Differential Equations and Their Applications Part 11 pdf
Differential Equations and Their Applications Part 11 pdf
... (A1), we see that a and fl are bounded, adapted pro- cesses. Therefore, subtracting (2.9) from (2.8), using (2.10) and (2 .11) , and noting the facts that u(t, x) = qo(t, x) and u(r, X(r)) _< ... unusual in applications to work from the beginning with the so-called "equivalent martingale measure," in the sense of Harrison and Kreps [1], and we do so. 198 Chapter 8. Applications ... results in Chapter 8 Applications of FBSDEs In this chapter we collect some interesting applications of FBSDEs. These applications appear in various fields of both theoretical and applied prob-...
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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
... 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 ... PC1)-~(PBIp + q). It is clear 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 ... 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...
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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). ... [0, T], c 9 (0, 1]. Here and in what follows, the constant C > 0 will be a generic one, de- pending only on L and T, and may change from line to line. By (1.24) and (1.15), we obtain E[Y~(T)I...
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Differential Equations and Their Applications Part 5 pptx
... (2 .11) [z(t,x,y,p)[ < C[p[, V(t,x,y,p) E [0, T] • R ~ • R "~ 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) ... 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)} < ... 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...
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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 ... 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.,...
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Differential Equations and Their Applications Part 7 docx
... solutions 111 We note that {A,B,a,b.c} satisfies (H),~ 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 ... '~) by (-,-)0, and the inner product and the norm in L2(IR ~) by 110 Chapter 5. Linear, Degenerate BSPDEs Theorem 2.3. Let m > 1 and (H)m hold for {A, B, a, b, c}. Let (1.6) and (2.2) hold. ... 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...
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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 ... (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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Differential Equations and Their Applications Part 9 pps
... = (hi, ai, hi, gi) 6 g[o, T] (i = 1, 2), set Ilrl - r211o(t) = lib1 - b211o(t) + I1Ol - a211o(t) (4.8) + Ilhl - h21lo(t) + Ilgl - g211o. Note that I1 IIo(t) is just a family of semi-norms ... (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 ... Ao and Co. To this end, we let L be the common Lipschitz constant for b, a, h and g. We note that (3.8) implies (1.7). Thus, it is enough to further have (3.9) a(T) + L2c(T) >_ 5, and...
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