The Annals of Applied Probability

Optimal reinsurance/investment problems for general insurance models

Yuping Liu and Jin Ma

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In this paper the utility optimization problem for a general insurance model is studied. The reserve process of the insurance company is described by a stochastic differential equation driven by a Brownian motion and a Poisson random measure, representing the randomness from the financial market and the insurance claims, respectively. The random safety loading and stochastic interest rates are allowed in the model so that the reserve process is non-Markovian in general. The insurance company can manage the reserves through both portfolios of the investment and a reinsurance policy to optimize a certain utility function, defined in a generic way. The main feature of the problem lies in the intrinsic constraint on the part of reinsurance policy, which is only proportional to the claim-size instead of the current level of reserve, and hence it is quite different from the optimal investment/consumption problem with constraints in finance. Necessary and sufficient conditions for both well posedness and solvability will be given by modifying the “duality method” in finance and with the help of the solvability of a special type of backward stochastic differential equations.

Article information

Ann. Appl. Probab., Volume 19, Number 4 (2009), 1495-1528.

First available in Project Euclid: 27 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91B28 91B30: Risk theory, insurance
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 93G20

Cramér–Lundburg reserve model proportional reinsurance optimal investment Girsanov transformation duality method backward stochastic differential equations


Liu, Yuping; Ma, Jin. Optimal reinsurance/investment problems for general insurance models. Ann. Appl. Probab. 19 (2009), no. 4, 1495--1528. doi:10.1214/08-AAP582.

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  • [1] Asmussen, S. and Nielsen, H. M. (1995). Ruin probabilities via local adjustment coefficients. J. Appl. Probab. 20 913–916.
  • [2] Bowers, N. L., Jr., Gerber, H. U., Hickman, J. C., Jones, D. A. and Nesbitt, C. J. (1997). Actuarial Mathematics. The Society of Actuaries, Schaumburg, IL.
  • [3] Bühlmann, H. (1970). Mathematical Methods in Risk Theory. Springer, Berlin.
  • [4] Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2 767–818.
  • [5] Cvitanić, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3 652–681.
  • [6] Gerber, H. U. (1970). Mathematical Methods in Risk Theory. Springer, Berlin.
  • [7] Højgaard, B. and Taksar, M. (1997). Optimal proportional reinsurance policies for diffusion models. Scand. Actuar. J. 2 166–180.
  • [8] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [9] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [10] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [11] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics (New York) 39. Springer, New York.
  • [12] Lepeltier, J. P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 32 425–430.
  • [13] Liu, Y. (2004). Some reinsurance/investment optimization problems for general risk reserve models. Ph.D. thesis, Purdue Univ.
  • [14] Ma, J. and Sun, X. (2003). Ruin probabilities for insurance models involving investments. Scand. Actuar. J. 3 217–237.
  • [15] Ma, J. and Yong, J. (1999). Forward–Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. 1702. Springer, Berlin.
  • [16] Protter, P. (1990). Stochastic Integration and Differential Equations: A New Approach. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [17] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
  • [18] Rong, S. (1997). On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl. 66 209–236.
  • [19] Tang, S. J. and Li, X. J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447–1475.
  • [20] Xue, X. X. (1992). Martingale representation for a class of processes with independent increments and its applications. In Applied Stochastic Analysis (New Brunswick, NJ, 1991). Lecture Notes in Control and Inform. Sci. 177 279–311. Springer, Berlin.