Differential Equations and Their Applications Part 12 pot

... both al and f12 are adapted processes and are uniformly bounded in (t, w). Similarly, by conditions (H1) (H3) and (H5'), we see that the process O(.,P(.)) is uniformly bounded and that ... x* ~=X(O) is the 212 Chapter 8. Applications of FBSDEs Proof. First consider the function 0"~ O -g. It is obvious that 0t = 0t, Op = Op - gp and Opp = Opp - gpp; and 0"...

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... (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...

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... 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 form...

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... Z2[2ds + E ([X1 - X 212 + [Y1 - y 212) ds P T < (L~ + T)eC~TE./n [C~lYl - Z 212 + (L~ + e)lZl - Z 212] ds /o /o + ~E IZl - Z212ds + E IY1 - Y2I 2ds + C~E IY1 - ~212ds. In the above, ... 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...

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... linear FBSDE (2 .12) is solvable for 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 ... 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) i...

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... (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...

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... -/3) EIZ(s) - Z(s)12ds ~t T (1.15) < C E{I:Y(s) - V(s)l 2 + [Y(s) - y(s)lLZ(s) - Z(s)l}ds __ c~ El~(s) - Y(s)12as + s 12( s) - Z(s)12as, where e > 0 is arbitrary and C~ depends on ... 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...

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... 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...

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... 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 ... and V' is denoted by (.,.)0, and the norms of V and V' are denoted by 1]" I[ and I[" []*, respectively. We know that the following holds: (3 .12) (u,v)0=(u,v...

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... (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,~) ... /o/ < ce {1 ~12 + El~ol 2 + E {INo(t)l 2 + I~o(t)l 2 + I'~o(t )12} dt} (2.18) +2-~E foT{If'(i)]2+l~(t)]2}dt // cUE 12( T)I 2~ + I~ol ~ + {12( t)l 2 + I'~o(...

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