Differential Equations and Their Applications Part 14 docx

... differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS 176, 200-217, Springer 1992. [3] Backward doubly stochastic differential equations and ... 27 (1993), 125 -144 . [9] Backward SDE and related g-expectation, preprint. [10] Adapted solution of backward stochastic differential equations and related partial differential equations, preprint. ... Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel, Dordrecht, Holland, 1987. Krylov, N. N., and Rozovskii, B. L., [1] Stochastic partial differential equations and diffusion...

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 ... 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 ... Gelfand triple H 1 (~n) ~+ L 2 (~n) ~_+ H-1 (IRa). Here, H-I(IR ~) is the dual space of Hl(~n), and the embeddings are dense and continuous. We denote the duality paring between H 1 (]R n) and...

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...

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...

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 ... that .4 and -41 are time-dependent matrix-valued functions and and ~ are {hvt}t>o-adapted processes. Further, under (4.11), the following SDE admits a unique strong solution: (4 .14) { dX ... 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...

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). ... [0, T], c 9 (0, 1]. Here and in what follows, the constant C > 0 will be a generic one, depending only on L and T, and may change from line to line. By (1.24) and (1.15), we obtain E[Y~(T)I...

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 ... following assumptions. (A1) d = n; and the functions b, or, h and g are smooth functions taking values in IR '~, IR m, Ill n• ]R "~• and Ill m, respectively, and with first order derivatives...

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 ... 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 deterministic, i.e.,...

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....

Differential Equations and Their Applications Part 9 pps

... -25, Now, we take a(t) and c(t) as in (3.7) and we require (3.13) d(t)+(3L + 2L2)lc(t)l = -Co(Co - 3L - 2L2)e C~ -Co(Co - 3L - 2L 2) ~ -5, Vt E [0,T], and (3 .14) c(t) = -Coe C~ <__ ... (5.30) and (5.32), we must have (5.29). Hence, we obtain (5.25). This shows that a strong bridge ~(t) has been constructed with K, 5 and s being given by (5.19) and (5.28), respectively, and ... with F is solvable 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...

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