Ito's Lemma Question

I have a question about the following Ito's Lemma problem:



X(t) has the following process, dX(t) = 0.4dt+0.2dZ(t) where {Z(t)} is a standard Brownian Motion.



Let Y(t) = e^(w[X(t)+0.02t])



Y(t) satisfies the following stochastic differential equation: d ln Y(t) = a(w)dt + B(w)dZ(t).



Determine a(1/2).



(just to state the obvious, a is alpha, B is beta, w is omega)



I did this problem without using Ito's Lemmma, but got the same answer. I am hoping someone can tell me if my method will always work or if I just got lucky. I will give it a shot at explaining what I did:

1. Took log of Y(t). [ln Y(t) = w(X(t)+0.02t)]



2. differentiated both sides of Y(t) with respect to t. [d ln Y(t) = w(dX(t)+0.02dt)]



3. Substituted 0.4dt +0.2dZ(t) into dY(t) for the d(x(t) term. [d ln Y(t) = w(0.42dt+0.2dZ(t))]



4. Substituted w(0.42dt+0.2dZ(t)) into the d ln Y(t) term of d ln Y(t) = a(w)dt + B(w)dZ(t).



[ w(0.42dt+0.2dZ(t)) = a(w)dt + B(w)dZ(t)] ---> [ 0.42wdt+0.2wdZ(t)) = a(w)dt + B(w)dZ(t)]



5. Evaluated new equation for w=1/2. [ 0.21dt+0.1dZ(t)) = a(1/2)dt + B(1/2)dZ(t)]



6. "Observed" that a(1/2)=0.21 and B(1/2)=0.1

Will this always work or did it just work out nicely for this problem?



Thanks for any input!

Ito's Lemma Question