# How to find the mean and variance of this stochastic process?

$I_t = \int_0^t e^{i W_s} dWs$ where $W_s$ is the standard brownian motion and $i$ is the complex number. Any help will be appreciated!

• Evaluate sounds too vague for me. You want a close formula or you want to simulate it... it would be nice to have more details. – MJ73550 Jan 26 '17 at 6:08
• What do you mean "it looks like"? What does the "It" refer to? – Gordon Jan 26 '17 at 16:24
• @Pandaaaaaaa you changed the question pretty drastically, how can you be sure this is what he wanted originally? – SRKX Feb 20 '17 at 1:08
• @SRKX I have seen the same problem somewhere. It asks for variance and mean. – Pandaaaaaaa Feb 20 '17 at 1:13

$$E[I_t|t=0]=0$$
To find the variance, let's write it into differential form $$dI_t =e^{iW_t}dW_t$$ Apply Ito's isometry $$Var(I_t)=\int_0^tE[e^{2iW_s}]ds$$ Apply MGF of normal $$Var(I_t)=\int_0^te^{\frac{1}{2}(2i)^2s}ds=\int_0^te^{-2s}ds=\frac{1-e^{-2t}}{2}$$ Please let me know if anything is incorrect.