratio ordering of terminal densities, and when these densities have bounded support, we bound the optimal boundary by that of a Gauss-Markov bridge. Although the stopping boundary need not be the graph of a function in general, we provide sufficient conditions under which this property holds, and identify strongly log-concave terminal densities that guarantee this structure. Numerical experiments illustrate representative boundary shapes.

On the optimal stopping of Gauss-Markov bridges with random pinning points

TL;DR

The paper advances the theory of optimal stopping for Gauss–Markov processes conditioned on a prescribed terminal distribution by mapping the problem to a randomized Brownian bridge through a time–space transformation. It proves the optimal stopping rule is the first passage into the stopping region and establishes Lipschitz regularity of the value function via coupling of terminal pins, along with likelihood-ratio based comparison results and boundary bounds under bounded-support priors. It provides structural results for the stopping boundary, identifying conditions under which it is a graph of a function, including Gaussian and strongly log-concave priors, and presents explicit integral equations characterizing the boundary in Gaussian and Dirac-delta prior cases. Numerical experiments illustrate diverse boundary geometries and demonstrate the practical computation of OS boundaries, supported by open-source code for reproducibility.

Abstract

We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random endpoint with a time-dependent payoff. We prove that the optimal rule is the first entry into the stopping region, and establish that the value function is Lipschitz continuous on compacts via a coupling of terminal pinning points across different initial conditions. A comparison theorems then order value functions according to likelihood-ratio ordering of terminal densities, and when these densities have bounded support, we bound the optimal boundary by that of a Gauss-Markov bridge. Although the stopping boundary need not be the graph of a function in general, we provide sufficient conditions under which this property holds, and identify strongly log-concave terminal densities that guarantee this structure. Numerical experiments illustrate representative boundary shapes.
Paper Structure (13 sections, 12 theorems, 71 equations, 2 figures, 1 algorithm)

This paper contains 13 sections, 12 theorems, 71 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Randomized Gauss--Markov bridges
  3. Gauss--Markov processes
  4. The standard BM representation
  5. Conditioning on a terminal density
  6. Randomized Brownian bridge
  7. The equivalent optimal stopping problems
  8. The equivalent optimal stopping problem of a randomized Brownian bridge
  9. Analysis of the transformed optimal stopping problem
  10. The single boundary case
  11. The Gaussian prior
  12. The Dirac delta prior
  13. Numerical results

Key Result

Proposition 1

For $s_1$, $s_2\in[0, 1)$ and $y_1$, $y_2 \in\mathbb{R}$, let $\pi^* = \pi_{(s_1,y_1),(s_2,y_2)}^*$ be Wasserstein-$1$ copula of $\nu_{s_1, y_1}$ and $\nu_{s_2, y_2}$. That is, where $\Pi(f, g)$ is the set of all couplings whose marginal densities are $f$ and $g$. Then, under Assumption asm:terminal_density,

Figures (2)

  • Figure 1: Numerical computation of the stopping and continuation regions for a BM forced to hit either $(1,1)$ or $(1,-1)$, each with equal probability, which corresponds to choosing the coefficients $\alpha \equiv 0$, $\beta\equiv 0$, and $\zeta \equiv 1$ in \ref{['eq:SDE_GM']}, and the prior terminal density $\widetilde{\nu}(z) = (\delta_{-1} + \delta_{1})/2$. The initial condition $x_0$ is specified in the caption of each image. The red and green areas indicate the stopping and continuation sets, respectively.
  • Figure 2: Numerical computation of the stopping and continuation regions associated to a GMP conditioned to adopt a positively-truncated normal terminal density $\nu(z) \propto \mathbbm{1}(z\geq 0)\phi(z;0,v_0(T)/2)$. Each image highlights the time-dependent effect of one coefficient, while keeping the other constant: $\alpha(t) = 2\sin(10\pi t)$ is used in image (a); image (b) considers $\beta(t) = -10 + 0.475(1 + \tanh(100(t - 0.5)))$; and image (c) takes $\zeta(t) = 0.25 + (4(t-0.5))^4$. $x_0 = 0$ was used for all images. The red and green areas indicates the stopping and continuation sets, respectively.

Theorems & Definitions (13)

  • Proposition 1: Wasserstein-$1$-Lipschitz continuity of the pinning point
  • Proposition 2: Equivalence of the OSPs
  • Proposition 3: Optimal stopping time characterization
  • Proposition 4: Lipschitz continuity of the value function
  • Proposition 5: The free-boundary problem
  • Remark 1
  • Proposition 6: Ordering of value functions
  • Corollary 1: Bounds for the value function with bounded-supported priors
  • Lemma 1: Brownian bridge maximal bounds
  • Proposition 7: Sufficient condition for single optimal stopping function
  • ...and 3 more