Backward stochastic differential equations with constraints on the gains-process
Jak{\v{s}}a Cvitani{\'c}, Ioannis Karatzas, and H. Mete Soner
Source: Ann. Probab. Volume 26, Number 4
(1998), 1522-1551.
Abstract
We consider backward stochastic differential equations with convex constraints on the gains (or intensity-of-noise) process. Existence and uniqueness of a minimal solution are established in the case of a drift coefficient which is Lipschitz continuous in the state and gains processes and convex in the gains process. It is also shown that the minimal solution can be characterized as the unique solution of a functional stochastic control-type equation. This representation is related to the penalization method for constructing solutions of stochastic differential equations, involves change of measure techniques, and employs notions and results from convex analysis, such as the support function of the convex set of constraints and its various properties.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1022855872
Mathematical Reviews number (MathSciNet): MR1675035
Digital Object Identifier: doi:10.1214/aop/1022855872
Zentralblatt MATH identifier: 0935.60039
References
Arkin, V. and Saxonov, M. (1979). Necessary optimality conditions for stochastic diferrential equations. Soviet Math. Doklady 20 1-15.
Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 57-83.
Mathematical Reviews (MathSciNet): MR98d:60110
Zentralblatt MATH: 0878.60036
Bensoussan, A. (1981). Lectures on stochastic control. Lecture Notes in Math. 972 1-62. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR705931
Digital Object Identifier: doi:10.1007/BFb0064859
Bismut, J. M. (1978). An introductory approach to duality in optimal stochastic control. SIAM Rev. 20 62-78.
Mathematical Reviews (MathSciNet): MR57:9256
Zentralblatt MATH: 0378.93049
Digital Object Identifier: doi:10.1137/1020004
JSTOR: links.jstor.org
Broadie, M., Cvitani´c, J. and Soner, H. M. (1998). Optimal replication of contingent claims under portfolio constraints. Rev. Financial Studies 11 59-79.
Buckdahn, R. and Hu, Y. (1996). Hedging contingent claims for a large investor in an incomplete market. Preprint.
Mathematical Reviews (MathSciNet): MR1618845
Zentralblatt MATH: 0904.90009
Digital Object Identifier: doi:10.1239/aap/1035228002
Project Euclid: euclid.aap/1035228002
Buckdahn, R. and Hu, Y. (1997). Pricing of American contingent claims with jump stock price and constrained portfolios. Preprint.
Mathematical Reviews (MathSciNet): MR1606470
Zentralblatt MATH: 0985.91026
Digital Object Identifier: doi:10.1287/moor.23.1.177
JSTOR: links.jstor.org
Cvitani´c, J. and Karatzas, I. (1992). Convex duality for constrained porfolio optimization. Ann. Appl. Probab. 2 767-818.
Cvitani´c, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3 652-681.
Mathematical Reviews (MathSciNet): MR95c:90022
Zentralblatt MATH: 0825.93958
Digital Object Identifier: doi:10.1214/aoap/1177005357
Project Euclid: euclid.aoap/1177005357
Cvitani´c, J. and Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 2024-2056.
Mathematical Reviews (MathSciNet): MR97h:93080
Zentralblatt MATH: 0876.60031
Digital Object Identifier: doi:10.1214/aop/1041903216
Project Euclid: euclid.aop/1041903216
Darling, R. W. R. and Pardoux, E. (1997). Backward SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25 1135-1159.
Mathematical Reviews (MathSciNet): MR98k:60095
Zentralblatt MATH: 0895.60067
Digital Object Identifier: doi:10.1214/aop/1024404508
Project Euclid: euclid.aop/1024404508
Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353-394.
Zentralblatt MATH: 0768.90006
Mathematical Reviews (MathSciNet): MR1162620
Digital Object Identifier: doi:10.2307/2951600
JSTOR: links.jstor.org
Elliott, R. J. (1990). The optimal control of diffusions. Appl. Math. Optim. 22 229-240.
Zentralblatt MATH: 0718.49013
Mathematical Reviews (MathSciNet): MR1068181
Digital Object Identifier: doi:10.1007/BF01447329
El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Probab. 25 702-737.
Mathematical Reviews (MathSciNet): MR98k:60096
Zentralblatt MATH: 0899.60047
Digital Object Identifier: doi:10.1214/aop/1024404416
Project Euclid: euclid.aop/1024404416
El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71.
Zentralblatt MATH: 0884.90035
Mathematical Reviews (MathSciNet): MR1434407
Digital Object Identifier: doi:10.1111/1467-9965.00022
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. Hamad ene, S. and Lepeltier, J. M. (1995a). Backward equations, stochastic control, and zerosum stochastic differential games. Stochastics Stochastics Rep. 54 221-231. Hamad ene, S. and Lepeltier, J. M. (1995b). Zero-sum stochastic differential games and backward equations. Systems Control Lett. 24 259-263.
Mathematical Reviews (MathSciNet): MR96h:90021
Zentralblatt MATH: 0831.90010
Digital Object Identifier: doi:10.1137/S0363012992232579
Haussmann, U. G. (1986). A Stochastic Maximum Principle for Optimal Control of Diffusions. Longman, Essex, U.K.
Mathematical Reviews (MathSciNet): MR88c:49022
Zentralblatt MATH: 0616.93076
It o, K. (1942). Differential equations determining Markov processes. Zenkoku Shij¯o S ¯ugaku Danwakai 1077 1352-1400 (in Japanese).
It o, K. (1946). On a stochastic integral equation. Proc. Imperial Acad. Tokyo 22 32-35.
Mathematical Reviews (MathSciNet): MR36958
Digital Object Identifier: doi:10.3792/pja/1195572371
Project Euclid: euclid.pja/1195572371
It o, K. (1951). On stochastic differential equations. Mem. Amer. Math. Soc. 4 1-51.
Mathematical Reviews (MathSciNet): MR40618
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet): MR92h:60127
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York. To appear.
Mathematical Reviews (MathSciNet): MR2000e:91076
Zentralblatt MATH: 0941.91032
Ma, J. and Cvitani´c, J. (1997). Reflected Forward-Backward SDEs and obstacle problems with boundary conditions. Preprint.
Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet): MR53:6729
Zentralblatt MATH: 0345.60026
Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55-61.
Mathematical Reviews (MathSciNet): MR91e:60171
Zentralblatt MATH: 0692.93064
Pardoux, E. and Peng, S. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 209-227.
Mathematical Reviews (MathSciNet): MR94m:60120
Zentralblatt MATH: 0792.60050
Digital Object Identifier: doi:10.1007/BF01192514
Pardoux, E. and Tang, S. (1996). The study of forward-backward stochastic differential equation and its application in quasilinear PDEs. Preprint.
Peng, S. (1990). A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28 966-979.
Zentralblatt MATH: 0712.93067
Mathematical Reviews (MathSciNet): MR1051633
Digital Object Identifier: doi:10.1137/0328054
Peng, S. (1991). Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37 61-74.
Zentralblatt MATH: 0739.60060
Mathematical Reviews (MathSciNet): MR1149116
Peng, S. (1993). Backwards stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27 125-144.
Mathematical Reviews (MathSciNet): MR1202528
Digital Object Identifier: doi:10.1007/BF01195978
Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
Mathematical Reviews (MathSciNet): MR43:445
Saksonov, M. (1989). The Variational Principles in Stochastic Control. Dushanbe State Educational Institute, Preprint.
