We define $$M_t=(B_t+t)^{-(B_t+\frac{1}{2}t)}$$ Where $B$ is a Brownian motion.
I must compute the stochastic differential of $M_t$, i.e. $dM_t$. I figured this should be possible with Itô's lemma, but I got stuck rather quickly.
I started off by setting $F(x,y)=(x+y)^{-(x-\frac{1}{2}y)}$, and then computing all necessary derivatives, i.e. $F_x,F_{xx},F_y,F_{yy},F_{xy},F_{yx}$. However, these derivatives get quite ugly, especially the double derivatives. In fact, they get so ugly that it got me doubting whether this was even the right approach. Is there a more ''elegant'' way to computing the stochastic differential of $M_t$?
Any help is appreciated!
$M_t=f(t,B_t)$ where $$f(t,x)=e^{-\left(x+\frac{t}{2}\right)\ln(x+t)}.$$
By Itô formula, $$dM_t=\left(\frac{\partial f}{\partial t}+\frac{\partial ^2f}{\partial x^2}\right)dt+\frac{\partial f}{\partial x}dB_t$$