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Potential kernels associated with a filtration and forward-backward SDEs. (English) Zbl 0929.60053

The author proves that the following two linear operators on \(L^{2}(\Omega)\), \[ V_{1}y=\int_{0}^{\infty}E(Z_{t}y\mid{\mathcal F}_{t}) dA_{t}, \qquad V_{2}y=\int_{0}^{\infty}E(Z_{t}y\mid{\mathcal F}_{t-}) dA_{t} \] are Markov potential kernels closing a sub-Markov resolvent. Here \((\Omega,{\mathcal F}, P; \{{\mathcal F}\}_{t\in [0,\infty]})\) is a filtered probability space satisfying the usual hypotheses, with \({\mathcal F}={\mathcal F}_{\infty}\), and where \(Z\) is a bounded positive stochastic process and \(A\) is a bounded adapted increasing process. The result extends those of N. Bouleau [in: Théorie du potentiel, Semin. Paris, No. 8. Lect. Notes Math. 1235, 39-53 (1987; Zbl 0618.60034)] and S. Martínez, G. Michon and J. San Martín [SIAM J. Matrix Anal. Appl. 15, No. 1, 98-106 (1994; Zbl 0798.15030)]. The method is based on the study of the forward-backward stochastic differential equations [see Y. Hu and S. Peng, Probab. Theory Relat. Fields 103, No. 2, 273-283 (1995; Zbl 0831.60065)], which can also be used to treat some nonlinear operators.

MSC:

60J45 Probabilistic potential theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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