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On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. (English) Zbl 1058.60042

Summary: We prove a result of existence and uniqueness of solutions to forward-backward stochastic differential equations, with non-degeneracy of the diffusion matrix and boundedness of the coefficients as functions of \(x\) as main assumptions. This result is proved in two steps. The first part studies the problem of existence and uniqueness over a small enough time duration, whereas the second one explains, by using the connection with quasi-linear parabolic system of PDEs, how we can deduce, from this local result, the existence and uniqueness of a solution over an arbitrarily prescribed time duration. Improving this method, we obtain a result of existence and uniqueness of classical solutions to non-degenerate quasi-linear parabolic systems of PDEs. This approach relaxes the regularity assumptions required on the coefficients by the four-step scheme.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Antonelli, F., Backward – forward stochastic differential equations, Ann. appl. probab., 3, 777-793, (1993) · Zbl 0780.60058
[2] Hu, Y., On the existence of solutions to one-dimensional forward – backward sdes, Stochastic anal. appl., 18, 101-111, (2000) · Zbl 0949.60070
[3] Hu, Y., On the solution of forward – backward SDEs with monotone and continuous coefficients, Non-linear anal., 42, 1-12, (2000) · Zbl 0970.34056
[4] Hu, Y.; Peng, S., Solution of forward – backward stochastic differential equations, Probab. theory related fields, 103, 273-283, (1995) · Zbl 0831.60065
[5] Hu, Y.; Yong, J., Forward – backward stochastic differential equations with nonsmooth coefficients, Stochastic process. appl., 87, 93-106, (2000) · Zbl 1045.60058
[6] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, Grundlehren der mathematischen wissenschaften, (1987), Springer Berlin, Heidelberg · Zbl 0635.60021
[7] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, Graduate texts in mathematics, (1988), Springer New York · Zbl 0638.60065
[8] Ladyzenskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasi-linear equations of parabolic type, Translation of mathematical monographs, (1968), American Mathematical Society Providence, RI
[9] Ma, J.; Yong, J., Solvability of forward – backward SDEs and the nodal set of hamilton – jacobi – bellman equations, Chinese ann. math. ser. B, 16, 279-298, (1995) · Zbl 0832.60067
[10] Ma, J.; Yong, J., Forward – backward stochastic differential equations and their applications, Lecture notes in mathematics, Vol. 1702, (1999), Springer Berlin · Zbl 0927.60004
[11] 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
[12] Nualart, D., The Malliavin calculus and related topics, (1995), Springer New York · Zbl 0837.60050
[13] Pardoux, É., 1999. BSDE’s, weak convergence and homogenization of semilinear PDE’s. In: Clarke, F.H., Stern, R.J. (Eds.). Nonlinear Analysis, Differential Equations and Control. Kluwer Academic Publishers, pp. 503-549. · Zbl 0959.60049
[14] Pardoux, É.; Peng, S., Backward stochastic differential equations and quasi-linear parabolic partial differential equations, (), 200-217 · Zbl 0766.60079
[15] Pardoux, É.; Tang, S., Forward – backward stochastic differential equations and quasilinear parabolic pdes, Probab. theory related fields, 114, 123-150, (1999) · Zbl 0943.60057
[16] Peng, S.; Wu, Z., Fully coupled forward – backward stochastic differential equations and applications to optimal control, SIAM J. control optim., 37, 3, 825-843, (1999) · Zbl 0931.60048
[17] Rogers, L.C.G.; Williams, D., Diffusions, Markov processes and martingales: Itô calculus, (1987), Wiley Chichester, UK
[18] 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
[19] Yong, J., Linear forward – backward stochastic differential equations, Appl. math. optim., 39, 93-119, (1999) · Zbl 0920.60045
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