Optimal Control of Ultradiffusion Processes with Application to Mathematical Finance

Document Type

Article

Publication Date

3-28-2014

Publication Title

International Journal of Computer Mathematics

Volume

92

Issue

2

First page number:

296

Last page number:

318

Abstract

We introduce the optimal control problem associated with ultradiffusion processes as a stochastic differential equation constrained optimization of the expected system performance over the set of feasible trajectories. The associated Bellman function is characterized as the solution to a Hamilton–Jacobi equation evaluated along an optimal process. For an important class of ultradiffusion processes, we define the value function in terms of the time and the natural state variables. Approximation solvability is shown and an application to mathematical finance demonstrates the applicability of the paradigm. In particular, we utilize a method-of-lines finite element method to approximate the value function of a European style call option in a market subject to asset liquidity risk (including limit orders) and brokerage fees.

Keywords

Optimal control; Ultradiffusion process; Limit orders; Asset liquidity; Transaction costs

Disciplines

Applied Mathematics | Finance and Financial Management

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