Representation theorems for generators of backward stochastic differential equations and their applications. (English) Zbl 1078.60043

Jiang, Long

Stochastic Processes Appl. 115, No. 12, 1883-1903 (2005).

Summary: We prove that the generator \(g\) of a backward stochastic differential equation (BSDE) can be represented by the solutions of the corresponding BSDEs at point \((t,y,z)\) if and only if \(t\) is a conditional Lebesgue point of generator \(g\) with parameters \((y,z)\). By this conclusion, we prove that, if \(g\) is a Lebesgue generator and \(g\) is independent of \(y\), then Jensen’s inequality for \(g\)-expectation holds if and only if \(g\) is super homogeneous; we also obtain a converse comparison theorem for deterministic generators of BSDEs.

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

Conditional Lebesgue point; Lebesgue generator; \(g\)-Expectation; Converse comparison theorem

References:

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