Differential Equations and Their Applications Part 7 docx

... 3.1. Let (2.3) hold and the following hold: (3.3) A 9 L~(O,T;L~ 176 BE L~/Oj:~, T', LO~/R,~.~ , ~n • d)), a 9 Lcff(O,T;L~ 176 b 9 L~ 176 176 c 9 L~(0, T; n~ 176 Then, the adapted ... 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 replace...

Ngày tải lên: 10/08/2014, 20:20

... differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS 176 , 200-2 17, Springer 1992. [3] Backward doubly stochastic differential equations and ... Stochastics, 3 (1 979 ), 1 27- 1 67. [2] Backward stochastic differential equations and applications, Proc. ICM, 1994. Pardoux, E., and Peng, S., [1] Adapted solu...

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... as g and satisfying the following: (1 .7) ~/(x, y) > O, V(x, y) 9 ~ x ~m, I f(x,y) O, if and only if y = g(x). w The value function 61 This proves (3.1). Next, let s E [0, T] and ... programming method and the HJB equation 57 Next, let (x) ~ ~/1 + [xl 2. It is not hard to check that both D (x) and D 2 (x) are uniformly bounded, thus applying It6's formula to (Y...

Ngày tải lên: 10/08/2014, 20:20

... "~ 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, by Lemma 2.1 there ... 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 <...

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 ... lemma. w Well-posedness of linear BSPDEs 1 07 We note that (H)m implies that the partial derivatives of A and B in x up to order (m + 1), and those of a, b and c up to order m are...

Ngày tải lên: 10/08/2014, 20:20

... with B(F; [0, T]) ~ r Then, for any 7 E 7- /[0, T] and x E ~'~, (1.5)r,.~,z admits at most one adapted solution. Moreover, for any 7, ~ E 7- /[0, T] and x, ~ E lR '~, the stability ... case that F1 -= (bl,cr1,hl ,71 ) and F2 (b2,cr2,h2,g2) are linked by a direct bridge. We now assume this. Next, for any 7 - w Method of continuation 1 47 (2.16)-(2. 17) and note (...

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... space M. The mean- ing of (3. 17) ' and (3.18)' are similar. Here, we should point out that (3. 17) implies m _> n and (3. 17) ' implies m ~ n. Hence, (M) and (M)' overlaps only ... not hard to see that under (3. 17) -(3.18), (3.21) implies (1.8) and (3.22) implies (1 .7) and (1.9)' (Note (1.8) implies (1.8)'). We see that the left hand side of...

Ngày tải lên: 10/08/2014, 20:20