Calculate expectation and variance of smooth function applied to brownian motion I applied a smoothing function to a Brownian equation and obtained a stochastic differential equation by using Ito's lemma. The smoothing function is $e^{Bt}$.
How do I get the expected value and variance of this function? Just looking for the required approach rather than a full fledged solution.
 A: It is my question also. I found a look like question  in "An Algorithmic Introduction to
Numerical Simulation of
Stochastic Differential
Equations" by prof. Desmond J. Higham
https://doi.org/10.1137/S0036144500378302
and it is about the expected value of $e^{t+0.5B_t}$ the research says that $\mathbb{E(x)}=e^{\frac 98 t}$I tried to find expect vale by direct definition but got nothing. I tell you about my experience. I use Ito difference formula to understand $\mathbb{E(x)}$.
take $g(t,x)=e^{t+0.5x}$ and $$dg=\frac{\partial g}{\partial t}dt+\frac{\partial g}{\partial x}dx+\frac12\frac{\partial^2 g}{\partial x^2}(dx)^2\\dg=
e^{t+0.5x}dt+0.5e^{t+0.5x}dB_t+\frac12(0.5)(0.5)e^{t+0.5x}(dB_t)^2=\\\frac98e^{t+0.5x}dt+0.5e^{t+0.5B_t}db_t\to \mathbb{E}(g)=e^{\frac98t}$$  so (I think similarly) when we want to find expected value for $e^{B_t}$ we can take $$g(t,x)=e^{x}\\It\hat{o} \space diff \space dg=0dt+1e^{B_t}dB_t+\frac{1}{2}e^{B_t}dt\\$$so it shows maybe ($\mathbb{E}(x)=e^{\frac12t})$ Iam not sure that is right approach or not, but hope it can help you.
