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, Ca is no more than C(1 + Ixl + 1 /5) where ... any s > 0, one can find a 5 = 5( ~) with the property that (4.33) 0 <_ VS'~(s,x,O~(s,x)) <_ cCo, V5 <_ (f(E). Then, by (3.28), (3.39) (with 5 = 0) and (4.33), we obtain 0 <...

Differential Equations and Their Applications Part 3 pptx

... [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 absolute constant (independent ... (3 .56 ) Y(s) ) ) ds. By (3.49), the above holds for all Z C 7/. Now, let 0 < 5 < T and take (3 .57 ) Z(8):oTeAT(T-s)(~)((s)X[T_~,T](S), S C [0, T]. This yields By It6's formula and...

Differential Equations and Their Applications Part 15 pptx

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Differential Equations and Their Applications Part 1 doc

... 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, Paris 1702 w Some examples 5 utility ... Equation 45 w Some Extensions 49 Chapter 3. Method of Optimal Control 51 w Solvability and the Associated Optimal Control Problem 51 w An optimal control problem 51 w Approximate Solvability 54 ...

Differential Equations and Their Applications Part 2 doc

... 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~, respectively, ... T], (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 and...

Differential Equations and Their Applications Part 4 ppsx

... 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 (5. 5), ... 67 Proof. Fix (s, x, y) E [0, T] x IR n x Rm, 5, 5, s, f > 0, and Z C Z[s, T]. Let Z~ (resp. Z$) be the 1 /5- (resp. 1 /5- ) truncation of Z; and (X, Y) (resp. A (X,Y)) the solution of (3.22)...

Differential Equations and Their Applications Part 6 ppt

... 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 equations ... ' Chapter 5 Linear Degenerate Backward Stochastic Partial Differential Equations w Formulation of the Problem We note that in the previous chapter, all the coefficients b, a, h and g are...

Differential Equations and Their Applications Part 7 docx

... 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 replaced by ~ and ... -C]ul 2. Combining (5. 6), (5. 9)- (5. 11) yields (5. 12) z ~ = z$ +z{~ +z~ +z$ > fR ~ { ((A - BBT)D(O~u),D(O~u)) + ~IB TD(Oau) + OC~q[2 }dx - C(lul~ + [ql~-l). Now, we sum (5. 12) up for all...

Differential Equations and Their Applications Part 8 pps

... 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 (f,g) and (f,~) ... let (6. 25) hold. Then (6.16) is satisfied and (6.17) holds. Proof. In the present case, (g, 0) is an adapted strong solution of (6.3). Then similar to the proof of Proposition 6 .5 and note...

Differential Equations and Their Applications Part 9 pps

... 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 holds. (5. 29) F(T) ... Construction of bridges 1 65 Thus, by (5. 15) , it follows that Then, by F"(T) > 0, together with (5. 30) and (5. 32), we must have (5. 29). Hence, we obtain (5. 25) . This shows that a strong...

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