The Annals of Applied Probability

Stochastic control problems for systems driven by normal martingales

Rainer Buckdahn, Jin Ma, and Catherine Rainer

Full-text: Open access

Abstract

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener–Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 2 (2008), 632-663.

Dates
First available in Project Euclid: 20 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1206018200

Digital Object Identifier
doi:10.1214/07-AAP467

Mathematical Reviews number (MathSciNet)
MR2399708

Zentralblatt MATH identifier
1141.93065

Subjects
Primary: 93E20: Optimal stochastic control
Secondary: 60G44: Martingales with continuous parameter 35K55: Nonlinear parabolic equations

Keywords
Normal martingales structure equation Bellman principle HJB equation partial differential/difference equation viscosity solution

Citation

Buckdahn, Rainer; Ma, Jin; Rainer, Catherine. Stochastic control problems for systems driven by normal martingales. Ann. Appl. Probab. 18 (2008), no. 2, 632--663. doi:10.1214/07-AAP467. https://projecteuclid.org/euclid.aoap/1206018200


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References

  • Alvarez, O. and Tourin, A. (1996). Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 293–317.
  • Amadori, A. L., Karlsen, K. H. and La Chioma, C. (2004). Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations. Stochastics Stochastics Rep. 76 147–177.
  • Attal, S. and Émery, M. (1994). Équations de structure pour les martingales vectorielles. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583 256–278. Springer, Berlin.
  • Attal, S. and Émery, M. (1996). Martingales d’Azéma bidimensionnelles. Hommage à P.-A. Meyer et J. Neveu. Astérisque 236 9–21.
  • Azéma, J. and Rainer, C. (1994). Sur l’équation de structure d[X, X]t=dtX+tdXt. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583 236–255. Springer, Berlin.
  • Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 53–83.
  • Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Soc. 27 1–67.
  • Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992). Probabilités et Pontentiel: Chapitres XVII à XXIV. Paris, Hermann.
  • Dritschel, M. and Protter, P. (1999). Complete markets with discontinuous security price. Finance Stoch. 3 203–214.
  • Émery, M. (1989). On the Azéma martingales. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 66–87. Springer, Berlin.
  • Émery, M. (2006). Chaotic representation property of certain Azéma martingales. Illinois J. Math. 50 395–411.
  • Fleming, W. H. and Souganidis, P. E. (1989). On the existence of value functions of two-player zero-sum stochastic differential games. Indiana Univ. Math. J. 38 293–313.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Kurtz, D. (2001). Une caractérisation des martingales d’Azéma bidimensionnelles de type II. Séminaire de Probabilités XXXV. Lecture Notes in Math. 1755 98–119. Springer, Berlin.
  • Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
  • Liu, Y. and Ma, J. (2007). Optimal reinsurance/investment for general insurance models. Preprint.
  • Ma, J., Protter, P. and San Martin, J. (1998). Anticipating integrals for a class of martingales. Bernoulli 4 81–114.
  • Meyer, P. A. (1989). Construction de solutions d’équations de structure. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 142–145. Springer, Berlin.
  • Phan, A. (2001). Martingales d’Azéma asymétriques. Description élémentaire et unicité. Séminaire de Probabilités XXXV. Lecture Notes in Math. 1755 48–86. Springer, Berlin.
  • Protter, P. (1990). Stochastic Integration and Differential Equations. A New Approach. Springer, New York.
  • Taviot, G. (1999). Martingales et équations de structure: Étude géométrique. Thèse, IRMA, Strasbourg.