Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

By Patrick Cheridito, H. M. Soner, Nizar Touzi, Nicolas Victoir. 2005

283 citations recorded by the source corpus.

For a d‐dimensional diffusion of the form dXt = μ(Xt)dt + σ(Xt)dWt and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) $$dY_{t} = f(t,X_{t}, Y_{t}, Z_{t}, \Gamma_{t}) dt + Z_t'\circ dX_{t}, \quad t \in [0,T),$$ $$dZ_{t} = A_{t} dt + \Gamma_{t}dX_{t}, \quad t \in [0,T),$$ $$Y_{T} = g(X_{T}).$$ If the associated PDE $$- v_{t}(t,x) + f(t,x,v(t,x), Dv(t,x), D^{2}v(t,x)) = 0,$$ $$(t,x) \in [0,T) \times {\cal R}^{d}, \quad v(T,x) = g(x),$$ has a sufficiently regular solution, then it follows directly from Itô's formula that the processes $$v(t,X_{t}), Dv(t,X_{t}), D^{2}v(t,X_t), {\cal L} Dv(t,X_{t}), \quad t \in [0,T],$$ solve the 2BSDE, where 𝓁 is the Dynkin operator of X without the drift term.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d0168407a337dbceb0d120560f542b667712927b

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Citations in the corpus

ROBUST UTILITY MAXIMIZATION IN NONDOMINATED MODELS WITH 2BSDE: THE UNCERTAIN VOLATILITY MODEL (2015, background): “In fact, the right objects to consider in this case are the so-called second-order BSDEs (2BSDEs for short) which were introduced for the first time by Cheridito et al. (2007).”
A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations (2010, methodology): “It should be emphasized that while our games are related to the processes in [12], our goals and methods are entirely different.”
Optimal stopping under adverse nonlinear expectation and related games (2012, unknown): “Regarding the literature on nonlinear expectations, we refer to [27, 29] and the references therein; for the related second-order backward stochastic differential equations (2BSDE) to [8, 32, 33], and in particular to [21, 22] for the reflected 2BSDE related to our problem; whereas for the uncertain”
Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations (2018, methodology): “, [3]) is to leverage the fundamental relationships (12) between second-order backward stochastic differential equations and fully-nonlinear second-order partial differential equations.”
Stochastic control for a class of nonlinear kernels and applications (2015, background): “More recently, motivated by obtaining probabilistic representations for fully non-linear PDEs, Soner, Touzi and Zhang [76, 78] (see also the earlier works [15] and [77]) introduced a notion of second-order BSDEs (2BSDEs for short), whose solutions could actually be written as a supremum, over a fami”
Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations (2012, background): “Furthermore, the second-order BSDE (2BSDE) is associated to a fully nonlinear PDE, and for such a relation see Cheritdito, Soner, Touzi & Victoir [6] and Soner, Touzi & Zhang [45].”
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, background): “stochastic control problems as well as second-order backward stochastic differential equations [4, 30] are naturally viscosity solutions of the corresponding path-dependent partial differential equation.”
Classical and Variational Differentiability of BSDEs with Quadratic Growth (2007, background): “For a more complete list of references see [ CSTV05 ] or [Kob00].”
Random G-expectations. (2010, background): “The Annals of Applied Probability 2013, Vol. 23, No. 5, 1755–1777 DOI: 10.1214/12-AAP885 © Institute of Mathematical Statistics, 2013 RANDOM G-EXPECTATIONS BY MARCEL NUTZ1”
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks (2019, methodology): “, [11, 21, 37, 49, 68, 105] for probabilistic approximation methods for PDEs using temporal discretizations of the associated second-order BSDEs cf.”
Dual Formulation of Second Order Target Problems (2010, background): “Indeed, [6] develops such a generalization to the second order and also proves a Markovian uniqueness result in an appropriate class.”
Error Criteria for Numerical Solutions of Backward SDEs (2010, background): “Moreover, they can be considered as general adjoint equations for stochastic optimal control problems, see [33] and the references therein, and yield representation formulae for quasi-linear PDEs [29] and even some fully non-linear parabolic PDEs [13].”
Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE (2012, methodology): “4 of [13] yields the convergence of the fourth component.”