Uniqueness of solutions to quadratic BSDEs with locally Lipschitz generator

Abstract

We study the uniqueness of solutions of backward stochastic differential equations (BSDEs), which generator verifies $ |F(t, y, z)| \leqslant \alpha_{t}+\beta_{t}|y|+\theta_{t}|z|+f(|y|)|z|^{2}$, where $α_t, β_t, θ_t$ are positive processes and the function f is positive, continuous and increasing. The uniqueness of solutions of such BSDEs is derived in two situations, when F is locally Lipschitz and when F is jointly convex. As a byproduct: we show the existence of viscosity solutions to the associated semilinear partial differential equations, which can contain nonlinearity that has quadratic growth in the gradient of the solution.

Keywords

Quadratic backward stochastic differential equations / Minimal and maximal solutions / Uniqueness of solutions / Comparison theorem / Partial differential equation / Viscosity solution

Acknowledgements

The authors would like to thank the referee for their valuable suggestions and comments, which have significantly contributed to the improvement of the paper.

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