×

Backward stochastic differential equations in a Lie group. (English) Zbl 0980.60085

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXXV. Berlin: Springer. Lect. Notes Math. 1755, 241-259 (2001).
Since the pioneer work of E. Pardoux and S. G. Peng [Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)] on nonlinear backward stochastic differential equations (BSDE) with values in \(R^n\), a lot of papers have been devoted to these equations and their various applications. The present paper is the first one to study backward stochastic differential equations with a solution process taking its values in a finite-dimensional Lie group. In a first step the authors are interested in getting group-valued martingales with prescribed terminal value: existence and uniqueness are proven for the three-dimensional Heisenberg group, for \((\Gamma)\)-groups and for nilpotent Lie groups. In a second step BSDE with general drift are studied. The main tools are the stochastic exponential and logarithm of Lie groups. First defined by M. Hakim-Dowek and D. Lépingle [in: Probabilités XX. Lect. Notes Math. 1204, 352-374 (1986; Zbl 0609.60009)], they allow exchanges between Lie group-valued semimartingales and \(R^n\)-valued semimartingales.
For the entire collection see [Zbl 0960.00020].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
22E25 Nilpotent and solvable Lie groups
PDFBibTeX XMLCite
Full Text: Numdam EuDML