January  2017, 2: 5 doi: 10.1186/s41546-017-0016-5

Backward stochastic differential equations with Young drift

1 Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany;

2 Department of Mathematics, University of Southern California, Los Angeles, California, USA

Received  October 28, 2016 Revised  March 26, 2017

Fund Project: This research was partially supported by the DAAD P.R.I.M.E. program and NSF grant DMS 1413717.

We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite q-variation with q ∈[1, 2). In contrast to previous work, we apply a direct fixpoint argument and do not rely on any type of flow decomposition. The resulting object is an effective tool to study semilinear rough partial differential equations via a Feynman-Kac type representation.
Citation: Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5
References:
[1]

Bismut, J-M:Conjugate Convex Functions in Optimal Stochastic Control. J.Math. Anal. Apl 44, 384-404(1973),

[2]

Caruana, M, Friz, PK, Oberhauser, H:A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. Henri Poincare (C) Non Linear Anal 28(1), 27-46 (2011),

[3]

Dan, C, et al:Robust filtering:correlated noise and multidimensional observation. Ann. Appl. Probab 23.5, 2139-2160 (2013),

[4]

Diehl, J, Friz, P:Backward stochastic differential equations with rough drivers. Ann. Probab. 40.4, 1715-1758 (2012),

[5]

Diehl, J, Oberhauser, H, Riedel, S:A Levy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations. Stochastic Process. Appl 125.1, 161-181 (2015),

[6]

Diehl, J, Friz, PK, Stannat, W:Stochastic partial differential equations:a rough path view (2014). arXiv preprint arXiv:1412.6557,

[7]

El Karoui, N, Peng, S, Quenez, M-C:Backward stochastic differential equations in finance. Math. Finance 7.1, 1-71 (1997),

[8]

Friz, PK, Victoir, NB:Multidimensional stochastic processes as rough paths:theory and applications, vol. 120. Cambridge University Press (2010),

[9]

Friz, P, Hairer, M:A course on rough paths. Springer Heidelberg (2014),

[10]

Guerra, J, Nualart, D:Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stochastic Anal. Appl 26.5, 1053-1075 (2008),

[11]

Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett 14.1, 55-61 (1990),

[12]

Pardoux, E, Peng, S:Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial differential equations and their applications, pp. 200-217. Springer Berlin Heidelberg (1992),

[13]

Stroock, DW, Karmakar, S:Lectures on topics in stochastic differential equations. Tata Institute of Fundamental Research, Bombay (1982),

[14]

Young, LC:An inequality of the Hölder type, connected with Stieltjes integration. Acta Math 67(1), 251-282 (1936),

show all references

References:
[1]

Bismut, J-M:Conjugate Convex Functions in Optimal Stochastic Control. J.Math. Anal. Apl 44, 384-404(1973),

[2]

Caruana, M, Friz, PK, Oberhauser, H:A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. Henri Poincare (C) Non Linear Anal 28(1), 27-46 (2011),

[3]

Dan, C, et al:Robust filtering:correlated noise and multidimensional observation. Ann. Appl. Probab 23.5, 2139-2160 (2013),

[4]

Diehl, J, Friz, P:Backward stochastic differential equations with rough drivers. Ann. Probab. 40.4, 1715-1758 (2012),

[5]

Diehl, J, Oberhauser, H, Riedel, S:A Levy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations. Stochastic Process. Appl 125.1, 161-181 (2015),

[6]

Diehl, J, Friz, PK, Stannat, W:Stochastic partial differential equations:a rough path view (2014). arXiv preprint arXiv:1412.6557,

[7]

El Karoui, N, Peng, S, Quenez, M-C:Backward stochastic differential equations in finance. Math. Finance 7.1, 1-71 (1997),

[8]

Friz, PK, Victoir, NB:Multidimensional stochastic processes as rough paths:theory and applications, vol. 120. Cambridge University Press (2010),

[9]

Friz, P, Hairer, M:A course on rough paths. Springer Heidelberg (2014),

[10]

Guerra, J, Nualart, D:Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stochastic Anal. Appl 26.5, 1053-1075 (2008),

[11]

Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett 14.1, 55-61 (1990),

[12]

Pardoux, E, Peng, S:Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial differential equations and their applications, pp. 200-217. Springer Berlin Heidelberg (1992),

[13]

Stroock, DW, Karmakar, S:Lectures on topics in stochastic differential equations. Tata Institute of Fundamental Research, Bombay (1982),

[14]

Young, LC:An inequality of the Hölder type, connected with Stieltjes integration. Acta Math 67(1), 251-282 (1936),

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