Journal of Applied Probability

Discounted optimal stopping for maxima of some jump-diffusion processes

Pavel V. Gapeev

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Abstract

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

Article information

Source
J. Appl. Probab. Volume 44, Number 3 (2007), 713-731.

Dates
First available: 13 September 2007

Permanent link to this document
http://projecteuclid.org/euclid.jap/1189717540

Digital Object Identifier
doi:10.1239/jap/1189717540

Mathematical Reviews number (MathSciNet)
MR2355587

Zentralblatt MATH identifier
1146.60037

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 34K10: Boundary value problems 91B70: Stochastic models
Secondary: 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 91B28

Keywords
Discounted optimal stopping problem Brownian motion compound Poisson process maximum process integro-differential free-boundary problem continuous and smooth fit normal reflection change-of-variable formula with local time on surfaces perpetual American lookback option

Citation

Gapeev, Pavel V. Discounted optimal stopping for maxima of some jump-diffusion processes. Journal of Applied Probability 44 (2007), no. 3, 713--731. doi:10.1239/jap/1189717540. http://projecteuclid.org/euclid.jap/1189717540.


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