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Dynamic minimisation of the commute time for a one-dimensional diffusion

Ma. Elena Hernández-Hernández, Saul Jacka

Abstract

Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel to $1$ and then return to the origin (the so-called commute time from 0 to 1). Substantially extending results in a previous paper, we consider a dynamic version of this problem where the control mechanism is related to the diffusion's drift via the corresponding scale function. We are only able to choose the drift at each point at the time of first visiting that point and the drift is constrained on a set of the form $[0,\ell)\cup(i,1]$. This leads to a type of stochastic control problem with infinite dimensional state.

Dynamic minimisation of the commute time for a one-dimensional diffusion

Abstract

Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel to and then return to the origin (the so-called commute time from 0 to 1). Substantially extending results in a previous paper, we consider a dynamic version of this problem where the control mechanism is related to the diffusion's drift via the corresponding scale function. We are only able to choose the drift at each point at the time of first visiting that point and the drift is constrained on a set of the form . This leads to a type of stochastic control problem with infinite dimensional state.
Paper Structure (11 sections, 6 theorems, 68 equations)

This paper contains 11 sections, 6 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The model and some notation
  3. The model
  4. The control setting
  5. Heuristic for the optimal strategy.
  6. The candidate optimal payoff
  7. Initial calculations
  8. A calculus of variations approach
  9. Further optimisation
  10. The dynamic solution
  11. Concluding remarks and further problems

Key Result

Lemma 3.4

Let $s\in \mathcal{M}$, then the payoff of the policy $s^*$ is given by where $H$ is defined by

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.1
  • ...and 6 more