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,~) ... ~(0) _~ K 00) , and either (1 .8) -(1.9) or (1 .8) '-(1.9)' hold: (1 .8) ((I)(T) g(x) g(Z) ' g(x) g(~) )>51x-~?, vx,~e~". (1.9) X w y ' h(t,O) h(...

Ngày tải lên: 10/08/2014, 20:20

... the function (8, x,y) J (8, x,y;ZJi,,Tl(.)) is continuous. Thus, by the definition of V, it is necessary that V(s, x, y) is upper semi-continuous. On the other hand, by (2.6) and (3.2), taking ... (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'...

Ngày tải lên: 10/08/2014, 20:20

... 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 ... B(t)T~,b ' (4.17) tc[O,T] I + 2 ( B(t)~ + C(t)~,h' - h) +2 (B(t)TF, 8& apos; - 8 > +(A(t) (8& apos; +8) ,8& apos;-~)} V0,0EM. The above means that if the perturba...

Ngày tải lên: 10/08/2014, 20:20

... 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...

Ngày tải lên: 10/08/2014, 20:20

... A2(s))dA; /o 8( 8) = hlz(s, Y2 (8) + af~ (8) ,'Z2(s) + 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 ... 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 differe...

Ngày tải lên: 10/08/2014, 20:20

... /~(s)=+l, Vs9 So+r /~(s)=0, 8k 80 1 {8 9 [8o ,8~ ] I Z(s) = 1)1 = ~ , 1 {8 9 [8o ,8~ ] l Z (8) -1}1 = 8~ - so, 2 Vs 9 (so + e,T]; k_>l, for some sequence Sk $ so and Sk < T - ~. Next, ... 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...

Ngày tải lên: 10/08/2014, 20:20

... "~ 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 <...

Ngày tải lên: 10/08/2014, 20:20

... 8( x + h) - 8( x) = 0, we have -Mh 2< lim 8~ (x)h < ~ 8~ (x)h < Mh 2. Dividing h and letting h + 0 we derive lim~-~oo 8~ (x) = 0. Further, note that 1 8~ - a(x ,8~ [-b(x ,8) Sx + h(x )8 ... boundedness of 8, 8~ , and 8~ z and the assumption (H1) then show that 8xz~ exists and is continuous and bounded as well. Thus apply Taylor's expansion to the third...

Ngày tải lên: 10/08/2014, 20:20

... 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 a...

Ngày tải lên: 10/08/2014, 20:20

... 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. 1 98 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 theoret...

Ngày tải lên: 10/08/2014, 20:20