On optimal stopping problems with positive discounting rates and related Laplace transforms of first hitting times in models with geometric Brownian motions

Gapeev, Pavel V. ORCID: 0000-0002-1346-2074 (2021) On optimal stopping problems with positive discounting rates and related Laplace transforms of first hitting times in models with geometric Brownian motions. In: Vervoort, R.W., Voinov, A.A., Evans, J.P. and Marshall, L., (eds.) MODSIM2021, 24th International Congress on Modelling and Simulation. Proceedings of the International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand Inc., 29 - 35. ISBN 9780987214386

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Abstract

We derive closed-form solutions to some optimal stopping problems for one-dimensional geometric Brownian motions with positive discounting rates. It is assumed that the original processes can be trapped or reflected or sticky at some fixed lower levels and the conditions on the gain functions imply that the the optimal stopping times turn out to be the first times at which the processes hit some upper level which are to be determined. The proof is based on the reduction of the original optimal stopping problems to the to the equivalent free-boundary problems and the solutions of the latter problems by means of the instantaneous stopping and smooth-fit conditions for the value functions at the optimal stopping boundaries. We also obtain explicit expressions for the Laplace transforms or moment generating functions (with positive exponents or parameters) of the first hitting times for the geometric Brownian motion of given upper levels under various conditions on the parameters of the model. In particular, we determine the upper bounds for the hitting levels and given positive exponents or parameters of the Laplace transforms for which the resulting expectations are finite under various relations between the parameters of the model. Moreover, we determine the upper bounds for the positive exponents or parameters of the Laplace transforms and given hitting levels for which the resulting expectations are finite under various relations between the parameters of the model. The main aim of this short article is to derive closed-form solutions to the optimal stopping problem of (2) for the geometric Brownian motion X defined in (1) with a positive exponential discounting rate λ > 0. We assume that the process X can be trapped or reflected or sticky at some level a > 0 and the gain function G(x) is a twice continuously differentiable positive and strictly increasing concave function on (0, ∞). Optimal stopping problems for one-dimensional diffusion processes with negative exponential discounting rates have been studied after Dynkin (1963) by many authors in the literature including Fakeev (1970), Mucci (1978), Salminen (1985), Øksendal and Reikvam (1998), Alvarez (2001), Dayanik and Karatzas (2003), and Lamberton and Zervos (2013) among others (we refer to Øksendal (1998, Chapter X), Peskir and Shiryaev (2006) and Gapeev and Lerche (2011) for further references). The consideration of optimal stopping problems for diffusions with positive discounting rates was initiated by Shepp and Shiryaev (1996) and then has been continued by other authors in the literature (we refer to Gapeev (2019) and Gapeev (2020) for further references). In this short article, we also present explicit expressions for the Laplace transforms (with positive exponents or parameters) of the first hitting times of given upper levels under various conditions on the parameters of the model (see Borodin and Salminen (2002, Part II) for other computations of the Laplace transforms of first hitting times).

Item Type:	Book Section
Additional Information:	© 2021 The Author and Modelling and Simulation Society of Australia and New Zealand Inc. (MSSANZ)
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	18 Oct 2021 10:27
Last Modified:	17 Dec 2024 10:42
URI:	http://eprints.lse.ac.uk/id/eprint/112456

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