Brownian motion and convergence in probability of step functions

1. 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 $g = \lambda_i 1_{[t_i,t_{i+1}]}$.

2. 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

• I am sorry to say that but if you have the least problem solving 1, there is not much point to try to attack 2. In other words, it is quite impossible at the moment to guess what would be an answer adapted to your needs. – Did May 9 '13 at 10:01

1. 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.