[Paper Review] Optimal Switching at Poisson Random Intervention Times

Gechun Liang, Wei Wei|Research Portal (King's College London)|Sep 22, 2013

Capital Investment and Risk Analysis11 references3 citations

TL;DR

This paper introduces a novel optimal switching problem where interventions occur at random Poisson arrival times, modeling exogenous constraints or information delays. It characterizes the value function and optimal switching strategy via an infinite-horizon backward stochastic differential equation (BSDE) system, proving the value function is Lipschitz continuous and establishing switching region structure via the comparison principle in a Markovian setting.

ABSTRACT

This paper introduces a new class of optimal switching problems, where the player is allowed to switch at a sequence of exogenous Poisson arrival times, and the underlying switching system is governed by an infinite horizon backward stochastic differential equation system. The value function and the optimal switching strategy are characterized by the solution of the underlying switching system. In a Markovian setting, the paper gives a complete description of the structure of switching regions by means of the comparison principle.

Motivation & Objective

• To model optimal switching under exogenous Poisson intervention times, reflecting liquidity or information constraints.
• To extend existing optimal stopping frameworks at Poisson times to the broader optimal switching context.
• To characterize the value function and optimal switching strategy through an infinite-horizon BSDE system.
• To establish the structure of switching regions in a Markovian setting using the comparison principle.
• To generalize prior results on optimal stopping at Poisson times to multi-regime switching problems.

Proposed method

• Models the switching system using an infinite-horizon backward stochastic differential equation (BSDE) system.
• Imposes switching constraints at random times generated by a Poisson process, representing exogenous intervention opportunities.
• Applies the comparison principle to analyze the structure of switching regions in the Markovian case.
• Uses a penalized version of the infinite-horizon BSDE system to derive existence and uniqueness of solutions.
• Establishes linear growth and Lipschitz continuity of the value function through a priori estimates on expected profits and switching costs.
• Employs ODE comparison techniques on finite intervals to prove subsolution and supersolution properties of the value function.

Experimental results

Research questions

• RQ1 How does the value function behave when switching is restricted to Poisson arrival times rather than arbitrary stopping times?
• RQ2 What is the structure of the switching regions in a Markovian setting under Poisson intervention constraints?
• RQ3 Can the value function be characterized via a system of infinite-horizon BSDEs under such constraints?
• RQ4 How does the comparison principle help in determining the optimal switching strategy in this framework?
• RQ5 What are the regularity properties (e.g., Lipschitz continuity) of the value function under this model?

Key findings

• The value function is Lipschitz continuous, with the Lipschitz constant bounded by $ \frac{C}{a_1 - b} $, where $ C $ is the Lipschitz constant of the profit function and $ a_1 > b $.
• The value function exhibits at most linear growth, with an upper bound of $ \frac{Cx}{a_1 - b} + \max_{j=1,2}[-g^{ij}] $, where $ x $ is the initial state.
• The subsolution property of the value function is proven via comparison principle on finite intervals, leading to $ \underline{v}^i(x^0) \leq 0 $ for all $ x^0 \in (0,\infty) $.
• The optimal switching strategy is characterized by the solution of the underlying infinite-horizon BSDE system.
• Switching regions are completely described in the Markovian case using the comparison principle applied to the associated ODE system.
• The model generalizes prior results on optimal stopping at Poisson times to the full optimal switching problem, including multi-regime and reversible investment settings.