BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

By P. Briand, Fulvia Confortola. 2007

118 citations recorded by the source corpus.

This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z. We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/242f722986b880d2a57e259ac7c82df281c2f6d7

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In Computational Theory and Mathematics

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Citations in the corpus

Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition (2011, unknown): “Then, according to [4, 1], X and (Y, Z) are differentiable with respect to x, we have ∇Yt = ∇g(XT )∇XT − ∫ T t ∇ZsdWs + ∫ T t ∇xf(s,Xs, Ys, Zs)∇Xs +∇yf(s,Xs, Ys, Zs)∇Ys +∇zf(s,Xs, Ys, Zs)∇Zsds, and Zt = ∇Yt(∇Xt)σ(t,Xt) a.”
Numerical simulation of BSDEs with drivers of quadratic growth (2010, methodology): “For their original version and their proof we refer to [2, 17] and [14].”
Numerical simulation of quadratic BSDEs (2013, unknown): “Moreover, the solution is known to have the following properties; see, for example, [1, 6, 34].”
A simple constructive approach to quadratic BSDEs with or without delay (2013, methodology): “The main argumentation of this proof follows the idea of [3], see p.”