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May 30th, 2015, 11:15 PM   #1
Joined: May 2015
From: Poland

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Wiener integral - is it Cauchy sequence in L^2 ?

$\displaystyle f \in L^2([0, \infty)) $ ,
$\displaystyle (f_n)_{n \ge 1 }$ - sequence of step functions ,
$\displaystyle f_n \rightarrow f $ in $\displaystyle L^2([0, \infty)) $

definition of a step function: $\displaystyle f(x)= \sum_{i=1}^{n}a_i I_{[t_{i-1}, t_i)}(x) $
Wiener integral: $\displaystyle I(f)= \sum_{i=1}^{n} a_i (B_{t_i}-B_{t_{i-1}})$

How to prove that $\displaystyle (I(f_n))_{n \ge 1}$ is Cauchy in $\displaystyle L^2 (\Omega, F, P)$ ?

Last edited by natalia0302; May 30th, 2015 at 11:26 PM.
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