Infinite horizon forward-backward stochastic differential equations. (English) Zbl 0997.60062

Peng, Shige; Shi, Yufeng

Stochastic Processes Appl. 85, No. 1, 75-92 (2000).

Let \(B\) be a standard \(d\)-dimensional Wiener process defined on a probability space \((\Omega ,\mathfrak F,P)\), let \((\mathfrak F_{t})\) be the (augmented) natural filtration of \(B\). An infinite horizon forward-backward stochastic differential equation \[ \begin{aligned} & dX(t) = b(t,X(t),Y(t),Z(t)) dt + \sigma (t,X(t), Y(t),Z(t)) dB(t), \tag{1} \\ -& dY(t) = f(t,X(t),Y(t),Z(t)) dt - Z(t) dB(t) \tag{2} \end{aligned} \] with an initial condition \( X(0) = x_0\) is studied. Considerable attention has been paid to the problem (1), (2) (with an additional condition \(Y(T) = \Phi (X(T))\)) on compact time intervals \([0,T]\); the paper under review, however, is devoted to the case \(T=+\infty \). The solution \((X,Y,Z)\) to (1), (2) should be an \((\mathfrak F_{t})\)-adapted process square integrable over \(\Omega \times \left [0,\infty \right [\). Let \(b,f: \Omega \times \mathbb R_{+}\times \mathbb R^{n}\times \mathbb R^{n}\times \mathbb R^{n\times d}\to \mathbb R^{n}\), \(\sigma : \Omega \times \mathbb R_{+}\times \mathbb R^{n}\times \mathbb R^{n}\times \mathbb R^{n\times d}\to \mathbb R^{n\times d}\) satisfy the following hypotheses: Set \(A(t,u) = (-f,b,\sigma)(t,(x,y,z))\) and assume that \(A(\cdot ,u)\) is an \((\mathfrak F_{t})\)-adapted process for each \(u\), \(A(\cdot ,0)\) is square integrable over \(\Omega \times \mathbb R_{+}\), \(A(t,\cdot)\) satisfies the Lipschitz condition uniformly in \(t\geq 0\), and the monotonicity hypothesis \(\langle A(t,u) - A(t,v), u-v\rangle \leq -\mu |u-v|^{2}\) for a \(\mu >0\) and all \(u,v\) holds. Then there exists a unique solution to (1), (2). Moreover, the authors prove a comparison theorem and investigate global asymptotical stability.

Reviewer: Jan Seidler (Praha)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

forward-backward stochastic differential equations; global exponential asymptotical stability; comparison theorems

References:

[1]	Antonelli, F., Backward-forward stochastic differential equations, Ann. Appl. Probab., 3, 777-793 (1993) · Zbl 0780.60058
[2]	Bensoussan, A., 1983a. Lectures on stochastic control, Nonlinear Filtering and Stochastic Control, Proceedings, Cortona.; Bensoussan, A., 1983a. Lectures on stochastic control, Nonlinear Filtering and Stochastic Control, Proceedings, Cortona.
[3]	Bensoussan, A., Stochastic maximum principle for distributed parameter system, J. Franklin Inst., 315, 387-406 (1983) · Zbl 0519.93042
[4]	Bensoussan, A., 1988. Perturbation Methods in Optimal Control Wiley, New York.; Bensoussan, A., 1988. Perturbation Methods in Optimal Control Wiley, New York. · Zbl 0648.49001
[5]	Bensoussan, A.; Peng, S., Singular perturbations in optimal control problems, Commande des systèmes complexes technologiques, APLL, 20, 359-381 (1986) · Zbl 0592.49015
[6]	Bismut, J. M., An introductory approach to duality in optimal stochastic control, SIAM Rev., 20, 62-78 (1978) · Zbl 0378.93049
[7]	Duffie, D.; Epstein, L., Asset pricing with stochastic differential utilities, Rev. Financial Stud., 5, 411-436 (1992)
[8]	Duffie, D.; Epstein, L., Stochastic differential utility, Econometrica, 60, 353-394 (1992) · Zbl 0763.90005
[9]	Duffie, D.; Ma, J.; Yong, J., Black’s consol rate conjecture, Ann. Appl. Probab., 5, 356-382 (1995) · Zbl 0830.60052
[10]	Haussmann, U. G., General necessary conditions for optimal control of stochastic systems, Math. Programming Stud., 6, 34-48 (1976) · Zbl 0369.93048
[11]	Hu, Y.; Peng, S., Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103, 273-283 (1995) · Zbl 0831.60065
[12]	Karoui, N. El.; Peng, S.; Quenez, M. C., Backward stochastic differential equation in finance, Math. Finance, 7, 1, 1-71 (1997) · Zbl 0884.90035
[13]	Kushner, H. J., Necessary conditions for continuous parameters stochastic optimization problems, SIAM. J. Control Optim., 10, 550-565 (1972) · Zbl 0242.93063
[14]	Ma, J.; Protter, P.; Yong, J., Solving forward-backward stochastic differential equations explicitly — a four step scheme, Probab. Theory Related Fields, 98, 339-359 (1994) · Zbl 0794.60056
[15]	Ma, J.; Yong, J., Solvability of forward-backward SDE’s and the nodal set of Hamilton-Jacobi-Bellman equations, Chinese Ann. Math., 16B, 279-298 (1995) · Zbl 0832.60067
[16]	Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14, 55-61 (1990) · Zbl 0692.93064
[17]	Pardoux, E., Peng, S., 1992. Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS, Vol. 176, Springer, Berlin, 200-217.; Pardoux, E., Peng, S., 1992. Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS, Vol. 176, Springer, Berlin, 200-217. · Zbl 0766.60079
[18]	Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 4, 966-979 (1990) · Zbl 0712.93067
[19]	Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics, 37, 61-74 (1991) · Zbl 0739.60060
[20]	Peng, S., A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation, Stochastics, 38, 119-134 (1992) · Zbl 0756.49015
[21]	Peng, S. (1992b). A Nonlinear Feynman-Kac Formula and Applications, in: Chen & Yong (Ed.), Proceedings of Symposium of System Sciences and Control Theory, World Scientific, Singapore, pp. 173-184.; Peng, S. (1992b). A Nonlinear Feynman-Kac Formula and Applications, in: Chen & Yong (Ed.), Proceedings of Symposium of System Sciences and Control Theory, World Scientific, Singapore, pp. 173-184.
[22]	Peng, S.; Wu, Z., Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37, 825-843 (1999) · Zbl 0931.60048
[23]	Yong, J., Finding adapted solutions of forward-backward stochastic differential equations — method of continuation, Probab. Theory Related Fields, 107, 537-572 (1997) · Zbl 0883.60053