BSDE associated with Lévy processes and application to PDIE (Q1812265)

The authors deal with \(\mathbb{R}\times \mathbb{R}\times l^{2}\)-valued solutions \((Y,U,Z)\) of the following backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Lévy process: \[ Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s^-},U_{s},Z_{s})ds -\int_{t}^{T}U_{s} dW_{s} -\sum_{i=1}^{\infty} \int_{t}^{T}Z_{s}^{(i)} dH_{s}^{(i)}, \] where \(\xi\) is a square integrable random variable, \(\{W_{t}\), \(t\in[0,T]\}\) is a standard Brownian motion on \(\mathbb{R}\) and \(\{H^{(i)}\}_{i\geq 1}\) denotes Teugel's martingales associated to a Lévy process \(\{L_{t}, t\in [0,T]\}\) of the form \(L_{t}=bt+\ell_{t}\), corresponding to a standard Lévy measure. Existence and uniqueness of solution is proved under globally Lipschitz condition on \(f\). For this a recently appeared martingale representation theorem is used [see \textit{D. Nualart} and \textit{W. Schoutens}, Bernoulli 5, 761-776 (2001; Zbl 0991.60045)]. Moreover, a comparison theorem and a stability result are proved. The case of locally Lipschitz coefficient is also considered, showing existence and uniqueness of solution assuming that the Lipschitz constant is bounded from above by \(\sqrt{\log N}\) on the ball centered at the origin and radius \(N\). These results extend the one's of \textit{K. Bahlali} [C. R. Acad. Sci., Paris, Sér. I 333, 481-486 (2001; Zbl 1010.60052) and Electron. Commun. Probab. 7, 169-180 (2002; Zbl 1008.60075)] to BSDEs driven by a Lévy process. Finally, a Clark-Ocone formula for functionals of a Lévy process is proved, linking BSDEs driven by a Lévy process and a family of partial differential integral equations.