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Strong solutions of forward-backward stochastic differential equations with measurable coefficients. (English) Zbl 1494.60066

The paper provides the existence of (strong) solutions to fully coupled systems of forward-backward stochastic differential equations (FBSDEs) with irregular coefficients, i.e., \begin{align*} X_t &= x+\int_0^t b(u,X_u,Y_u,Z_u)\, d u + \int_0^t \sigma\, d W_u, \\ Y_t &= h(X_T)+\int_t^T g(u,X_u,Y_u,Z_u)\, du - \int_t^T Z_u \, d W_u, \quad t \in [0,T], \end{align*} where \(b\), \(g\) and \(h\) are measurable functions, uniformly continuous in \((y, z)\) and \(W\) is a multi-dimensional Brownian motion. The authors’ approach to prove the existence of solutions is based on approximating the functions \(b\), \(g\) and \(h\) by smooth functions and to use ideas from Malliavin calculus, notably a compactness principle. Furthermore, despite the irregularity of the coefficients, it is shown that solutions of the such FBSDE are differentiable in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
60H07 Stochastic calculus of variations and the Malliavin calculus
35D40 Viscosity solutions to PDEs
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[1] Amann, H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, (Schmeisser, H.; H. Triebel, T., Function Spaces, Differential Operators and Nonlinear Analysis. Vol. 133 (1993), Teubner-Texte zur Mathematik), 9-126 · Zbl 0810.35037
[2] Andreucci, D., Degenerate parabolic equations with initial data measures, Trans. Amer. Math. Soc., 349, 3911-3923 (1997) · Zbl 0885.35056
[3] Ankirchner, S.; Imkeller, P.; Dos Reis, G., Classical and variational differentiability of BSDEs with quadratic growth, Electron. J. Probab., 12, 1418-1453 (2007) · Zbl 1138.60042
[4] Antonelli, F.; Hamadène, S., Existence of solutions of backward-forward SDEs with continuous monotone coefficients, Statist. Probab. Lett., 76, 14, 1559-1569 (2006) · Zbl 1101.60038
[5] Bahlali, K., Flows of homeomorphisms of stochastic differential equations with measurable drifts, Stochastics, 67, 1, 53-82 (1999) · Zbl 0937.60059
[6] Bahlali, K.; Djehiche, B.; Mezerdi, B., On the stochastic maximum principle in optimal control of degenerate diffusions with Lipschitz coefficients, Appl. Math. Optim., 56, 364-378 (2007) · Zbl 1135.60323
[7] Bahlali, S.; Djehihe, B.; Mezerdi, B., The relaxed stochastic maximum principle in singular optimal control of diffusions, SIAM J. Control Optim., 46, 2, 427-444 (2007) · Zbl 1141.93063
[8] Bahlali, K.; Eddahbi, M.; Ouknine, Y., Quadratic BSDE with \(\mathbb{L}^2\)-terminal data: Krylov’s estimates, itô-Krylov’s formula and existence results, Ann. Probab., 45, 4, 2377-2397 (2017) · Zbl 1379.60057
[9] Brezis, H.; Cazenave, T., A nonlinear heat equation with singular initial data, J. Anal. Math., 68, 277-304 (1996) · Zbl 0868.35058
[10] Buckdahn, R.; Engelbert, H.-J., A backward stochastic differential equation without strong solution, Teor. Veroyatn. Primen., 50, 2, 390-396 (2005) · Zbl 1090.60051
[11] Carmona, R.; Delarue, F., (Probabilistic Theory of Mean Field Games with Applications. I. Probabilistic Theory of Mean Field Games with Applications. I, Probability Theory and Stochastic Modelling, vol. 83 (2018), Springer, Cham), xxv+713, ISBN: 978-3-319-56437-1; 978-3-319-58920-6. Mean field FBSDEs, control, and games · Zbl 1422.91014
[12] Cvitanic, J.; Zhang, J., Contract Theory in Continuos Time Models (2012), Springer Finance: Springer Finance Heidelberg
[13] Da Prato, G.; Malliavin, P.; Nualart, D., Compact families of Wiener functionals, C. R. Acad. Sci. Paris, 315, 1287-1291 (1992) · Zbl 0782.60002
[14] Dareiotis, K. A.; Gyöngy, Y., A comparison principle for stochastic integro-differential equations, Potential Anal., 41, 1203-1222 (2014) · Zbl 1307.60093
[15] Delarue, F., On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Process. Appl., 99, 209-286 (2002) · Zbl 1058.60042
[16] Delarue, F., Estimates of the solutions of a system of quasi-linear PDEs. a probabilistic scheme, Séminaire de Probab., XXXVII, 290-332 (2003) · Zbl 1055.35029
[17] Delarue, F.; Guatteri, G., Weak existence and uniqueness for forward-backward SDEs, Stochastic Process. Appl., 116, 1712-1742 (2006) · Zbl 1113.60059
[18] Denis, L.; Matoussi, A.; Stoica, L., \( L^p\) estimates for the uniform norm of solutions of quasilinear SPDE’s, Probab. Theory Related Fields, 133, 437-463 (2005) · Zbl 1085.60043
[19] Denis, L.; Matoussi, A.; Zhang, J., Maximum principle for quasilinear stochastic PDEs with obstacle, Electron. J. Probab., 19, 1-32 (2014) · Zbl 1310.60093
[20] Denis, L.; Stoica, L., A general analytical result for non-linear s.p.d.e.’s and applications, Electron. J. Probab., 9, 674-709 (2004) · Zbl 1067.60048
[21] Fromm, A.; Imkeller, P.; Prömel, D. J., An FBSDE approach of the skorokhod embedding problem for Gaussian processes with non-linear drift, Electron. J. Probab., 20, 127, 1-38 (2015) · Zbl 1332.60081
[22] Giga, Y., Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62, 186-212 (1986) · Zbl 0577.35058
[23] Heyne, G.; Kupper, M.; Tangpi, L., Portfolio optimization under nonlinear utility, Int. J. Theor. Appl. Finance, 19, 5 (2016) · Zbl 1396.91689
[24] Horst, U.; Hu, Y.; Imkeller, P.; Réveillac, A.; Zhang, J., Forward backward systems for expected utility maximization, Stochastic Process. Appl., 124, 5, 1813-1848 (2014) · Zbl 1329.60182
[25] Hoshino, H.; Yamada, Y., Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34, 475-494 (1991) · Zbl 0757.35033
[26] Ikehata, R.; Suzuki, T., Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26, 475-491 (1996) · Zbl 0873.35010
[27] Ishige, K.; Kawakami, T.; Okabe, S., Existence of solutions for a higher-order semilinear parabolic equation with singular initial data, Ann. Inst. Henri Poincare C Anal. Non Linéaire, 37, 1185-1209 (2020) · Zbl 1454.35222
[28] Issoglio, E., A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis, J. Differential Equations, 267, 5976-6003 (2019) · Zbl 1447.35193
[29] Issoglio, E.; Jing, S., Forward-backward SDEs with distributional coefficients, Stochastic Process. Appl., 130, 47-78 (2020) · Zbl 1443.60062
[30] Karoui, N. E.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7, 1-77 (1997) · Zbl 0884.90035
[31] Kim, I.; Kim, K.-H.; Lim, S., An \(L_q ( L_p )\)-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306, 123-176 (2017) · Zbl 1361.35196
[32] Kim, D.; Krylov, N. V., Parabolic equations with measurable coefficients, Potential Anal., 26, 4, 345-361 (2007) · Zbl 1124.35024
[33] Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab, 28, 2, 558-602 (2000) · Zbl 1044.60045
[34] Krylov, N. V., An analytic approach to SPDEs, (Carmona, R.; Rozovsky, B., Stochastic Partial Differential Equations: Six Perspectives (1999), American Mathematical Society) · Zbl 0933.60073
[35] Krylov, N.; Röckner, M., Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields, 131, 154-196 (2005) · Zbl 1072.60050
[36] Kupper, M.; Luo, P.; Tangpi, L., Multidimensional Markovian FBSDEs with superquadratic growth, Stoch. Proc. Appl. Appear. (2018)
[37] Ladyzhenskaya, O.; Solonnikov, V.; Ural’tseva, N., (Linear and Quasi-Linear Equations of Parabolic Type. Linear and Quasi-Linear Equations of Parabolic Type, Translation of Mathematical Monographs (1968), American Mathematical Society) · Zbl 0174.15403
[38] Laurière, M.; Tangpi, L., Convergence of large population games to mean field games with interaction through the controls, Preprint (2020)
[39] Leoni, G.; Morini, M., Necessary and sufficient conditions for the chain rule in \(W_{l o c}^{1 , 1} ( \mathbb{R}^N ; \mathbb{R}^d )\) and \(B V_{l o c} ( \mathbb{R}^N ; \mathbb{R}^d )\), J. Eur. Math. Soc., 9, 219-252 (2005) · Zbl 1135.26011
[40] Lieberman, G., Second Order Parabolic Differential Equations (1996), World Scientific Publishing Co. Inc. · Zbl 0884.35001
[41] Luo, P.; Tangpi, L., Solvability of coupled FBSDEs with diagonally quadratic generators, Stoch. Dyn., 17, 6, Article 1750043 pp. (2017) · Zbl 1372.60084
[42] 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
[43] Ma, J.; Zhang, J., On weak solutions of forward-backward SDEs, Probab. Theory Related Fields, 151, 475-507 (2011) · Zbl 1235.60067
[44] Ma, J.; Zhang, J.; Zheng, Z., Weak solutions for forward-backward SDEs - a martingale problem approach, Ann. Probab., 36, 6, 2092-2125 (2008) · Zbl 1154.60045
[45] Matoussi, A.; Possamaï, D.; Sabbagh, Z., Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs, Probab. Theory Related Fields, 174, 177-233 (2019) · Zbl 1447.60116
[46] Menoukeu-Pamen, O.; Meyer-Brandis, T.; Nilssen, T.; Proske, F.; Zhang, T., A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s, Math. Ann., 357, 2, 761-799 (2013) · Zbl 1282.60057
[47] Menoukeu-Pamen, O.; Tangpi, L., Strong solutions of some one-dimensional sde’s with random unbounded drifts, SIAM J. Math. Anal., 51, 5, 4105-4141 (2019) · Zbl 1423.60063
[48] Mezerdi, B., Necessary conditions for optimality for a diffusion with a non-smooth drift, Stochastics, 24, 305-326 (1988) · Zbl 0651.93077
[49] Mikami, T.; Thieullen, M., Duality theorem for the stochastic optimal control problem, Stochastic Process. Appl., 116, 12, 1815-1835 (2006) · Zbl 1118.93056
[50] Mohammed, S. E.A.; Nilssen, T.; Proske, F., Sobolev differentiable stochastic flows for sde’s with singular coeffcients: Applications to the stochastic transport equation, Ann. Probab., 43, 3, 1535-1576 (2015) · Zbl 1333.60127
[51] Nualart, D., The Malliavin Calculus and Related Topics (2006), Springer Berlin · Zbl 1099.60003
[52] Palagachev, D. K., Quasilinear elliptic equations with VMO coefficients, Trans. A.M.S., 2481-2493 (1995) · Zbl 0833.35048
[53] Pardoux, E.; Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, (Rozuvskii, B.; Sowers, R., Stochastic Partial Differential Equations and their Applications. Vol. 176 (1992), Springer: Springer Berlin, New York), 200-217 · Zbl 0766.60079
[54] Pardoux, E.; Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114, 2, 123-150 (1999) · Zbl 0943.60057
[55] Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979 (1990) · Zbl 0712.93067
[56] 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
[57] Sattingeri, D. H., On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30, 148-1721 (1968) · Zbl 0159.39102
[58] Shang, H.; Li, F., On the Cauchy problem for the evolution \(p\)-Laplacian equations with gradient term and source and measures as initial data, Nonlinear Anal. TMA, 72, 3396-3411 (2010) · Zbl 1184.35186
[59] Takahashi, J., Solvability of a semilinear parabolic equation with measures as initial data, (Gazzola, F.; Ishige, K.; Nitsch, C.; Salani, P., Geometric Properties for Parabolic and Elliptic PDE’s (2015), Springer), 257-276 · Zbl 1349.35208
[60] Üstünel, A. S.; Zakai, M., (Transformation of Measure on Wiener Space. Transformation of Measure on Wiener Space, Springer Monographs in Mathematics (2000), Springer) · Zbl 0938.46045
[61] Weissler, F. B., Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32, 277-296 (1979) · Zbl 0419.47031
[62] Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034
[63] Zhang, X., \( L^p\)-theory of semi-linear SPDEs on general measure spaces and applications, J. Funct. Anal., 239, 44-75 (2006) · Zbl 1128.60307
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