Differential Equations and Their Applications Part 13 pot
Differential Equations and Their Applications Part 13 pot
... (A1) and (A2), one can show that w i as io + co, uniformly in i and k. In particular, if we fix the mesh size h > 0, and let R = ioh, then the quantities (2.8) maxlu(tk,x~) wk[ and max ... {Vik}, and that of 0., b, b and their partial derivatives. Consequently, ! k 1 k 1 ' (h + At), Vk, i. (2.36) 1(4)}1 < C~{l~ I+ I~:~ I} +63 Use of the maximum principle and the ... and Bo(u,v)~k are analogous to ai,k (b0)~k and (Bo)i ,~ k except that U k-~ and Vi k-~ are replaced by u k-~ and v k-~. Estimating the error {(e~) k} and {(e~) k) in the same fashion as in Lemma...
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Differential Equations and Their Applications Part 12 pot
... t < s < T, and is c~dl~g as well. We have the following theorem. Theorem 4.11. Assume (H6) and (H7). Let X, fo, 7r be defined by (4.31) and (4.34), respectively; and let X(s) = O(s,x(s)), ... (4.1). We assume the following (H6) b and a are independent of 7r and are time-homogeneous; g _> 0 and belongs boundedly to C 2+~ for some a E (0, 1); and r is uniformly bounded. Define x(t) ... case the PDE (4.11) is degenerate, and the function g is not bounded, the solution/9 to (4.11) and its partial derivatives could blow up as p approaches to 01R d and infinity. Therefore some modification...
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Differential Equations and Their Applications Part 1 doc
... the stochastic differential Jin Ma Jiongmin Yong Forward-Backward Stochastic Differential Equations and Their Applications Springer Lecture Notes in Mathematics Editors: A. Dold, Heidelberg ... into the stock, and puts viii Preface strong degeneracy in the sense of stochastic partial differential equations. Such BSPDEs can be used to generalize the Feynman-Kac formula and even the ... Deutsche Bibliothek - CIP-Einheitsaufnahme Ma, Jin: Foreward backward stochastic differential equations and their applications / Jin Ma ; Jiongmin Yong. - Berlin, Heidelberg ; New York ; Barcelona...
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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, ... 3.2. Assume that b, a, h and ~ satisfy (2.6). Assume further that a and ~ are continuous in (t, x, y) uniformly in x, for each w E ~; and that g C C 2 M C~(R~; R m) and is deterministic. Suppose...
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Differential Equations and Their Applications Part 3 pptx
... all g C H if and only if (3.17) and (3.19) hold. In this case, the adapted solution to (2.12) is unique (for any given g E H). Proof. Theorems 3.2 and 3.3 tell us that (3.17) and (3.19) are ... Let F = I E ~m• and ~ = m. Then (1.1) is solvable for all b, a, b, ~, x and g satisfying (1.3) if and only if (2.11) is solvable for all c We note that by Theorem 1.2, F I and g = m imply ... matrix-valued function and p : [0, T] • ~ + ]R m is an {Svt)t>_0-adapted process. We are going to derive the equations for P(.) and p(-). First of all, from (4.1) and the terminal condition...
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Differential Equations and Their Applications Part 4 ppsx
... (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, ~5,~ (s, x, y) and Va'~(s,x,y) ... similar to that of Proposition 3.1 and the proof of (ii) and (iii) are by now standard, which we omit here for simplicity of presentation (see Yong-Zhou [1] and Fleming-Soner [1], for details). ... Proposition 1.5. Let (HI) hold. Then (1 .13) implies (1.18); conversely, if V(x, .) is continuous, and (H2) holds, then (1.18) implies (1 .13) . Proof. That condition (1 .13) implies (1.18) is obvious....
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Differential Equations and Their Applications Part 5 pptx
... 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)} < ... "~ 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 ... the forward part is still n dimensional. We can now consider (1.1) with m = 1. Here W is an n-dimensional standard Brownian motion, b, a, h and g take values in IR n, ~n• IR and IR, respectively....
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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 ~" and ... between (1.1) and (2.12) holds for adapted strong and weak solutions, respectively. On the other hand, from (2 .13) we see easily that the group {A, B, a, b, c} satisfies (H)m if and only if...
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Differential Equations and Their Applications Part 7 docx
... 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 ... J0 + z s (4 .13) 1-1_ <m + BT[D(O~uk)] + O~q k 2}dxdt fo k 2 Igkl:}. <CE{ rlf (t)lm+ Note that the constant C > 0 in (4 .13) only depends on T, m and Kin. From (4 .13) , we may assume ... have the Gelfand triple V r H = H' ~ + V'. We denote the inner product and the induced norm of H by (-, ")0 and [-10, respectively. The duality paring between V and V' is...
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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 (~, ... 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....
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