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**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.

\[ Y_t=\xi+\int_t^Tg(s,Y_s,Z_s) \, ds-\int_t^TZ_s \,dB_s+(A_T-A_t), \quad t\in [0,T], \]

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.

Reviewer: Rainer Buckdahn (Brest)

### MSC:

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

### Citations:

Zbl 0495.60005
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\textit{S. Fan} et al., J. Appl. Math. Comput. 24, No. 1--2, 427--435 (2007; Zbl 1128.60045)

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### References:

[1] | F..Pardoux and S.G. Peng,Adapted solution of a backward stochastic differential equation, Systems Control Letters 14(1990), 55–61. · Zbl 0692.93064 |

[2] | S.G.Peng,Nonlinear expectation, nonlinear evaluation and risk measures, Lectures in CIME-EMS school, Bressanone, 2003. |

[3] | X.R. Mao,Adapted solutions of backward stochastic differential equations with no-Lipschitz cofficients, Stochastic Process and Their Applications 58(1995), 281–292. · Zbl 0835.60049 |

[4] | Zh.G. Cao and J.A. Yan,A comparison theorem for solutions of backward stochastic differential equations, Advances in Mathematics Vol. 28(1999), No. 4, 304–308. · Zbl 1054.60505 |

[5] | L. Jiang,On Jensen’s inequality of bivariate function for g-expectation, Journal of Shandong University Vol. 38(2003), No. 5, 13–22. (In Chinese). |

[6] | L. Jiang and Z.J. Chen,On Jensen’s inequality for g-expectation, China.Ann.Math. Vol. 25B(2004), No. 3, 401–412. · Zbl 1062.60057 |

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