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

By Lixing, Jin

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Yt+ = 2 t s 1 This proof is analogue to Shi’s proof from page 23 to 25 in [11]. 30 t Rearranging the equation above, we have ˆ Yt+ ˆ + t T ⟨ ⟩ 1(Ys >0) d M s T Ys+ [f 1 (s, Ys1 , L(M 1 )s − f 2 (s, Ys2 , L(M 2 )s )]ds t ˆ T + 2 + (ξ ) − 2 Ys+ dMs . 4) t Next we show that ´T t Ys+ dMs is a martingale. By using Burkholder-Davis-Gundy inequality, we have ˆ E[ sup | t0 ≤t≤T t Ys+ dMs [ˆ |] ≤ CE ( 0 [ ≤ CE T | Ys+ |2 t0 ) 1 2 s T Yˆs+ t0 ≤s≤T ] ⟩ d M (ˆ sup [ ⟨ ] ⟨ ⟩ ) 21 d M 0 ⟩ ] s ⟨ ≤ CE sup Yˆs+ M T t ≤s≤T { 0[ ] [⟨ ⟩ ]} 2 C + ≤ E sup Yˆs +E M 2 s t0 ≤s≤T { [ ] [ ] [⟨ ⟩ ]} C 1 2 2 2 2E sup Ys + 2E sup Ys +E M , ≤ 2 T t0 ≤s≤T t0 ≤s≤T 1 2 < ∞ where C is a positive constant.

It could be applied to the theory of forward-backward stochastic differential equa- 35 tions and Backward stochastic differential equation. In finance area, it has intimate connection with European option pricing problems and stochastic control problems. We believe that with the further studies in this approach, more properties will be found. 36 Bibliography [1] Pardoux, E . G. (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.

Then from the above inequality, ˆ T 1 g(t) ≤ + ) g(T − s)ds K T −t ˆ t 1 2 = (2KC2 + ) g(s)ds. K 0 (2KC22 Gronwall’s inequality ensures that g(t) = 0, t0 ≤ t ≤ T . Hence, Yt+ = 0, t0 ≤ t ≤ T , which is equivalent to say that Yt1 ≤ Yt2 . That completes the proof. 33 Chapter 4 CONCLUSIONS We are interested in solving the following BSDE dYt = −f (t, Yt , L (M )t ) dt + dMt , with terminal condition YT = ξ on a filtered probability space (Ω, F, Ft , P ), where L is a deterministic mapping which sends M to an adapted process L (M ), where M is a martingale to be determined.

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