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Nonlinear Analysis, Differential Equations and Control pp 407–446Cite as

Controlled Markov processes and mathematical finance

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Part of the book series: NATO Science Series ((ASIC,volume 528))

Abstract

These lectures are concerned with optimal control of Markov diffusion processes with complete state information, and with some applications in financial economics. Problems on a finite time horizon, and on an infinite horizon with discounted cost or ergodic (average cost per unit time) criterion are considered. We also consider risk sensitive stochastic control on an infinite horizon, with expected exponential-of-integral cost criteria. These problems are related, through a logarithmic transformation, to infinite horizon ergodic stochastic control and stochastic differential games. Risk sensitive control provides a link between stochastic and deterministic (robust control) approaches to disturbance attenuation. This is done by considering the robust control model as a small noise intensity limit of a corresponding risk sensitive model. In mathematical finance, we consider some models of optimal portfolio choice which are extensions of the classical Merton model. Problems of optimal long-term growth of expected utility of wealth are reformulated as infinite horizon risk sensitive control problems. Explicit solutions are given in absence of investment control constraints. In addition, a robust control approach to the Black-Scholes formula for pricing stock options is mentioned.

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Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Wendell H. Fleming

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  1. Wendell H. Fleming
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Editor information

Editors and Affiliations

  1. Institut Gérard Desargues, Université Claude Bernard (Lyon 1), Villeurbanne, France

    F. H. Clarke

  2. Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada

    R. J. Stern

  3. Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada

    G. Sabidussi

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© 1999 Springer Science+Business Media Dordrecht

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Fleming, W.H. (1999). Controlled Markov processes and mathematical finance. In: Clarke, F.H., Stern, R.J., Sabidussi, G. (eds) Nonlinear Analysis, Differential Equations and Control. NATO Science Series, vol 528. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4560-2_7

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  • DOI: https://doi.org/10.1007/978-94-011-4560-2_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-5666-0

  • Online ISBN: 978-94-011-4560-2

  • eBook Packages: Springer Book Archive

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