Is there a Stochastic process like the following:

Question

I want to model a Stochastic process that has the following description:

The process starts with a zero initial value at time $t=0$ and proceeds in time as the Brownian motion till time $t=T$. We know that the distribution at any time $t$, in the interval $0 \le t \le T$, is $\sim N(0, t)$.

Therefore, at $t=T$, the sample paths have the distribution $\sim N(0,T)$, the process now proceeds in time in such a manner that, at $t=2T$ it returns to the zero value (the value with which it started of at $t=0$ ).

I think I can say that for $T< t \le 2T$, the distribution at any time $t$, is $\sim N(0, 2T-t)$ so that at the end of this interval the distribution reduces to a deterministic value: $N(0,0) \rightarrow 0$.

The question is, Is there a standard process like this? If so, what is the calculus involving such a process? Can we implement the differential equation for a diffusion process in this case?

Please help! Thanks in advance.

Answer 1

There are many processes like this. One of them is

$$ X_t=\begin{cases} W_t, & t\leq T\\ W_{2T-t}, & T< t\leq 2T \end{cases} $$

or another one

$$ X_t=\begin{cases} W_t, & t\leq T\\ \sqrt{2-t/T}W_T, & T< t<2T. \end{cases} $$

Or this

$$ X_t=\begin{cases} W_t, & t\leq T\\ \sqrt{2T/t-1}W_t, & T< t<2T. \end{cases} $$