Applying Itô's Lemma to a Geometric Brownian Motion and Brownian Bridge Ratio

Summary

The document sets up a ratio between a geometric Brownian motion with time-varying volatility and a Brownian bridge, then asks how to derive its stochastic differential. The bridge has a drift that pulls its value toward a fixed endpoint as the terminal time approaches. The two processes are driven by independent Brownian motions, so their instantaneous cross-variation is zero.

The author recalls the quotient form of Itô's lemma and asks how the multidimensional covariance terms apply. This is a derivation question rather than a worked solution: no resulting drift or diffusion terms, numerical example, or validation are included. Applying Itô's lemma requires accounting for the second-order variation of the denominator as well as the numerator; the independence assumption removes the mixed variation term. The setup also presumes the denominator remains nonzero where the ratio is evaluated.

Key ideas

• The ratio combines a time-varying-volatility geometric Brownian motion with a Brownian bridge.
• Itô's lemma for a quotient includes second-order terms from the denominator.
• Independent driving Brownian motions imply zero instantaneous cross-variation.
• The document poses the derivation but does not give the resulting stochastic differential.

Tags

# Change of numeraire : quotient


# Change of numeraire : quotient












Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge :

$dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$

$dX_2(t) = \frac{b - X_2(t)}{T - t} dt + \sigma_2 dZ(t)$

Assuming that $w$ and $z$ are independant standard BM, how can I get $d\left(\frac{X_1}{X_2}\right)$ ?

I know, from Ito's lemma, that $d\left(\frac{X}{Y}\right) = \frac{dX}{Y} - \frac{XdY}{Y^2} - \frac{dXdY}{Y^2} + \frac{XdY^2}{Y^3}$. I also know that some $\sigma_{ij}$ appear in the multidimensionnal extension for Ito, but i don't understand how it works ...

Thanks in advance for your help

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