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.