Backward stochastic differential equations with rough drivers

Abstract

Backward stochastic differential equations (BSDEs) in the sense of Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200-217, 1992] provide a non-Markovian extension to certain classes of nonlinear partial differential equations; the non-linearity is expressed in the so-called driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons [Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14, no. 2, 215-310, 1998] and so obtain a notion of "BSDE with rough driver". Existence, uniqueness and a version of Lyons' limit theorem in this context are established. Our main tool, aside from rough path analysis, is the stability theory for quadratic BSDEs due to Kobylanski [Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab., 28(2):558-602, 2000].

A priori estimates for rough PDEs with application to rough conservation laws

“…• Flow transformations applied to viscosity formulation of fully non-linear RPDEs (including Backward rough differential equations) have been studied in a series of work by Friz and coauthors: Diehl and Friz [19], Friz and Oberhauser [25], Caruana and Friz [7], Diehl, Friz and Oberhauser [20], Caruana, Friz and Oberhauser [8] and finally Friz, Gassiat, Lions and Souganidis [24]. • Rough formulations of evolution heat equation with multiplicative noise (with varying degree of success) have been considered by Gubinelli and Tindel [36], Deya, Gubinelli and Tindel [17], Teichmann [63], Hu and Nualart [47] and Garrido-Atienza, Lu and Schmalfuss [29].…”

Numerical Schemes for Rough Parabolic Equations

“…As reported in [8, Subsection 2.6], the situation where both γ < 1/4 and x generates a k-rough path with k ≥ 4, is likely to raise additional issues as far as the space parameter γ ′ is concerned (when x is a fBm, these hypotheses cannot cover the case where H < 1/4 anyway, see [3]). For the time being, it seems that the only approach able to deal with the condition γ < 1/4 in (1) is the BSDE/viscosity-strategy initiated in [10]. At this point, we cannot guarantee that the latter (theoretical) treatment remains consistent with our space-time discretization methods, though.…”

Backward stochastic differential equations with Young drift

“…Interestingly, the flow decomposition used in (Diehl and Friz 2012) leads to a transformed BSDE that is quadratic in Z. Hence, also there the terminal condition needs to be bounded.…”

“…In (Diehl et al 2014), these results are extended to show that the limit actually solves an integral equation. Semilinear equations like (8) are investigated in (Diehl and Friz 2012). Again, the convergence of solutions corresponding to smooth approximations of η is shown via a transformation of the rough PDE.…”

“…Such equations have previously been studied in (Diehl and Friz 2012). In that work η is even allowed to be a rough path, i.e., every q ≥ 1 is feasible.…”