Open Access
2019 Discretionary stopping of stochastic differential equations with generalised drift
Mihail Zervos, Neofytos Rodosthenous, Pui Chan Lon, Thomas Bernhardt
Electron. J. Probab. 24: 1-39 (2019). DOI: 10.1214/19-EJP377


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.


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Mihail Zervos. Neofytos Rodosthenous. Pui Chan Lon. Thomas Bernhardt. "Discretionary stopping of stochastic differential equations with generalised drift." Electron. J. Probab. 24 1 - 39, 2019.


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

zbMATH: 07142934
MathSciNet: MR4041000
Digital Object Identifier: 10.1214/19-EJP377

Primary: 60G40
Secondary: 60J55 , 60J60 , 91G80

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

Vol.24 • 2019
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