(Applied mathematical sciences 15) martin braun (auth ) differential equations and their applications an introduction to applied mathematics springer new york (1978)

Differential Equations and Their Applications Part 15 pptx

...

Differential Equations and Their Applications Part 1 doc

... 89 92 98 10 3 10 3 10 6 11 1 11 1 11 3 11 8 12 6 13 0 13 7 13 7 14 0 14 0 14 3 14 8 14 8 14 9 15 1 15 4 15 8 15 8 16 1 16 9 16 9 17 1 18 1 18 2 18 6 19 0 19 3 19 3 19 7 2 01 207 210 219 226 Contents w w w w xiii ... time-reverse feature seen in (1. 18) Again, once a solution of (1. 19) can be determined, then U(c) = Y(0) defines a unitiliy function An interesting variation of (1. 18) and (1. 19) is their finite horizon ... Springer, 19 99 (Lecture notes in mathematics ; 17 02) ISBN 3-540-65960-9 Mathematics Subject Classification (19 91) : Primary: 60H10, 15 , 20, 30; 93E03; Secondary: 35K15, 20, 45, 65; 65M06, 12 , 15 , 25;...

Differential Equations and Their Applications Part 2 doc

... EIX~ (t) - X2 (t) l2 _< E (5.7) f ~t2LlXl - - X2[k]Xl( 22 1 q-I]I1 ] 72[ q-[Z1 - Z21] JW(L([X1 -X2[-[-[rl - Y2D-~Lo[Z1 - Z2[ )2} d8 ~E {Ce([Xl -X2 [2 ~.-]r1-r2 12) -[-(L2-~-c)[Z1 -Z2 12} ds, where ... cE IZ1 - Z2[2ds + E P < (L~ + T)eC~TE./n /o + ~E + C~E IF1 - Y21~as f T ([X1 - X2 12 + [Y1 - y2 12) ds T [C~lYl - Z2 12 + (L~ + e)lZl - Z2 12] ds IZl - Z212ds + E /o IY1 - Y2I 2ds IY1 - ~21 2ds In the ... IYl(t) - Y2(t)l 2, we have (note (5.1) and (5.8)) ElY,(t) - Y2(t)l + E j(t T I~1 - 2: l~ds < L~EIXI(T) - X2(T)I T + 2LE[ It [Y1 - ~ [ ( I X ~ - X~l + [Y~ - Y21 + Iz1 - Z2[)ds _< LI~EIXI(T) - X2(T)I...

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 necessary ... Z(s) )12ds < K(Sk - o s0) /? ElZ(s)12ds o Hence, (3. 32)- (3. 33) and (3. 35) imply sk - SOE[ll - ~ ( s ) l ~ + I1 + ~(s0)lJ - o ( s k - so) (3. 36) J 80 This is impossible ... a ~ E 74, such that (3. 54) r~ = Then, from (3. 49) and (3. 53) , o = E(v,~ZI (3. 55) r ~(s)dW(s) we have = E(~,(O,~) ( X ( r ) ~ ~,Y(T) ) ) This yields (3. 56) Y(s) ) ) ds By (3. 49), the above holds...

Differential Equations and Their Applications Part 4 ppsx

... (1. 24) EIX~(t)I will be a generic one, depending only on L and T, and may change from line to line By (1. 24) and ... y = Y(0) IRTM Then (1. 14) holds which gives (x,y) N ( V ) and (1.13) follows 54 Chapter Method of Optimal Control Conversely, if (1. 14) holds with some (x,y) E R n • Rm and Z(-) E Z[0, T], then ... uniformly in _> and s e [0, T] Next, for fixed (s, x, y) [0, T] x IR'~ x l{ m, s _> 0, and > > 0, by (3. 24) , we have o 0, and So > 0,...

Differential Equations and Their Applications Part 5 pptx

... discussion L e m m a 5. 1 Let (H1) and (H4) hold Then there exists a constant C > O, depending only on L and T, such that for all 5, r >_ O, and (s, x, y) E [0, T] • IR2, it holds that (5. 2) ~Js'~(s,x,y) ... [0,T], uniformly (5. 4) and (5. 5) that inf ZCh~[s,T] = ] ( x , y) + > f ( x , y) - inf ZEZ~[s,T] c(1 + E L II(t, X(t), Y(t), Z(t))dt proving the lemma [] Next, for any x E ]R and r > 0, we define ... set r = Recall that by Proposition 3.6 and Theorem 3.7, for any p > 0, and fixed x E IR, we can first choose 5, e > depending only on x and Q~(1), so that (5. 7) O...

Differential Equations and Their Applications Part 6 ppt

... which are C in x and C 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 T > O, (0 ... FBSDEs 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 ... oK(t, X(t)K))dt + a(X(t) K, OK (t, X(t)K))dW(t), (3 .66 ) x K ( o ) = z dX(t) = b(X(t), O(X(t)))dt + a(X(t), O(X(t)))dW(t), (3 .67 ) X(0) = x By It6's formula, we have EIXK(t) - X(t)l =E /o [2(xK X,b(XK,OK(s,...

Differential Equations and Their Applications Part 7 docx

... (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 2.3 for BSPDE (1.5) with a and b replaced by ~ and b ... following result T h e o r e m 3.1 Let (2.3) hold and the following hold: A L~(O,T;L~ 176 B E L~/Oj:~,T', LO~/R,~.~ ~ n • d)), , (3.3) a Lcff(O,T;L~ 176 b L~ 176 176 c L~(0, T; n ~ Then, the adapted weak ... 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 denoted by ( , ) , and the...

Differential Equations and Their Applications Part 8 pps

... Let T > and = (: c and C : [0, T] if there exist some constants K, > 0, such that { c(T) _< ~(0) _~ K (1.7) A(t) o, vt Io, T], 00) , and either (1 .8) -(1.9) or (1 .8) '-(1.9)' hold: (1 .8) ((I)(T) ... (2.11) and working on (v,p) for the transformed equations [] Our main comparison result is the following T h e o r e m 6.2 Let (1.6), (2.2) and (H),~ hold for (6.2) and (6.3) Let (f,g) and (f,~) ... the case that and ~ are not necessarily smooth enough, we may make approximation [] P r o p o s i t i o n 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

... (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, and s being given by (5. 19) and (5.28), respectively, and we may ... any x E 0 (9 the set of inward normals to O at x by (1.1) h/'x = {~: I '1 = 1, and (% x - y) < O, Vy e (9) It is clear that if the boundary 0( .9 is smooth (say, C1), then for any x E 0 (9, the set ... [0, T]) for suitable choice of and Co Again, we let L be the common Lipschitz constant for b, a, h and g We will choose and Co so that (3.21) a(T) + 23 + c(T)L > 5, and h(t)]xI + c(t)iy] + c(t)]z]...

Differential Equations and Their Applications Part 11 pdf

... nondecreasing and RCLL paths, such that C(0) = and C(T) < c~, a.s.-P; and E f J 17r(s)12ds < c~ Clearly, under suitable conditions, for a given strategy (Tr,C) and the initial values p > and x >_ ... interval It2, T], we see that (3.8) and (3.9) will look the same, with I1" I1~ being replaced by I1" Itt2,~; but (3.6) and (3.7) become e-AT EIXTI + (3.6)' A 1112 1h21,;~ ...

Differential Equations and Their Applications Part 12 pot

... that 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 there ... X, and neither we know that it is even a semimartingale Let us now take a closer look First notice that Lemma 4.9 and the boundedness of r(.) and E [~n(s)12ds < C; E If(s, xn(s),X~(s),~n(s))12ds ... case the PDE (4.11) is degenerate, and the function g is not bounded, the solution/9 to (4.11) and its partial derivatives could blow up as p approaches to 01Rd and infinity Therefore some modification...

Differential Equations and Their Applications Part 13 pot

... {Vik}, and that of 0., b, b and their partial derivatives Consequently, (2.36) ! k 1(4)}1 < C~{l~k I+ I~:~ I} +63 ' (h + At), Vk, i Use of the maximum principle and the estimates (2.33), (2.35) and ... satisfies xh~(x) - h(x) 7_ O, and that § and are true interest rate and volatility such that they are {~'t}t_>o-adapted random fields satisfying ~(t,x) > r(t,x), and ~(t,x) > ~(t,x), V(t,x) Then, ... volatility, and denote it by a a(t, x), which is C in x; and assume that the interest rate is deterministic and independent of the stock price By the conclusion (iii) in the previous part we know...

Differential Equations and Their Applications Part 14 docx

... Buckdahn, R., and Pardoux, E., [1] Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastics Reports, 60 (1997), 57-83 Bellman, R., and Wing, ... 125 -144 [9] Backward SDE and related g-expectation, preprint [10] Adapted solution of backward stochastic differential equations and related partial differential equations, preprint Peng, S., and ... Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel, Dordrecht, Holland, 1987 Krylov, N N., and Rozovskii, B L., [1] Stochastic partial differential equations and diffusion processes,...

Xem thêm