Differential Equations and Their Applications Part 7 docx

... 3.1. Let (2.3) hold and the following hold: (3.3) A 9 L~(O,T;L~ 176 BE L~/Oj:~, T', LO~/R,~.~ , ~n • d)), a 9 Lcff(O,T;L~ 176 b 9 L~ 176 176 c 9 L~(0, T; n~ 176 Then, the adapted ... 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...

Differential Equations and Their Applications Part 14 docx

... differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS 176 , 200-2 17, Springer 1992. [3] Backward doubly stochastic differential equations and ... Stochastics, 3 (1 979 ), 1 27- 1 67. [2] Backward stochastic differential equations and applications, Proc. ICM, 1994. Pardoux, E., and Peng, S., [1] Adapted solution of a backward stochastic differential ... degenerate backward SPDE, with applications, Stochastic Processes and their Applications, 70 (19 97) , 59-84. [3] On linear, degenerate backward stochastic partial differential equations, Probab. Theory...

Differential Equations and Their Applications Part 1 doc

... FBSDEs) in finance theory have been the stochastic differential Jin Ma Jiongmin Yong Forward-Backward Stochastic Differential Equations and Their Applications Springer Lecture Notes in Mathematics ... 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, ... g and x satisfying (1.3), (1.1) admits an adapted solution (X, Y, Z) E 2t4[0, T]. Then (1.4) 7~ (dl - GCi) D_ T~(F) + 7~ (/91) + 7~ (GD1), where T~(S) is the range of operator S. In particular,...

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 ... L2r(~;~ ~) and 3 = Ho • H = n~r (f~; ]Rn+m). Define /( /CZ =r '~(s)-l(C - M1C1)Z(s)ds (3. 37) /( + ~(T) ~(s)-lClZ(s)dW(s), Z 9 7t. Then, by (3 .7) ,/C is a bounded linear operator and/ C ... "~ and Z C 7- /, such that (3.5) g = (O, I)O(T) ( ~ ) y + lCZ, and define (X, Y) by (3.2). Then (X, Y, Z) E AA[0, T] is an adapted solution of (2.12). Hence, the study of operators O(T) and...

Differential Equations and Their Applications Part 4 ppsx

... as g and satisfying the following: (1 .7) ~/(x, y) > O, V(x, y) 9 ~ x ~m, I f(x,y) O, if and only if y = g(x). w The value function 61 This proves (3.1). Next, let s E [0, T] and ... programming method and the HJB equation 57 Next, let (x) ~ ~/1 + [xl 2. It is not hard to check that both D (x) and D 2 (x) are uniformly bounded, thus applying It6's formula to (Yc(t)), and note ... (Y) 1 + lYl, we have by Gronwall's inequality and (1.25) that (1. 27) E(Y~(t))<_C(I+Ixl), Vt e [0,TI, s e (0,1]. In particular, (1. 27) leads to the boundedness of the set {IY~ (0)1}~>o....

Differential Equations and Their Applications Part 5 pptx

... "~ 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 ... 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)} < ... bounded. Let (1 .7) admit a classical solution O(t, x) with bounded O~ and 0~. Let func- tions b and a be uniformly Lipschitz continuous in (x, y, z) with b(t, O, O, O) and a(t, O, O, O)...

Differential Equations and Their Applications Part 6 ppt

... 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 ... lemma. w Well-posedness of linear BSPDEs 1 07 We note that (H)m implies that the partial derivatives of A and B in x up to order (m + 1), and those of a, b and c up to order m are bounded uniformly ... We note here that conditions (2.6) and (2 .7) together with (1.9) are still weaker than the super-parabolicity condition (1 .7) . For example, if n > d and B is an (n • d) matrix, then BB...

Differential Equations and Their Applications Part 8 pps

... with B(F; [0, T]) ~ r Then, for any 7 E 7- /[0, T] and x E ~'~, (1.5)r,.~,z admits at most one adapted solution. Moreover, for any 7, ~ E 7- /[0, T] and x, ~ E lR '~, the stability ... case that F1 -= (bl,cr1,hl ,71 ) and F2 (b2,cr2,h2,g2) are linked by a direct bridge. We now assume this. Next, for any 7 - w Method of continuation 1 47 (2.16)-(2. 17) and note (2.11), we have ... T] and 7 = (bo, ao, ho,go) E ~/[0, T], we can naturally define (1.4) F+~/=(b+bo,~+ao,h+ho,g+go) E H[0, T]. Now, for any F - (b, a, h, g) C H[0, T], 7 - (b0, Cro, ho, go) E 7/ [0, T] and...

Differential Equations and Their Applications Part 9 pps

... space M. The mean- ing of (3. 17) ' and (3.18)' are similar. Here, we should point out that (3. 17) implies m _> n and (3. 17) ' implies m ~ n. Hence, (M) and (M)' overlaps only ... not hard to see that under (3. 17) -(3.18), (3.21) implies (1.8) and (3.22) implies (1 .7) and (1.9)' (Note (1.8) implies (1.8)'). We see that the left hand side of (3.22) can be controlled ... C 7/ [0, T] (see (1.3)) and x E IRn. We have the following result. Lemma 5.1. Let T > O, Then, the two-point boundary value problem (5.1) is uniquely solvable for all V E 7/ [0, T] if and...

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