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Markov chains under nonlinear expectation

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  • Max Nendel

Abstract

In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

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  • Max Nendel, 2021. "Markov chains under nonlinear expectation," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 474-507, January.
    • Handle: RePEc:bla:mathfi:v:31:y:2021:i:1:p:474-507
    DOI: 10.1111/mafi.12289
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    References listed on IDEAS

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    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Cohen, Samuel N., 2012. "Representing filtration consistent nonlinear expectations as g-expectations in general probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1601-1626.
    3. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    4. Cohen, Samuel N. & Elliott, Robert J., 2010. "A general theory of finite state Backward Stochastic Difference Equations," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 442-466, April.
    5. Denk, Robert & Kupper, Michael & Nendel, Max, 2020. "A semigroup approach to nonlinear Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1616-1642.
    6. Renato Pelessoni & Paolo Vicig, 2003. "Convex Imprecise Previsions for Risk Measurement," Risk and Insurance 0309001, University Library of Munich, Germany.
    7. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    8. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    9. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    10. Samuel N. Cohen & Robert J. Elliott, 2008. "Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions," Papers 0810.0055, arXiv.org, revised Jan 2010.
    11. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    12. Marco Avellaneda & Antonio ParAS, 1996. "Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 21-52.
    13. Marco Avellaneda & Robert Buff, 1999. "Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(1), pages 1-18.
    14. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
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