Probabilistic Interpretation for a System of Quasilinear Parabolic Partial Differential-Algebraic Equations: The Classical Solution

Abstract

In the present paper, by introducing a family of coupled forward-backward stochastic differential equations (FBSDEs for short), a probabilistic interpretation for a system consisting of m second order quasilinear (and possibly degenerate) parabolic partial differential equations and (m × d) algebraic equations is given in the sense of the classical solution. For solving the problem, an L^p-estimate (p > 2) for coupled FBSDEs on large time durations in the monotonicity framework is established, and a new method to analyze the regularity of solutions to FBSDEs is introduced. The new method avoids the use of Kolmogorov’s continuity theorem and only employs L²-estimates and L⁴-estimates to obtain the desired regularity.

Keywords

Forward-backward stochastic differential equation / Monotonicity condition / Parabolic partial differential equation / Classical solution / 60H10 / 35K59 / 35C99

References

[1] Antonelli F. Backward-forward stochastic differential equations. Ann. Appl. Probab., 1993, 3(3): 777-793.

[2] Barles G, Buckdahn R, Pardoux É. Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep., 1997, 60(1–2): 57-83.

[3] Briand P, Delyon B, Hu Y, et al. . L^p solutions of backward stochastic differential equation. Stochastic Process. Appl, 2003, 108(1): 109-129.

[4] Courant R, Hilbert D. Methods of mathematical physics, Vol. II, Partial differential equations, 1962, New York-London. Interscience Publishers

[5] Darling R W R, Pardoux É. Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab., 1997, 25(3): 1135-1159.

[6] Delarue F. On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stochastic Process. Appl, 2002, 99(2): 209-286.

[7] El Karoui N. Peng, S. and Quenez, M. C, Backward stochastic differential equations in finance. Math. Finance, 1997, 7(1): 1-71.

[8] Feng C, Wang X, Zhao H. Quasi-linear PDEs and forward-backward stochastic differential equations: Weak solutions. J. Differential Equations, 2018, 264(2): 959-1018.

[9] Hu M, Ji S, Xue X. A global stochastic maximum principle for fully coupled forward-backward stochastic systems. SIAM J. Control Optim., 2018, 56(6): 4309-4335.

[10] Hu Y, Peng S. Solution of forward-backward stochastic differential equations. Probab. Theory Related Fields, 1995, 103(2): 273-283.

[11] Kobylanski M. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab., 2000, 28(2): 558-602.

[12] Li J, Wei Q. Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim., 2014, 52(3): 1622-1662.

[13] Li J, Wei Q. L^p estimates for fully coupled FBSDEs with jumps. Stochastic Process. Appl., 2014, 124(4): 1582-1611.

[14] Li N, Yu Z. Forward-backward stochastic differential equations and linear-quadratic generalized Stackelberg games. SIAM J. Control Optim., 2018, 56(6): 4148-4180.

[15] Ma J, Protter P, Yong J. Solving forward-backward stochastic differential equations explicitly–a four step scheme. Probab. Theory Related Fields, 1994, 98(3): 339-359.

[16] Ma J, Wu Z, Zhang D, Zhang J. On well-posedness of forward-backward SDEs–a unified approach. Ann. Appl. Probab., 2015, 25(4): 2168-2214.

[17] Nualart D. The Malliavin calculus and related topics. Probability and its Applications (New York), 2006Second editionBerlin. Springer-Verlag

[18] Pardoux É. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. Stochastic analysis and related topics, VI (Geilo, 1996), 1998, Boston, MA. Birkhäuser Boston79-127. 42

[19] Pardoux É, Peng S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 1992, Berlin. Springer-Verlag200-217. 176

[20] Pardoux É, Pradeilles F, Rao Z. Probabilistic interpretation of a system of semi-linear parabolic partial differential equations. Ann. Inst. H. Poincaré Probab. Statist., 1997, 33(4): 467-490.

[21] Pardoux É, Răşcanu A. Stochastic differential equations, backward SDEs, partial differential equations, 2014, Cham. Springer-Verlag. 69

[22] Pardoux É, Tang S. Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields, 1999, 114(2): 123-150.

[23] Peng S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep., 1991, 37(1–2): 61-74. DOI:

[24] Peng S. Open problems on backward stochastic differential equations. Control of distributed parameter and stochastic systems (Hangzhou, 1998), 1999, Boston, MA. Kluwer Acad. Publ.265-273.

[25] Peng S, Wu Z. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim., 1999, 37(3): 825-843.

[26] Wu Z. Adapted solutions of forward-backward stochastic differential equations and their parameter dependence. Chinese J. Contemp. Math., 1998, 19(1): 9-18. DOI:

[27] Wu Z, Yu Z. Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations. Stochastic Process. Appl., 2014, 124(12): 3921-3947.

[28] Xie B, Yu Z. An exploration of L^p-theory for forward-backward stochastic differential equations with random coefficients on small durations. J. Math. Anal. Appl., 2020, 483(2): 12364218 pp

[29] Xie, B. and Yu, Z., L^p-estimate for linear forward-backward stochastic differential equations, Acta Math. Sin., Engl. Series., (2023, to appear), https://doi.org/10.1007/s10114-023-1147-5.

[30] Yong J. Finding adapted solutions of forward-backward stochastic differential equations: Method of continuation. Probab. Theory Related Fields, 1997, 107(4): 537-572.

[31] Yong J. Forward-backward stochastic differential equations with mixed initial-terminal conditions. Trans. Amer. Math. Soc, 2010, 362(2): 1047-1096.

[32] Zhang J. Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory, 2017, New York. Springer-Verlag.

[33] Zhang Q, Zhao H. Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients. J. Theoret. Probab., 2012, 25(2): 396-423.

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