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Thus EXt t0EWt dt 0 and EX2t Et 0t 0WuWv dv du t 0t 0EWuWv dv du t 0t 0 min u v dv du using the covariance of the Brownian motion in the last equality. A key concept is the notion of quadratic variation. Stochastic Integral Inequality Mathematics Stack Exchange Notice that the random fluctuation rates ξ j in the sum 3 are independent of the Brownian increments Wt j1Wt j that they multiply. Ito integral of brownian motion . If f is a step process. The idea is to use Fubinis theorem to interchange expectations with respect to the Brownian path with the integral. Easy to check mean is zero and variance is. This is a consequence of the independent increments. Integral of Brownian motion wrt. Although continuous the paths seem to exhibit rapid infinitesimal movement up. 312015 The construction of an abstract stochastic integral a b X t d B t where a b R B is a Brownian motion and X is an adapted stochastic process in a vector lattice is an indispensable step to further d