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

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...

Ngày tải lên: 10/08/2014, 20:20

... 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