A New Approach to BSDE (Backward Stochastic Differential by Lixing, Jin

By Lixing, Jin

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1990), Adapted solution of a backward stochastic differential equation, System and Control Letters, 14(1), 55-61. M. G. Martingale representations for diffusion processes and backward stochastic differential equations. , Lyons, T. M. (2009), Backward stochastic dynamics on a filtered probability space. G. C. (1997), Backward stochastic differential equations in finance. [5] Tang, S. and Li, X,. Maximum principle for optimal control of distributed parameter stochastic systems with random jumps, Differential equations, dynamical systems, and control science, 152, 1994, 867-890.

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M [11]. Our contribution in this aspect is the the following lemma which fills in the gap in their proof. If L satisfies local, homogeneous and K − Lipschitz conditions, then for any 34 nonnegative predictable process Y ˆ t1 KE[ ⟨ Ys d M − M 1 t2 2 ˆ ⟩ s ]≥E t1 Ys2 | L[t2 ,t1 ] (M 1 )s − L[t2 ,t1 ] (M 2 )s |2 ds, t2 where K − Lipschitz condition property means there is a positive constant K such that for any 0 ≤ t2 ≤ t1 ≤ T and any M1 , M2 ∈ M2 ([t2 , t1 ], Rn ), ⟨ ⟩ ⟨ ⟩ KE[ M 1 − M 2 t1 − M 1 − M 2 t2 ] ≥ ˆ t1 | L(M 1 )s − L(M 2 )s |2 ds.

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