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

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P. , Backward stochastic differential equations with continuous coefficient, Statistics and Probability Letters, 32(4), 1997, 425430. , Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28(2), 2000, 558-602. [22] Duffie, D. , Stochastic differential utility, Econometrica, 60, 1992, 353-394. , Hamadene, S. , BSDEs and applications, Indifference pricing: theory and applications, Priceton University Press, 2009, 267-320. [24] El Karoui, N.

0 Therefore, Yt is a continuous semimartingale. For t ∈ [t0 , T ], ´T Ys+ [f 1 (s, Ys1 , L(M 1 )s − f 2 (s, Ys2 , L(M 2 )s )]ds ⟨ ⟩ ´T ´T +(ξ + )2 − 2 t Ys+ dMs − t 1(Ys >0) d M . 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.

BSDE with quadratic growth and unbounded terminal value, Probability Theory and Related Fields, 136(4), 2006, 604-618. S. Probability Theory. An Advanced Course. New York, SpringerVerlag, 1995. , The Theory of Stochastic Processes. London, Chapman & Hall, 1980. [28] Thang, S. , 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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