Computation of distorted probabilities for diffusion processes via stochastic control methods. (English) Zbl 1103.65305

Summary: We study distorted survival probabilities related to risks in incomplete markets. The risks are modeled as diffusion processes, and the distortions are of general type. We establish a connection between distorted survival probabilities of the original risk process and distortion-free survival probabilities of new pseudo risk diffusions; the latter turns out to be diffusions with killing or splitting rates related, respectively, to concave and convex distortions. The main tools come from the theories of stochastic control, stochastic differential games, and non-linear partial differential equations.


65C50 Other computational problems in probability (MSC2010)
60J60 Diffusion processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
91A15 Stochastic games, stochastic differential games
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
91B26 Auctions, bargaining, bidding and selling, and other market models
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[1] Artzner, P., Delbaen, F., Eber, J.M., Heath, D., 1998. A characterization of measures of risk. Working paper. Department of Mathematics, ULP Strasbourg. · Zbl 0980.91042
[2] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-654, (1973) · Zbl 1092.91524
[3] Chateauneuf, A.; Kast, R.; Lapied, A., Choquet pricing for financial markets with frictions, Mathematical finance, 6, 323-330, (1996) · Zbl 0915.90011
[4] Crandall, M.; Lions, P.-L., Viscosity solutions of hamilton – jacobi equations, Transactions of the American mathematical society, 277, 1-42, (1983) · Zbl 0599.35024
[5] Davis, M.; Panas, V.; Zariphopoulou, T., European option pricing with transaction costs, SIAM journal on control and optimization, 31, 470-493, (1993) · Zbl 0779.90011
[6] Denneberg, D., 1994. Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht. · Zbl 0826.28002
[7] Durrett, R., 1996. Stochastic Calculus: A Practical Introduction. CRC Press, Boca Raton, FL. · Zbl 0856.60002
[8] El Karoui, N.; Quenez, M.-C., Dynamic programming and pricing of contingent claims in an incomplete market, SIAM journal on control and optimization, 33, 29-66, (1995) · Zbl 0831.90010
[9] Fleming, W.H., Soner, H.M., 1993. Controlled Markov processes and viscosity solutions. Applications of Mathematics, Vol. 25, Springer, New York. · Zbl 0773.60070
[10] Fleming, W.H.; Souganidis, P.E., On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana university mathematics journal, 38, 293-314, (1989) · Zbl 0686.90049
[11] Friedlin, M., 1985. Functional Integration and Partial Differential Equations. Princeton University Press, Princeton, NJ. · Zbl 0479.35011
[12] Gihman, I., Skorohod, A., 1972. Stochastic Differential Equations. Springer, Berlin. · Zbl 0242.60003
[13] Grundy, B.D., Wiener, Z., 1998. The analysis of deltas, state prices and VaR: a new approach. Working paper. University of Pennsylvania and Hebrew University.
[14] Ishii, H.; Lions, P.-L., Viscosity solutions of fully non-linear second-order elliptic partial differential equations, Journal of differential equations, 83, 26-78, (1990) · Zbl 0708.35031
[15] Jouini, E.; Kallal, H., Arbitrage in securities markets with short-sales constraints, Mathematical finance, 5, 197-232, (1995) · Zbl 0866.90032
[16] Jouini, E.; Kallal, H., Martingales and arbitrage in securities markets with transaction costs, Journal of economic theory, 66, 178-197, (1995) · Zbl 0830.90020
[17] Kellison, S.G., 1991. The Theory of Interest, 2nd Edition. Richard D. Irwin, Homewood, IL.
[18] Lions, P.-L., Optimal control of diffusion processes and hamilton – jacobi – bellman equations. part I. the dynamic programming principle and applications, Communications in PDE, 8, 1101-1174, (1983) · Zbl 0716.49022
[19] Lions, P.-L., Optimal control of diffusion processes and hamilton – jacobi – bellman equations. part II. viscosity solutions and uniqueness, Communications in PDE, 8, 1229-1276, (1983) · Zbl 0716.49023
[20] Norberg, R., 1997. Stochastic calculus in actuarial science: Ito’s revolution — our revelation. Working paper. University of Copenhagen.
[21] Von Neumann, J., Morgenstern, O., 1944. Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ. · Zbl 0063.05930
[22] Wang, S.S., Premium calculation by transforming the layer premium density, ASTIN bulletin, 26, 71-92, (1996)
[23] Wang, S.S.; Young, V.R., Risk-adjusted credibility premiums and distorted probabilities, Scandinavian actuarial journal, 1998, 143-165, (1998) · Zbl 1043.91512
[24] Wang, S.S.; Young, V.R.; Panjer, H.H., Axiomatic characterization of insurance prices, Insurance: mathematics and economics, 21, 173-183, (1997) · Zbl 0959.62099
[25] Yaari, M.E., The dual theory of choice under risk, Econometrica, 55, 95-115, (1987) · Zbl 0616.90005
[26] Young, V.R., Zariphopoulou, T., 1998. Stochastic variational formulae for jump processes and distorted probabilities. Working paper.
[27] Zariphopoulou, T., Investment-consumption models with transaction fees and Markov-chain parameters, SIAM journal on control and optimization, 30, 613-636, (1992) · Zbl 0784.90027
[28] Zariphopoulou, T., Investment-consumption models with constraints, SIAM journal on control and optimization, 32, 59-85, (1994) · Zbl 0790.90007
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