Differential Equations and Their Applications Part 4 ppsx

... exactly the same as (4. 23). The associated BSDE is now replaced by the following: (5.3) dp = [B - PB]pdt + E qidWi (t), t e [0, r], p(T) = g. Also, (4. 13), (4. 14) and (4. 8) are now replaced ... terms involving h and a. Then it 54 Chapter 3. Method of Optimal Control Conversely, if (1. 14) holds with some (x,y) E R n • Rm and Z(-) E Z[0, T], then (X(.), Y(.), Z(-)) E Jl4[0, T] is an adapted ... Let (H1) and (H3) hold. Then V~'~(s,x,y) is semi- concave uniformly in s 9 [0, T], 5 9 (0, 1] and c 9 [0, 1]. In particular, there exists a constant C > O, such that (3 .41 ) AyVh'~(s,x,y)...

Differential Equations and Their Applications Part 1 doc

... analysis 240 w The approximating solutions {u(n)}~=l 244 w General case 245 w Numerical scheme 247 w Error analysis 248 w Numerical Approximation of the Forward SDE 250 Comments and Remarks ... from the relevant protective laws and regulations and theretbre free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 106501 74 41/3 143 - 543 210 - Printed on acid-free paper ... Deutsche Bibliothek - CIP-Einheitsaufnahme Ma, Jin: Foreward backward stochastic differential equations and their applications / Jin Ma ; Jiongmin Yong. - Berlin, Heidelberg ; New York ; Barcelona...

Differential Equations and Their Applications Part 2 doc

... Lf, r }2 O(t) <_ ~{ r , (4. 19) Consequently, (4. 20) Hence, (4. 16) follows from (4. 15) and (4. 20). vt e [0, T]. (4. 21) Now, applying the above result to (4. 13), we obtain EIY(t ) - Y(t)l ... hand side of (4. 15) 2LO(t). Then, by (4. 15), 0'(t) = -~(t)r _> -r ~/2LO(t), (4. 17) which yields (4. 18) { 4~ }' _> - v~U~r Noting O(T) = O, we have .T -v/~ _> -4~ ... from (4. 25) that (note (4. 27)) T < + f0 te[0,T] + 2E sup I (Y(s),Z(s)dW(s)) tC[0,T] (4. 28) .lj (/o -F- C1 (E sup IY(t)l 2) E tE[0,T] J Now (4. 23) follows easily from (4. 28) and (4. 27)....

Differential Equations and Their Applications Part 3 pptx

... (4. 22), (3.19) holds, then (4. 11) holds and the linear FBSDE (2.12) (with C = O) is uniquely solvable with the repre- sentation given by (4. 14) , (4. 1) and (4. 8). w A Riccati type equation 47 ... Theorem 4. 1. Let (4. 10) admits a solution P(.) such that (4. 11) holds. Then FBSDE (2.12) admits a unique adapted solution (X, II, Z) C 3,l[0, T] which is determined by (4. 14) , (4. 1) and (4. 8). ... closed. Also, by (3 .4) and Lemma 3.1, 7~(/C) C_ A/(E) (since C = .41 61). Thus, to show (3 .46 ), it suffices to show that (3 .48 ) H(E) We now prove (3 .48 ). Take 7/E O=E(~,~CZ) (3 .49 ) = E (rl,...

Differential Equations and Their Applications Part 5 pptx

... we obtain eAT ho (4. 31) w(s, x) <_ A - A ~' V(s, x) E [0, T] x ]R n. Combining (4. 29) and (4. 31), one obtains (4. 23). Proof of Theorem 4. 3. We define (note (3. 24) ) A~ A A0 e~T ... 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)} < ... useful. Now, let us assume that both V and 0 are smooth and we find an equation that is satisfied by 8 (so that (4. 4) holds). To this end, we define (4. 7) w(s, x) = V(s, x, O(s, x)), V(s,...

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 ... bounded partial derivatives and there exist constants A, # > 0, and some continuous increasing function u: [0, oc) + [0, cx)), such that (3 .4) AI <_ a(x,y)a(x,y) T <_ #I, (x,y) 9 4 n...

Differential Equations and Their Applications Part 7 docx

... C~(~n), we have r ICI~) -1 c S. Thus, there exists a ~o E S, such that (4 4) Z -1 I~l<m Combining (4. 33) and (4. 34) , using Parseval's formula again, we obtain (F,C)m = (G,Dr Vr E C~(~). ... e L~(O, T; C~), 0 ~ and (4. 4) holds for all multi-index a. w Existence of adapted solutions 123 Let ~ > 0 be fixed and k _ ~. For any ~ E H~ C C~(]Rn), from (4. 8) and the fact Pk~o = ~, ... dual space of Hl(~n), and the embeddings are dense and continuous. We denote the duality paring between H 1 (]R n) and H-I(~ '~) by (-,-)0, and the inner product and the norm in L2(IR...

Differential Equations and Their Applications Part 8 pps

... (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,~) ... (2.2) and (H),~ (for some m _> 1) hold for (6.2) and (6.3). Then by Theorem 2.3, for any pairs (f, g) and (f, ~) satisfying (2. 14) , there exist unique adapted weak solutions (u, q) and (~, ... that (6.2) and (3.2) are a little differenl since the operators s and J ~4 are defined a little differently. However, by the discussion at the end of w we know that Theorems 2.1, 2.2 and 2.3 hold...

Differential Equations and Their Applications Part 9 pps

... able and F admits a strong bridge, then all the equations "nearby" are solvable. This is a kind of stability result. Remark 4. 3. We see from (4. 14) and (4. 15) that the condition (4. 10) ... E ~. Clearly, it holds (4. 5) F3 = F1 + F2. By the remark right after Corollary 2 .4, we know that B(F3; [0, T]) = r Thus, it follows from (3.5) and (4. 3) that F1 and F2 are not linked by a ... following simple proposition. 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]....

Differential Equations and Their Applications Part 11 pdf

... ~b(t, p, ~, ~); (4. 3) ~(t,p,x, Tr) ~=Tra(t,p,x, Tc), for (t,p,x,~) E [0, T] • IR 3. We can rewrite (4. 1) and (4. 2) as ' Po(t) ~ foo t } = exp/ r(s)ds , /o (4. 4) P(t) = p+ P(s){b(s,P(s),X(s),Tr(s))ds ... (Tr, C) and the initial values p > 0 and x >_ 0 the SDEs (4. 4) and (4. 5) have unique strong solutions, which will be denoted by P = pp,z,~,c and X = X p,~,~,C, whenever the dependence ... that of Theorem 3 .4, so we only sketch it. Let (tl, Xl) and (t2, x2) be given, and let )( = X t~'~ -X t2,~. Assume first tl > t2, and recall the norms I1" IIt,x and ]'[t,~,~...

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