Expectation of the stochastic process

Question

Consider the following stochastic process.

• $X(t)=Y(t-1)$ where $Y(t-1)\sim Normal(X(t-1)+1,\sigma^2)$
• $X(0)=1$

Now I am interested in the $E[X(n)]$. Intuitively, I think $E[X(n)]$ is equal to $n$. Is this the correct answer? If it is, how can I show this mathematically?

Answer 1

It’s simple induction. From $E[X(t)]=E[X(t-1)]+1 $ you can easily get the desired result.