Differential Equations and Their Applications Part 15 pptx

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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 ... follows, we will not distinguish (2.12) and (2.14), and we will let { d~(t) = A~(t)dt + Al~(t)dW(t), t E [0, T], (2 .15) ~(0) = I. If we call (X,Y) the state and Z the control, (2.12) is called...

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 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, ... a2(8))aa. Clearly, a and fl are {Ft}t_>o-adapted processes, and are both uniformly bounded, thanks to (6.2). In particular, /3 satisfies the so-called Novikov condition, and therefore the...

Differential Equations and Their Applications Part 4 ppsx

... 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 ... 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). ... (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)...

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 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 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

... under (5 .15) and (5.28), the fol- lowing holds. (5.29) F(T) - g2 f(O) >_ O. In fact, by the choice of 6 and by (5.27), (5.30) F(O) - g2 f(O) = ff'(O) = O. On the other hand, 1(1+~)2e2~T ... T]) for suitable choice of 5 and Co. Again, we let L be the common Lipschitz constant for b, a, h and g. We will choose 5 and Co so that w Properties of the bridges 155 Proof. (i) The convexity ... is solv- 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...

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