Electronic Journal of Probability

Discretionary stopping of stochastic differential equations with generalised drift

Mihail Zervos, Neofytos Rodosthenous, Pui Chan Lon, and Thomas Bernhardt

Full-text: Open access

Abstract

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem’s value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem’s data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case shows that the so-called “principle of smooth fit” does not hold in general for optimal stopping problems involving solutions to SDEs with generalised drift.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 140, 39 pp.

Dates
Received: 30 March 2019
Accepted: 16 October 2019
First available in Project Euclid: 5 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1575514915

Digital Object Identifier
doi:10.1214/19-EJP377

Mathematical Reviews number (MathSciNet)
MR4041000

Zentralblatt MATH identifier
07142934

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
optimal stopping stochastic differential equations with generalised drift skew Brownian motion variational inequalities perpetual American options

Rights
Creative Commons Attribution 4.0 International License.

Citation

Zervos, Mihail; Rodosthenous, Neofytos; Lon, Pui Chan; Bernhardt, Thomas. Discretionary stopping of stochastic differential equations with generalised drift. Electron. J. Probab. 24 (2019), paper no. 140, 39 pp. doi:10.1214/19-EJP377. https://projecteuclid.org/euclid.ejp/1575514915


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