Showing that $X^2 + Y^2$ is a deterministic process with stochastic differentials

Question

Let $X$ and $Y$ solve the equations $$dX = aXdt - YdW$$ $$dY = aYdt + XdW$$ with $X(0), Y(0)$ given. I manage to show by Ito calculus that $Z = X^2 + Y^2$ solves $$ dZ = [2aZ + Z]dt$$ But what I really want to show is that $Z$ is deterministic, how do I do that? What relation does deterministicity have with stochastic differentials? Do we only have to eliminate the dW part, or also the dt part? Or do I have to somehow solve the equation now and get a formula for Z with hopefully only depends on $t$, and if so, how to do this?

Answer 1

Using Ito's lemma , you get that $dZ = [2aZ + Z]dt$, which is an Ito process without stochastic term. Applying Ito's lemma again on $U_t=log(Z_t)$, you get $$dU_t=(2a+1)dt$$ or $$U_t=U_0+(2a+1)t$$ Furthermore, the initial condition on Z is non-random, therefore $$Z(t)=Z(0)e^{(2a+1)t}$$

which is deterministic