Differential Equations and Their Applications Part 6 ppt

... 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 ~&...

Ngày tải lên: 10/08/2014, 20:20

... I)e'A'(T-s).A1 is analytic and hence bounded over [0, T]. From (3 .64 ), we obtain (3 .65 ) Z(s) = -B(s) {X (s) ) \Y(s) ' Then, (X, Y) is the solution of (3 .66 ) { a.e. s E [0, T], a.s. ... to the conditions: C = 0, C = 0 and A1C1 + B1 = 0. Note that A1, B1 and C1 are not necessarily zero. Lemma 3 .6. Let C = 0 and let (3.19) hold. Then (3. 46) ~(K:) = {~...

Ngày tải lên: 10/08/2014, 20:20

... 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)} < ... that, for all 6, e >_ 0, one has (5 .6) {y C IR: V~'~ (0, x, y) < r) C_ Q~ (r). From now on we set r = 1. Recall that by Proposition 3 .6 and Theorem 3.7, for any p >...

Ngày tải lên: 10/08/2014, 20:20

Ngày tải lên: 10/08/2014, 20:20

... 3-540 -65 960 -9 Mathematics Subject Classification (1991): Primary: 60 H10, 15, 20, 30; 93E03; Secondary: 35K15, 20, 45, 65 ; 65 M 06, 12, 15, 25; 65 U05; 90A09, 10, 12, 16 ISSN 0075-8434 ISBN 3-540 -65 960 -9 ... National Science Fundation grant #DMS-93015 16 and the United States Office of Naval Research grant #N00014- 96- 1-0 262 ; and JY would like to acknowledge partia...

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... to (6. 6). (In fact, it can be shown by using Four Step Scheme of Chapter 6 that this is the unique adapted solution to (6. 6)!) Now let g2(x) = x + 1. Then one checks that X2(t) - Z2(t) and ... (5 .6) Xi(0) = x. It is standard that under our conditions, (5 .6) admits a unique (strong) solution Xi C L2(~t;C([O,T];]Rn)). By It6's formula and the Lipschitz continuity of b...

Ngày tải lên: 10/08/2014, 20:20

... Proposition 3 .6. Let (H1) hold. Then (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, ... (s, x, y) and Va'~(s,x,y) are continuous in (s,x,y) 9 [0, T] x Rn • Rm. (ii) For 6 > 0 and ~ >_ O, Va'e(s, x, y) is the unique viscosity solution of (3.25), and...

Ngày tải lên: 10/08/2014, 20:20

... 3.1. Let (2.3) hold and the following hold: (3.3) A 9 L~(O,T;L~1 76 BE L~/Oj:~, T', LO~/R,~.~ , ~n • d)), a 9 Lcff(O,T;L~1 76 b 9 L~1 761 76 c 9 L~(0, T; n~1 76 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 rep...

Ngày tải lên: 10/08/2014, 20:20

... V(t,x) e[O,T]• n, a.s. (6. 26) fo(t,x) > O, go(x) >_ O, Then (6. 16) is satisfied and thus (6. 17) holds. Proof. By Corollary 6. 3 and Lemma 6. 4, ~ is convex and nonnegative. Thus, ... solution of (6. 3) and (6. 18) holds. Consequently, (6. 22) follows from (6. 17). [] Let us make an observation on Theorem 6. 2. Suppose (~,~) is an adapted strong solution of...

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... C~ (3. 26) < -6 + Khe T + 25KCoe c~ <_ 2~ - 6, and (3.27) a(T) + 2~ + c(T)L 2 = 5 + 2~ - 25CoeC~ L 2 > 6. Since/~ > 0, (3. 26) and (3.27) can be achieved by letting 5 > 0 be ... Proposition 4.4. Let T > O, F - (b,a,h,g) 6 H[0, T] and 9 6 Bi(r; [0,T]). Let f E R and f ~(t) = e2Zto(t), t E [0, T], (4.18) = (b -flx, a,h - fly, g) 6 H[0,T]. Then, ~...

Ngày tải lên: 10/08/2014, 20:20