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A class of solvable multidimensional stopping problems in the presence of Knightian uncertainty

Part of: Markov processes Mathematical finance Stochastic processes Sequential methods

Published online by Cambridge University Press:  01 July 2021

Luis H. R. Alvarez E. Open the ORCID record for Luis H. R. Alvarez E. [Opens in a new window]  and
Sören Christensen
Luis H. R. Alvarez E.*
Affiliation:
University of Turku
Sören Christensen*
Affiliation:
Christian-Albrechts-Universität zu Kiel
*
*Postal address: Department of Accounting and Finance, Turku School of Economics, FIN-20014 University of Turku, Finland. Email address: lhralv@utu.fi
**Postal address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany. Email address: christensen@math.uni-kiel.de
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Abstract

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity-averse decision-maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a general characterization of the value of the optimal timing policy and the worst-case measure in terms of a family of explicitly identified excessive functions generating an appropriate class of supermartingales. In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting. Somewhat surprisingly, we find that ambiguity may lead to stationarity in models which typically do not possess stationary behavior. In this way, our results indicate that ambiguity may act as a stabilizing mechanism.

Keywords

κ-ambiguitymultidimensional Brownian motiondiffusion processesBessel processes

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 400 - 424
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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