[Paper Review] Expected Supremum Representation of the Value of a Singular Stochastic Control Problem

Luis H. R. Alvarez E., Pekka Matomäki|arXiv (Cornell University)|Aug 12, 2015

Capital Investment and Risk Analysis30 references3 citations

TL;DR

This paper establishes a novel representation of the value function in singular stochastic control problems as an expected supremum of a derived function of the controlled diffusion process. By equating the value of a constant-boundary reflection policy to the expected supremum of an unknown function, the authors derive the representing function explicitly and show that optimality corresponds to the threshold where the derivative of this function vanishes—coinciding with the standard C²-smoothness condition.

ABSTRACT

We consider the problem of representing the value of singular stochastic control problems of linear diffusions as expected suprema. Setting the value accrued from following a standard reflection policy equal with the expected value of a unknown function at the running supremum of the underlying is shown to result into a functional equation from which the unknown function can be explicitly derived. We also consider the stopping problem associated with the considered singular stochastic control problem and present a similar representation as an expected supremum in terms of characteristics of the control problem.

Motivation & Objective

• To represent the value of singular stochastic control problems as an expected supremum of a function of the running supremum of the underlying diffusion.
• To derive the representing function explicitly in a linear diffusion setting, enabling closed-form analysis.
• To establish a connection between the smoothness condition in singular control and the vanishing of the derivative of the representing function.
• To extend the representation to the associated optimal stopping problem, showing similar expected supremum structure.

Proposed method

• Formulate the value of a constant-boundary reflection policy as an integral of the revenue flow discounted by a constant boundary.
• Express the expected value of the unknown function of the running supremum at an exponentially distributed random time using known first-passage distributions.
• Equate the two expressions to form a functional equation that determines the representing function explicitly.
• Use the known distribution of the supremum of a diffusion killed at 0 to compute the expected supremum term.
• Derive the representing function by solving the resulting identity, leveraging properties of scale functions and speed measures in linear diffusions.
• Verify that the optimal boundary corresponds to the point where the derivative of the representing function vanishes, aligning with standard C²-smoothness conditions.

Experimental results

Research questions

• RQ1 Can the value of a singular stochastic control problem be represented as an expected supremum of a function of the running supremum of the underlying process?
• RQ2 What explicit form does the representing function take in a linear diffusion setting with constant reflection boundaries?
• RQ3 Does the optimality of the reflection boundary correspond to a specific property of the representing function, such as the vanishing of its derivative?
• RQ4 How does the representing function for the singular control problem relate to the value function of the associated optimal stopping problem?
• RQ5 Can a similar expected supremum representation be established for the marginal value function in singular control problems?

Key findings

• The representing function for the singular control problem vanishes at the reflection boundary regardless of whether the boundary is optimal or not, contrasting with optimal stopping problems.
• Optimality in the singular control problem is achieved precisely at the threshold where the derivative of the representing function vanishes, which coincides with the standard C²-smoothness condition across the boundary.
• The representing function is explicitly derived in closed form for a class of linear diffusions, including cases with power and exponential components.
• For the optimal boundary, the representing function remains nonnegative across the state space, while suboptimal boundaries lead to regions where the function becomes negative.
• A similar expected supremum representation is established for the marginal value function, which corresponds to the value of an associated optimal stopping problem under certain conditions.
• The connection between the representing functions of the control and stopping problems explains the differing regularity conditions required for boundary characterization.