Chris P Tsokos's Random Integral Equations with Applications to Stochastic PDF

By Chris P Tsokos

ISBN-10: 3540056602

ISBN-13: 9783540056607

ISBN-10: 3540369929

ISBN-13: 9783540369929

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Contained We shall (B,D) linear first in section in C c ( R + , L 2 ( ~ , A , P ) ) and and prove a lemma operator spaces is a d m i s s i b l e operator Xn(t;w)eB (TXn)(t;w)Dy(t;w) Since linear with operator We m u s t in B , X n ( t ; ~ ) ÷ x ( t ; w ) But since T: C c ( R + , L 2 ( ~ , A , P ) ) ~ C c ( R + , L 2 ( ~ , A , P ) ) y(t;w) uniqueness and by the c l o s e d - g r a p h bounded. there . theorem that Therefore, K>0 that infimum operator T. of all such constants the proof. it is such that K is called of the T is closed, I I (Tx)(t;m) I ID _< K I Ix(t;~) IIB.

Then the nth term of the series converges to zero as n ÷ ~, and there exists an N > 0 such that for k > N, we have 1 {/~Ixk(t;~)-x(t;~) 12dp(~)} ~ < i, so that for k > N, we have {/nlXk(t;~)-x(t;~) I2 dP(~) } 1 {/~[xk(t;w)-x(t;w) I2 dP (e) } Therefore, N 1 E {S~IXn(t;~)-x(t;~) I2 dP(~)} ~ + ~ {f~IXn(t;~)-x(t;w) 12dp(~)} n=0 n=N+l 1 co oo < - ~ n=0 {f~IXn(t;~)-x(t;~) I2 dp(~)} ~ < ~ - I-IK < -55- Hence, 1 N { E l X n ( t ; ~ ) - x ( t ; w ) 12} [ + S n=0 < - Since x n for each fixed (t;w)-x(t;~), I-IK x(t;w) the d i f f e r e n c e space, E l X n ( t ; ~ ) - x ( t ; w ) I2 " t E R+ X n ( t ; w ) , since is in the B a n a c h S n=N+l 2Q and h e n c e s L2(~q,ArP), of e l e m e n t s so is in a B a n a c h for each n = 0 , .

For every M = 1,2 .... , there solution operator h(t;~) I IDM + KI I f(t,x(t;~)) I IBM [0,M] C R+, completing Thus, random = Thus, (i) and + /~k(t,T;~)f(T,x(T;~))dTl in SM, that is, at least one random x(t;~) by condition that the composite (ii) and the last hypothesis Therefore, mapping < S M and the conditions satisfied. of D M. 2, are + /~k(t,T;~)f(T,x(T;~))dT, from D M into B M and bounded, Lemma and hence I I-I IDM, respectively. 1), is at least one bounded x(t;~), t g [0,M]. 1). 2 the u n i q u e n e s s spaces of T s o k o s defined defined of the r a n d o m [4] and does above.

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Random Integral Equations with Applications to Stochastic Systems by Chris P Tsokos

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