zbMATH — the first resource for mathematics

Anticipated backward stochastic differential equations. (English) Zbl 1186.60053
Let \(T>0\) denote a finite time horizon and \(W=(W_t)_{t\in[0,T]}\) be a \(d\)-dimensional Brownian motion. The authors consider two functions \(\delta_i:[0,T]\rightarrow \mathbb{R}_+,\, i=1,2,\) for which there is some constant \(K>0\) such that, for both of them, \(s+\delta_i(s)\leq T+K,\, s\in[0,T],\) and \(\int_t^Tg(s\delta_i(s))ds\leq C_K\int_t^{T+K}g(s)ds,\, i=1,2,\) for all non negative, integrable function \(g\).
The authors investigate in their paper the following new type of (\(m\)-dimensional) backward stochastic differential equations (BSDE):
\[ dY_t=-f(t,Y_t,Z_t,Y_{t+\delta(t)},Z_{t+\zeta(t)})dt+Z_tdW_t,\, t\in[0,T], \] where they put \(Y_t=\xi_t,\, Z_t=\eta_t,\, t\in[T,T+K],\) for given stochastic processes \(\xi\in{\mathcal S}_{\mathbb{F}}^2(T,T+K;\mathbb{R}^{m}),\, \eta\in L_{\mathbb{F}}^2(T,T+K;\mathbb{R}^{m\times d}).\) This type of BSDE generalizes the by now well known BSDE introduced by E. Pardoux and S. Peng in their pioneering paper of 1990 [Syst. Control Lett. 14, No. 1, 55–61 (1990; Zbl 0692.93064)]. Taking into account that a BSDE is solved backward, beginning from its time horizon \(T\), the above BSDE can be regarded as a translation of the concept of stochastic differential equations (SDE) with delay into the context of backward stochastic equations.
But the link between these anticipated BSDE and SDE with delay is even narrower as the authors show with their proof of duality between both types of equation. Concerning the above anticipated BSDE, the authors prove that under adequate assumptions on the driving coefficient \(f\) which generalize those for standard BSDE and include in particular the assumption that \(f(s,y,z,\xi,\eta)\) is \({\mathcal F}^W_s\)-measurable, \(s\in[0,T],\) for all square integrable random variables \(\xi,\eta,\) there exists a unique solution \((Y,Z)\). Moreover, for the one-dimensional anticipated BSDE (\(m=1\)) they establish a comparison principle and prove also a strict comparison result. Finally, their paper is completed by the study of a control problem in which the controlled state process is a linear SDE with delay, and whose cost functional is described by an anticipated BSDE.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
93E03 Stochastic systems in control theory (general)
Full Text: DOI arXiv
[1] Beneš, V. E. (1970). Existence of optimal strategies based on specified information, for a class of stochastic decision problems. SIAM J. Control Optim. 8 179-188. · Zbl 0195.48202 · doi:10.1137/0308012
[2] Beneš, V. E. (1971). Existence of optimal stochastic control laws. SIAM J. Control Optim. 9 446-472. · Zbl 0219.93029 · doi:10.1137/0309034
[3] Cao, Z. G. and Yan, J.-A. (1999). A comparison theorem for solutions of backward stochastic differential equations. Adv. Math. ( China ) 28 304-308. · Zbl 1054.60505
[4] Cvitanić, J. and Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 2024-2056. · Zbl 0876.60031 · doi:10.1214/aop/1041903216
[5] Dellacherie, C. (1972). Capacités et Processus Stochastiques. Ergebnisse der Mathematik und ihrer Grenzgebiete 67 . Springer, Berlin. · Zbl 0246.60032
[6] 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. · Zbl 0899.60047 · doi:10.1214/aop/1024404416
[7] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. · Zbl 0884.90035 · doi:10.1111/1467-9965.00022
[8] Hu, Y. and Peng, S. (2006). On the comparison theorem for multidimensional BSDEs. C. R. Math. Acad. Sci. Paris Ser. I 343 135-140. · Zbl 1098.60052 · doi:10.1016/j.crma.2006.05.019
[9] Lepeltier, J.-P. and Martín, J. S. (2004). Backward SDEs with two barriers and continuous coefficient: An existence result. J. Appl. Probab. 41 162-175. · Zbl 1051.60066 · doi:10.1239/jap/1077134675
[10] Lin, Q. Q. (2001). A comparison theorem for backward stochastic differential equations. J. Huazhong Univ. Sci. Tech. 29 1-3. · Zbl 0467.93062
[11] Liu, J. and Ren, J. (2002). Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 56 93-100. · Zbl 1004.60063 · doi:10.1016/S0167-7152(01)00178-X
[12] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory Related Fields 113 473-499. · Zbl 0953.60059 · doi:10.1007/s004400050214
[13] Peng, S. (2004). Nonlinear expectations, nonlinear evaluations and risk measures. In Stochastic Methods in Finance. Lecture Notes in Math. 1856 165-253. Springer, Berlin. · Zbl 1127.91032
[14] Peng, S. and Xu, M. Y. (2005). The smallest g -supermartingale and reflected BSDE with single and double L 2 obstacles. Ann. Inst. H. Poincaré Probab. Statist. 41 605-630. · Zbl 1071.60049 · doi:10.1016/j.anihpb.2004.12.002 · numdam:AIHPB_2005__41_3_605_0 · eudml:77860
[15] Situ, R. (1999). Comparison theorem of solutions to BSDE with jumps, and viscosity solution to a generalized Hamilton-Jacobi-Bellman equation. In Control of Distributed Parameter and Stochastic Systems ( Hangzhou , 1998) 275-282. Kluwer Academic, Boston, MA. · Zbl 0981.93077
[16] Zhang, T. S. (2003). A comparison theorem for solutions of backward stochastic differential equations with two reflecting barriers and its applications. In Probabilistic Methods in Fluids 324-331. World Science, River Edge, NJ. · Zbl 1062.60058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.