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Finance and Stochastics
Published in: Finance and Stochastics 3/2016

01-07-2016

Consumption-investment problem with transaction costs for Lévy-driven price processes

Authors: Dimitri De Vallière, Yuri Kabanov, Emmanuel Lépinette

Published in: Finance and Stochastics | Issue 3/2016

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Abstract

We consider an optimal control problem for a linear stochastic integro-differential equation with conic constraints on the phase variable and with the control of singular–regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs, where the prices of the assets are given by a geometric Lévy process, and the investor is allowed to take short positions. We prove that the Bellman function of the problem is a viscosity solution of an HJB equation. A uniqueness theorem for the solution of the latter is established. Special attention is paid to the dynamic programming principle.
previous article Robust pricing and hedging under trading restrictions and the emergence of local martingale models
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Literature
1.
Aït-Sahalia, Y., Cacho-Diaz, J., Hurd, T.: Portfolio choice with jumps: A closed form solution. Ann. Appl. Probab. 19, 556–584 (2009) MathSciNetCrossRefMATH
2.
3.
Akian, M., Menaldi, J.L., Sulem, A.: On an investment-consumption model with transaction costs. SIAM J. Control Optim. 34, 329–364 (1996) MathSciNetCrossRefMATH
4.
Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, 293–317 (1996) MathSciNetMATH
5.
Arisawa, M.: A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, 695–711 (2006) MathSciNetCrossRefMATH
6.
Arisawa, M.: A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators. J. Math. Pures Appl. 89, 567–574 (2008) MathSciNetCrossRefMATH
7.
Aubin, J.-P.: Optima and Equilibria. An Introduction to Nonlinear Analysis. Springer, Berlin–Heidelberg–New York (1993) MATH
8.
Barles, G., Chasseigne, E., Imbert, C.: The Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57, 213–246 (2008) MathSciNetCrossRefMATH
9.
Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: Viscosity solutions’ theory revisited. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, 567–585 (2008) MathSciNetCrossRefMATH
10.
Benth, F.E., Karlsen, K.H., Reikvam, K.: Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs. Stoch. Stoch. Rep. 74, 517–569 (2002) MathSciNetCrossRefMATH
11.
Benth, F.E., Karlsen, K.H., Reikvam, K.: Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution. Finance Stoch. 5, 447–467 (2001) MathSciNetCrossRefMATH
12.
13.
Crandall, M., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 277, 1–42 (1983) MathSciNetCrossRefMATH
14.
Cvitanić, J., Polimenis, V., Zapatero, F.: Optimal portfolio allocation with higher moments. Ann. Finance 4, 1–28 (2008) CrossRefMATH
15.
16.
Emmer, S., Klüppelberg, C.: Optimal portfolios when stock prices follow an exponential Lévy process. Finance Stoch. 8, 17–44 (2004) MathSciNetCrossRefMATH
17.
Framstad, N.C., Øksendal, B., Sulem, A.: Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs. J. Math. Econ. 35, 233–257 (2001) MathSciNetCrossRefMATH
18.
19.
Jacobsen, E.R., Karlsen, K.H.: Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Equ. 212, 278–318 (2005) MathSciNetCrossRefMATH
20.
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2002) MATH
21.
22.
Kabanov, Yu.M., Klüppelberg, C.: A geometric approach to portfolio optimization in models with transaction costs. Finance Stoch. 8, 207–227 (2004) MathSciNetCrossRefMATH
23.
Kabanov, Yu.M., Safarian, M.: Markets with Transaction Costs: Mathematical Theory. Springer, Berlin–Heidelberg–New York (2009) MATH
24.
25.
Pham, H.: Optimal stopping of controlled jump-diffusion processes: A viscosity solutions approach. J. Math. Syst. Estim. Control 8, 1–27 (1998)
26.
Sayah, A.: Equations d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels, I, II. Commun. Partial Differ. Equ. 16, 1057–1093 (1991) CrossRefMATH
27.
28.
29.
Soner, H.M.: Optimal control of jump-Markov processes and viscosity solutions. In: Fleming, W., Lions, P.-L. (eds.) Stochastic Differential Systems, Stochastic Control Theory and Applications. IMA Vols. in Mathematics and Its Applications, vol. 10, pp. 501–511. Springer, New York (1988) CrossRef
Metadata
Title
Consumption-investment problem with transaction costs for Lévy-driven price processes
Authors
Dimitri De Vallière
Yuri Kabanov
Emmanuel Lépinette
Publication date
01-07-2016
Publisher
Springer Berlin Heidelberg
Published in
Finance and Stochastics / Issue 3/2016
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-016-0303-5

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