Continuous dependence properties on solutions of backward stochastic differential equation. (English) Zbl 1128.60045

Let \(B\) be a Brownian motion, \(A\) a square integrable \(\mathbb{F}^B=({\mathcal F}_t^B)\)-adapted càdlàg process and \(g(=g(\omega,t,y,z)): \Omega\times[0,T]\times\mathbb R\times\mathbb R^d\rightarrow\mathbb R\) a progressively measurable driver that is Lipschitz in \(z\) and such that \(| g(\omega,t,y,z)-g(\omega,t,y',z)|^2\leq \rho(| y-y'|^2)\) for some concave function \(\rho\) with \(\rho(0)=0,\, \rho(u)>0\) for \(u>0\) and \(\int_{0^+}\rho(u)^{-1}\,du=+\infty\) (this condition is well known from the study of stochastic differential equations; see, e.g., [N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Amsterdam-Oxford-New York: North-Holland Publishing Company (1981; Zbl 0495.60005)]. In their paper the authors consider the backward stochastic differential equation (BSDE)
\[ Y_t=\xi+\int_t^Tg(s,Y_s,Z_s) \, ds-\int_t^TZ_s \,dB_s+(A_T-A_t), \quad t\in [0,T], \]
and they study the continuous dependence in \(L^2\) but also the \(P\)-almost sure one of its solution \((Y,Z)\) on the terminal condition \(\xi\in L^2({\mathcal F}_T^B)\). While the \(L^2\)-continuity of \((Y,Z)\) in \(\xi\) is the result of straight-forward BSDE-estimates, the \(P\)-almost sure one is studied for increasing (resp., decreasing) sequences of terminal values and it is easily deduced from the comparison theorem for BSDEs, proved by Cao and Yan (1999) in the framework described above.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)


Zbl 0495.60005
Full Text: DOI


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