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Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions

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  • Seydel, Roland C.

Abstract

General theorems for existence and uniqueness of viscosity solutions for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVI) with integral term are established. Such nonlinear partial integro-differential equations (PIDE) arise in the study of combined impulse and stochastic control for jump-diffusion processes. The HJBQVI consists of an HJB part (for stochastic control) combined with a nonlocal impulse intervention term. Existence results are proved via stochastic means, whereas our uniqueness (comparison) results adapt techniques from viscosity solution theory. This paper, to our knowledge is the first treating rigorously impulse control for jump-diffusion processes in a general viscosity solution framework; the jump part may have infinite activity. In the proofs, no prior continuity of the value function is assumed, quadratic costs are allowed, and elliptic and parabolic results are presented for solutions possibly unbounded at infinity.

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  • Seydel, Roland C., 2009. "Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3719-3748, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3719-3748
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    1. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    2. Pierre Bernhard, 2006. "On The Singularities Of An Impulsive Differential Game Arising In Mathematical Finance," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 219-229.
    3. Martin Dahlgren & Ralf Korn, 2005. "The Swing Option On The Stock Market," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 123-139.
    4. Ralf Korn, 1998. "Portfolio optimisation with strictly positive transaction costs and impulse control," Finance and Stochastics, Springer, vol. 2(2), pages 85-114.
    5. Stanley Pliska & Kiyoshi Suzuki, 2004. "Optimal tracking for asset allocation with fixed and proportional transaction costs," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 233-243.
    6. Mundaca, Gabriela & Oksendal, Bernt, 1998. "Optimal stochastic intervention control with application to the exchange rate," Journal of Mathematical Economics, Elsevier, vol. 29(2), pages 225-243, March.
    7. Ralf Korn, 1999. "Some applications of impulse control in mathematical finance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(3), pages 493-518, December.
    8. Abel Cadenillas & Fernando Zapatero, 2000. "Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rates and Reserves," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 141-156, April.
    9. repec:dau:papers:123456789/332 is not listed on IDEAS
    10. Kristin Reikvam & Fred Espen Benth & Kenneth Hvistendahl Karlsen, 2001. "Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 447-467.
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    5. Aïd, René & Campi, Luciano & Langrené, Nicolas & Pham, Huyên, 2014. "A probabilistic numerical method for optimal multiple switching problems in high dimension," LSE Research Online Documents on Economics 63011, London School of Economics and Political Science, LSE Library.
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    10. Martin Hunting & Jostein Paulsen, 2013. "Optimal dividend policies with transaction costs for a class of jump-diffusion processes," Finance and Stochastics, Springer, vol. 17(1), pages 73-106, January.

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