Differential Equations and Their Applications Part 5 pptx

... discussion. Lemma 5. 1. Let (H1) and (H4) hold. Then there exists a constant C > O, depending only on L and T, such that for all 5, r >_ O, and (s, x, y) E [0, T] • IR 2, it holds that (5. 2) ~Js'~(s,x,y) ... and their derivatives. Therefore using assumptions (H1) and (H4), and noting that supt IZ(k)(t)l <_ sup Is (k)} < ~, we see that, for fixed 5,...

Ngày tải lên: 10/08/2014, 20:20

... [0,T- 6]. (3 .52 ) result in /]( ) (3 .58 ) 5 K ZlC(s)l 2 -5 < K EIf(s)J 2 -5 By (3.19), we obtain (3 .59 ) 43 Consequently, (3 .56 ) and (/: ) 1/2 EIZ(r) 12dr ~ 1/2ds -5 _sElff(r)lUdr) ... -5 _sElff(r)lUdr) ds. T fTT T _ E[((r)l~dr ds -5 -5 1 T T s Thus, it follows that (3.60) f El((s)12ds < K5 El((s)12ds, .IT -5 5 with K > 0 being an a...

Ngày tải lên: 10/08/2014, 20:20

Ngày tải lên: 10/08/2014, 20:20

... 3 -54 0- 659 60-9 Mathematics Subject Classification (1991): Primary: 60H10, 15, 20, 30; 93E03; Secondary: 35K 15, 20, 45, 65; 65M06, 12, 15, 25; 65U 05; 90A09, 10, 12, 16 ISSN 00 75- 8434 ISBN 3 -54 0- 659 60-9 ... Stochastic Differential Equations and Their Applications Springer Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier,...

Ngày tải lên: 10/08/2014, 20:20

... 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...

Ngày tải lên: 10/08/2014, 20:20

... V 5& apos;~(s,x,y) >VS'~(s,x,y) >0, V5,6>0, k&apos ;52 '~is, x,y ) > l /5& quot;~(s,x,y), V52 >__ 51 > O, ~ >_ 0, V~2'~(s,x,y) >_ VS~'~(s,x,y), V52 ... (4. 25) such that (5. 8) [I - P(t)C~] -1 is bounded for t e [0, T], 1 < i < d, and the FBSDE (5. 1) admits a unique adapted solution (X, Y, Z) e ~4[0, T] which can be represented by...

Ngày tải lên: 10/08/2014, 20:20

... 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 ~" and ... 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...

Ngày tải lên: 10/08/2014, 20:20

... and the induced norm of H by (-, ")0 and [-10, respectively. The duality paring between V and V' is denoted by (.,.)0, and the norms of V and V' are denoted by 1]" I[ and ... 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...

Ngày tải lên: 10/08/2014, 20:20

... the case that and ~ are not necessarily smooth enough, we may make approximation. [] Proposition 6 .5. Let A, B, -5, b and -~ be independent of x. Let f and be convex in x and nonnegative. ... (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...

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... AT g2( 52 + #2) T (5. 24) + -~- + + 3) - g2~ A 2 1 g# 392# 2 + 25 52 + 45 7 - g2-~. Thus, we need (note (5. 23)) F(T)~= 1+~) e 2~T + ~ ~(1 + 52 (5. 25) 1 g# 3g2# 2 + + ~ 25 52 453 > ... + g2). (5. 27) 0 < F(0) = Hence, in what follows, we take 3 (5. 28) ~ - 45( 1 + g2) Then, (5. 26) holds. Next, we claim that under (5. 15) and (5. 28), the fol- lowing ho...

Ngày tải lên: 10/08/2014, 20:20