For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = \lambda_i 1_{[t_i,t_{i+1}]}$.
If $g_n$ is a sequence of step functions such that $g_n \rightarrow g$ in $L^2$ does $\int_0^a g_n(s)dB_s$ converge? Is it converging in probability? In $L^2$ ? Almost surely?
My intuition is that for 2 it does converge to the result in 1, but would I use dominated convergence theorem and try to argue something about boundedness? Help please?!!! Thanks
By definition of the stochastic integral, we have $$\int_0^T 1_{[s,t)}(r) \, dB_r = B_t-B_s$$ for all $0 \leq s \leq t \leq T$. Use this and the linearity of the integral, to compute the stochastic integral of the given step function.
Start thinking about this part, if you are really sure about the first question! Hint Apply Itô's isometry to show the convergence of the stochastic integrals.