Ito formula technique.

Question

What is the best way to apply Ito formula to

$$ ||\bar{X_t}-X_t||^2 e^{-C(h(\bar{X_t})+h(X_t))} $$

(Do I just have to brute force use the Ito multi-dimensional formula, or are there some shortcuts.)

Where

• $||a||=(\sum a_i^2)^{1/2}$
• $C$ is constant

• h is some nice $C^{\infty}$ function.

• The processes $X_t$ and $\bar{X}_t$ are N-dimensional with common Brownian Motions i.e For $i=1,...,N$ $$d\bar{X}^i_t=\tilde{b}^i(\bar{X}_t)dt+dB^i_t $$ and $$dX^i_t=b^i(X_t)dt+dB^i_t $$

($B^i-$ independent for each $i$).

Answer 1

This is how I will tackle the problem.

1. First rewrite $G_t = ||\bar{X_t}-X_t||^2 e^{-C(h(\bar{X_t})+h(X_t))}$ as $G_t=||Y_t||^2Z_tI_t$ where $Y_t = \bar{X_t}-X_t, Z_t = e^{-Ch(\bar{X}_t)}$ and $I_t = e^{-Ch({X}_t)}$.

2. Then to have the dynamic of $G_t$, apply the integration by parts formula whereby:

   • Given the dynamics of $X_t$ and $\bar{X}_t$, we have $||Y_t||^2 = ||\int_0^t \left(\tilde{b}(\bar{X}_s) - {b}({X}_s)\right)ds||^2 $.
   • The dynamic of $I_t$ and $Z_t$ are practically the same, or more precisely the dynamic of $h(\bar{X}_t)$ and $h(X_t)$ are practically the same.
   • The fact that the Brownians are uncorrelated should simply a lot the quadratic variation computation.