# Problem finding correct SDE for Stochastic Process

I am really struggling to come up with the correct SDE for the stochastic process:

$$Y(t) = a[Z(t)]^2$$

where $$Z(t)$$ is a Brownian Motion. According to my Prof, the SDE is:

$$dY(t) = adt + 2aZ(t)dZt$$

Can anyone explain how he got to that solution? Thanks in advance, any help appreciated

## 1 Answer

If you apply Ito to $$Y_t=aZ_t^2$$ it simple to arrive at $$dY_t$$:

$$f(t,z):=az^2 \\ dY_t = f_t(t,Z_t)dt+f_z(t,Z_t)dZ_t+\frac{1}{2}f_{zz}(t,Z_t)*[dZ_t]^2 \\ dY_t = 0*dt+ 2aZ_t dZ_t+\frac{1}{2}2a[dZ_t]^2 \\ dY_t = 2aZ_t dZ_t+adt \\$$ Which is your desired result. Recall that $$dZ_t^2=dt$$.

• Thanks a lot! Just to make sure, lets suppose it said that $Z(t)$ followed a geometric Brownian Motion instead of only a brownian Motion. Would it then be allowed to change $dZ_t$ with: $\begin{equation} dZ_t=\mu Z_t dt + \sigma Z_t dW_t \end{equation}$ Thanks a lot again – MikeHeimlich May 11 '19 at 16:09
• Yes, that is correct – Sanjay May 11 '19 at 17:59