Abstract
In this paper, we prove a stochastic Fubini theorem by solving a special backward stochastic differential equation (BSDE, for short) which is different from the existing techniques. As an application, we obtain the well-posedness of a class of BSDEs with the Itô integral in drift term under a subtle Lipschitz condition.
Highlights
Introduction and the main result GivenT >, let (, F, Ft, P; t ≥ ) be a complete filtration space and F = {Ft; t ≥ } be a filtration satisfying the usual conditions which are generated by the following two mutually independent stochastic processes:(i) a d-dimensional Brownian motion {B(t); t ≥ }; (ii) a Poisson random measure N on R+ × E, where E = Rl – { } with the Borel σ -fieldB(E). λ is the intensity (Lévy measure) of N with the property that ∧ |z| λ(dz) < ∞E and μ is the compensator of N with μ(dt, dz) = dtλ(dz)
{N((, t] × A) = (N – μ)((, t] × A), Ft; t ≥ } is a compensated Poisson process which is a càdlàg martingale for all A ∈ B(E) satisfying λ(A) < ∞
4 Conclusions In this paper, we consider a stochastic Fubini theorem with two time variables
Summary
The following stochastic Fubini theorem holds: t t φ(x, s) dM(s) dμ(x) = Under suitable conditions, we obtain the well-posedness of the following two BSDEs: Ts yT – y(t) =
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