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...

Ngày tải lên: 10/08/2014, 20:20

... 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...

Ngày tải lên: 10/08/2014, 20:20

... 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 ha...

Ngày tải lên: 10/08/2014, 20:20

... (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) ,...

Ngày tải lên: 10/08/2014, 20:20

... 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 <_...

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

... 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...

Ngày tải lên: 10/08/2014, 20:20

... (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 solut...

Ngày tải lên: 10/08/2014, 20:20

... 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 a...

Ngày tải lên: 10/08/2014, 20:20

... ~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 =...

Ngày tải lên: 10/08/2014, 20:20