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Therefore, ( (B(t E n i ) )  ( B(tin 1 ))2  Var[ B(tin )  B(tin 1 )] n   (tin  tin 1 )  t  0  t. i 1 Let us take the variance of ( (B(t Var n i  ( B(t n i )  B(tin 1 ))2 : ) )  B(tin 1 ))2   3(tin  tin 1 )2  3 max (tin  tin 1 ) t  3t n , and as n   ,  n  0, Var ( B(tin )  B(tin 1 ))2  0 . 12) Stochastic Differential Equations and Related Inverse Problems 31 Summarizing the results, E(  ( B(tin )  B(tin 1 ))2 )  t , and Var (  ( B(tin )  B(tin 1 ))2 )  0 as n   .

1) i 1 where  n  max(ti  ti  1 ) . 1 i  n If Vf([a,b]) is finite such as in continuous differentiable functions then f is called a function of finite variation on [a,b]. Variation of a function is a measure of the total change in the function value within the interval considered. 7 Klebaner (1998)) is that a function of finite variation can only have a countable number of jumps. Furthermore, if f is a continuous function, f  exists and  f (t ) dt  then f is a function Computational Modelling of Multi-Scale Non-Fickian Dispersion in Porous Media - An Approach Based on Stochastic Calculus 24 of finite variation.

There are weakly converging stochastic processes and we do not discuss the weak convergence criteria as they are not relevant to the development of the material in this book. 7) i 1 where tin ’s represents partitions of the interval, a  t0n  t1n  t2n ....  tnn  b ,   max(tin  tin 1 ), and 1 i  n tin 1   in  tin . Riemann integral is used extensively in standard calculus where continuous functions are the main concern. The integral converges regardless of the chosen  in within [ tin 1 , tin ].

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