Ito process with a brownian motion with drift

Question

i have a problem with this exercise, considered the following Ito process

$$dX_t = dt + dB_t, X_0 = 0$$ dBt increment of Brownian motion

• Find the distribution of $X(t)$, $E[X_t]$; $E[X_t^2]$; $Var[X_t]$; $Cov(X_t; X_s)$; $P(X_t > t)$.

in partcular i have no idea how to solve $E[X_t^2]$ and $Cov(X_t, X_s)$. Thanks for the help

Answer 1

Integrate on both sides

$$X_t-X_0=t+B_t-B_0$$

$$X_t=t+B_t$$

$B_t$ is normally distributed, with mean $0$ and variance $t$, therefore $X_t$ is normally distributed with mean $t$ and variance $t$.

Thus, $$E(X_t)=t$$ $$var(X_t)=t$$ $$E(X_t^2)=var(X_t)+E(X_t)^2=t+t^2$$ $$cov(X_t,X_s)=cov(t+B_t,s+B_s)=cov(B_t,B_s)=min(t,s)$$ $$P(X_t >t)=P(t+B_t >t)=P(B_t>0)=\frac12$$