Differential Equations and Their Applications Part 14 docx

... differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS 176, 200-217, Springer 1992. [3] Backward doubly stochastic differential equations and ... 27 (1993), 125 -144 . [9] Backward SDE and related g-expectation, preprint. [10] Adapted solution of backward stochastic differential equations and re- lated partial dif...

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... if and only if {A,B,~d,b,c} satisfies (H),~, where ~ and b are given by (1.4). Thus, we have the exact statements as Theorems 2.1, 2.2 and 2.3 for BSPDE (1.5) with a and b replaced by ~ and ... have the Gelfand triple V r H = H' ~ + V'. We denote the inner product and the induced norm of H by (-, ")0 and [-10, respectively. The duality paring between V a...

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... the stochastic differential Jin Ma Jiongmin Yong Forward-Backward Stochastic Differential Equations and Their Applications Springer Lecture Notes in Mathematics Editors: A. Dold, Heidelberg ... into the stock, and puts viii Preface strong degeneracy in the sense of stochastic partial differential equations. Such BSPDEs can be used to generalize the Feynman-Kac form...

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... for ordinary differential equations do not necessarily admit solutions. On the other hand, an FBSDE can be viewed as a two-point boundary value problem for stochastic differential equations, ... suitable sizes, b, a, b and ~ are stochastic processes and g is a random variable. We are looking for {gvt}t>0-adapted processes X(.), Y(-) and Z(-), valued in ]R n, ]R m and IR...

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... all g C H if and only if (3.17) and (3.19) hold. In this case, the adapted solution to (2.12) is unique (for any given g E H). Proof. Theorems 3.2 and 3.3 tell us that (3.17) and (3.19) are ... that .4 and -41 are time-dependent matrix-valued functions and and ~ are {hvt}t>o-adapted processes. Further, under (4.11), the following SDE admits a unique strong solution: (4 ....

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... (i) ~,~(s, ~, y) and V ~,~(s, x, y) are continuous in (x, y) 9 ~" • ~m, uniformly in s C [0, T] and 6, E >_ O; For fixed 6 > 0 and e > O, ~5,~ (s, x, y) and Va'~(s,x,y) ... similar to that of Proposition 3.1 and the proof of (ii) and (iii) are by now standard, which we omit here for simplicity of presentation (see Yong-Zhou [1] and Fleming-Soner [1], for...

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... of the coefficients of the system (2.1) (i.e., b, h, a, and 3(z) - z) and their derivatives. Therefore using assumptions (H1) and (H4), and noting that supt IZ(k)(t)l <_ sup Is (k)} < ... "~ x R m• Now, we see that (2.6) and (2.8) follow from (A1) and (A2); (2.7) follows from (A1), (2.2) and (2.11); and (2.9)-(2.10) follow from (A1) and (2.2). Therefore,...

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... with random coefficients, i.e., b, a, h and g are possibly depending on w E ~ explicitly, then it will lead to the study of general degenerate nonlinear backward partial differential equations ... in x and C 1 in t with H51der continuous v,~s and vt of exponent a and a12, respectively. Moreover, we have (3.54) IwR(t,x)l < M, (t,x) e [0,~) • BR, and for any Xo C ~&...

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... (2.11) and working on (v,p) for the transformed equations. [] Our main comparison result is the following. Theorem 6.2. Let (1.6), (2.2) and (H),~ hold for (6.2) and (6.3). Let (f,g) and (f,~) ... (2.2) and (H),~ (for some m _> 1) hold for (6.2) and (6.3). Then by Theorem 2.3, for any pairs (f, g) and (f, ~) satisfying (2 .14) , there exist unique adapted weak solut...

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... -25, Now, we take a(t) and c(t) as in (3.7) and we require (3.13) d(t)+(3L + 2L2)lc(t)l = -Co(Co - 3L - 2L2)e C~ -Co(Co - 3L - 2L 2) ~ -5, Vt E [0,T], and (3 .14) c(t) = -Coe C~ <__ ... (5.30) and (5.32), we must have (5.29). Hence, we obtain (5.25). This shows that a strong bridge ~(t) has been constructed with K, 5 and s being given by (5.19) and (5.28), respe...

Ngày tải lên: 10/08/2014, 20:20