What are the dynamics of the reverse of this FX process?

Assuming the dynamics of the exchange rate between two currencies at time $t$ is given by:

$$dX_t=\Delta r X_t dt+ σ X_t dW_t$$

Is the FX Reverse process $\frac{1}{X_t}$ a brownian motion?

How can Ito's Lemma be applied to prove that?

• Plese write down your model for $FX$ - then on can look at $1/FX$. – Ric Nov 20 '14 at 14:18
• 1/Fx is supposed to be brownian process. How to prove it. – user13524 Nov 20 '14 at 16:45
• Its a GBM Process not a brownian. See below. – Drew Nov 20 '14 at 19:30
• I tried to keep you notation but frankly the way you expressed your question and added details in comments were not very community-friendly. Please pay some attention to formatting and clarity next time. – SRKX Nov 21 '14 at 1:00

Well, if you assume Fx is a Brownian Motion $W_t$ then $\frac{1}{X_t} = -\frac{1}{X^2_t} \bullet X_t + \frac{1}{X^3_t} \bullet \langle X\rangle_t = -\frac{1}{X^2_t} \bullet X_t + \frac{1}{X^3_t} \bullet \sigma^2 X^2_t t$.

So $d\bigl(\frac{1}{X_t}\bigr) = -\frac{1}{X_t} [(\Delta r + \sigma^2 ) dt + \sigma dW_t ]$

Setting $\frac{1}{X_t} = M_t$, we see it is not a Brownian Motion, but a Geometric Brownian Motion. I think thats the proof you were asked.

This is Yor's notation where $H \bullet X = \int H dX_s$

• Nice solution! I also think that it is important that it is GBM, which is a positive process. With BM we could run into serious problems at zero. – Ric Nov 21 '14 at 7:24
• I changed it to $\Delta r$ on seeing the change made above. – Drew Nov 21 '14 at 15:13
• Your quadratic appears missing the fraction $\frac{1}{2}$. – Gordon Apr 1 '15 at 15:54