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Time-dependent backward stochastic evolution equations. (English) Zbl 1139.60026
Given a time-dependent unbounded linear operator A=(A t ) t0 on a separable Hilbert space H, generating a semigroup (U s,t ) st0 L(H) which is strongly continuous in (s,t), and given a cylindrical Brownian motion W defined over some probability space (Ω,,P), the author of the present paper investigates the H-valued backward stochastic differential equation (BSDE) dY t =-{A t Y t +f(t,Y t ,Z t )}dt+Z t dW t ,t[0,T],Y T =ξL 2 (Ω, T W ,P;H). Supposing that the ( t W )-progressively measurable coefficient f(t,ω,y,z) is Lipschitz in z and such that |f(t,y,z)-f(t,y ' ,z)|c(|y-y ' |),y,y ' H, for some non increasing concave function c:R + * R + * with c(0+)=0 and 0 + 1 c -1 (t)dt=+, the author shows the existence and the uniqueness for this BSDE. Afterwards he proves that the process Y has continuous trajectories. The author’s work concerns a subject which enjoys a great interest since the pioneering paper by Y. Hu and S. Peng [Stochastic Anal. Appl. 9, No. 4, 445–459 (1991; Zbl 0736.60051)], and a lot of generalizations and applications of the equation considered by Hu and Peng (the above BSDE with time-independent unbounded linear operator A) have been studied since then. The present paper employs standard methods for its generalization.
MSC:
60H10Stochastic ordinary differential equations
60H15Stochastic partial differential equations