Minimizing shortfall risk and applications to finance and insurance problems. (English) Zbl 1015.93071

The author studies the problem of minimizing the shortfall risk defined as the expectation of the shortfall \((B-X_{T}^{x,\theta})_+\) (weighted by some loss function) with the following controlled process: \[ X_t^{x,\theta}=x+\int_0^t\theta_u dS_u+H_t^{\theta}, \] where \(B\) is a given nonnegative measurable random variable, \(S\) is a semimartingale, \(\Theta\) is the set of the control process, \(\theta\) is a convex subset of \(L(S)\) and \((H^{\theta}:\theta\in \Theta)\) is a concave family of adapted processes with finite variation, \(t\in [0,T].\)
An existence result to this optimization problem is stated, and some qualitative properties of the associated value function are shown. A verification theorem in terms of a dual control problem is established. Some applications to hedging problems in constrained portfolios, large investor and reinsurance models are given.
The approach for solving these control problems uses probabilistic methods rather than PDE methods via the Bellman equation. This allows relaxing the assumption of a Markov state process required in the PDE approach.


93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
49K45 Optimality conditions for problems involving randomness
91G80 Financial applications of other theories
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[1] BRÉMAUD, P. (1981). Point Processes and Queues. Springer, New York.
[2] CUOCO, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J. Econom. Theory 72 33-73. · Zbl 0883.90050
[3] CUOCO, D. and CVITANI Ć, J. (1998). Optimal consumption choice for a large investor. J. Econom. Dynam. Control 22 401-436. · Zbl 0902.90031
[4] CVITANI Ć, J. (2000). Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. 38 1050-1066. · Zbl 1034.91037
[5] CVITANI Ć, J. and KARATZAS, I. (1999). On dynamic measures of risk. Finance Stoch. 3 451-482. · Zbl 0982.91030
[6] CVITANI Ć, J., SCHACHERMAYER, W. and WANG, H. (2000). Utility maximization in incomplete markets with random endowment. Finance Stoch. · Zbl 1422.91647
[7] DELBAEN, F. and SCHACHERMAYER, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463-520. · Zbl 0865.90014
[8] EL KAROUI, N. and JEANBLANC, M. (1998). Optimization of consumption with labor income. Finance Stoch. 4 409-440. · Zbl 0930.60050
[9] EL KAROUI, N. and QUENEZ, M. C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29-66. · Zbl 0831.90010
[10] FLEMING, W. and SONER, M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York. · Zbl 0773.60070
[11] FÖLLMER, H. and KRAMKOV, D. (1997). Optional decomposition under constraints. Probab. Theory Related Fields 109 1-25. · Zbl 0882.60063
[12] FÖLLMER, H. and LEUKERT, P. (1999). Quantile hedging. Finance Stoch. 3 251-273. · Zbl 0977.91019
[13] FÖLLMER, H. and LEUKERT, P. (2000). Efficient hedging: cost versus shortfall risk. Finance Stoch. 4 117-146. · Zbl 0956.60074
[14] FRIEDMAN, A. (1975). Stochastic Differential Equations and Applications 1. Academic Press, New York. · Zbl 0323.60056
[15] HEATH, D. (1995). A continuous-time version of Kulldorf’s result.
[16] JACOD, J. (1979). Calcul stochastique et problémes de martingales. Lecture Notes in Math. 714. Springer, Berlin. · Zbl 0414.60053
[17] KARATZAS, I. and SHREVE, S. (1998). Methods of Mathematical Finance. Springer, New York. · Zbl 0941.91032
[18] KRAMKOV, D. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields 105 459-479. · Zbl 0853.60041
[19] KRAMKOV, D. and SCHACHERMAYER, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904-950. · Zbl 0967.91017
[20] KULLDORF, M. (1993). Optimal control of favourable games with a time-limit. SIAM J. Control Optim. 31 52-69. · Zbl 0770.90099
[21] MÉMIN, J. (1980). Espaces de semimartingales et changement de probabilité. Z. Wahrsch. Verw. Gebiete 52 9-39.
[22] PHAM, H. (2000). Dynamic Lp-hedging in discrete-time under cone constraints. SIAM J. Control Optim. 38 665-682. · Zbl 0964.91022
[23] SCHWEIZER, M. (1994). Approximating random variables by stochastic integrals. Ann. Probab. 22 1536-1575. · Zbl 0814.60041
[24] SPIVAK, G. and CVITANI Ć, J. (1999). Maximizing the probability of a perfect hedge. Ann. Appl. Probab. 9 1303-1328. · Zbl 0966.91042
[25] TOUZI, N. (2000). Optimal insurance demand under marked point processes shocks. Ann. Appl. Probab. 10 283-312. · Zbl 1161.91419
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