Computational Mdlg. of Multi-scale Non-Fickian Dispersion in by D. Kulasiri

By D. Kulasiri

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As stated earlier I[ X ]( ) is a stochastic process and it has the following properties (see, for example, ksendal (1998) for more details): 1. Linearity If X(t) and Y(t) are predictable processes and  and  as some constants, then I[ X   Y ]( )   I[ X ]( )   I[Y ]( ) . 5) E( I[ X ]( ))  0 . 6) 2. Zero mean Property 3. Isometry Property T E[(  X(t ) dB(t ))2 ]  S T  E(X (t )) dt . 7) S The isometry property says that the expected value of the square of Ito integral is the integral with respect to t of the expectation of the square of the process X (t).

I[ X ]( ) is also a stochastic process in its own right and have properties originating from the definition of the integral. It is natural to expect I[ X ]( ) to be equal to c( B(t ,  )  B(s ,  )) when X (t ,  ) is a Stochastic Differential Equations and Related Inverse Problems 33 constant c. 1) i 0 c , t S where X (t )  {c0 , t  t  t i  0, , n  1 . i i i 1 The time interval [S,T] has been discretized into n intervals : S  t0  t1    tn  T . Using the property of independent increments of Brownian motion, we can show that the mean of I[ X ]( ) is zero and, n1 Variance  Var ( I[ X ])   ci2 (ti  1  ti ) .

I i i 1 The time interval [S,T] has been discretized into n intervals : S  t0  t1    tn  T . Using the property of independent increments of Brownian motion, we can show that the mean of I[ X ]( ) is zero and, n1 Variance  Var ( I[ X ])   ci2 (ti  1  ti ) . 2) i 0 It turns out that if X (t ,  ) is a continuous stochastic process and its future values are solely dependent on the information of this process only up to t, Ito integral I[ X ]( ) exists. The future states on a stochastic process, X (t ,  ) , is only dependent on Ft then it is called an adapted process.

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