In the book "Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing" of Henry-Labordère, page 77 :
Considering the market model composed of $n$ Ito processes :
$$ dx^i_t=b^i(t,x)dt+\sigma^i(t,x)dW_t^i, $$
with
- $d<W^i,W^j>_t=\rho_{ij}(t)$.
Under the risk neutral probability, assuming the risk free rate is null, the price at $t$ of an European option on $x$ with payoff $f(x_T)$ can be written as :
$$ C(\alpha, t, T)=\mathbb{E}(f(x_T)|\mathcal{F}_t), $$
with
- $\alpha$ the initial condition.
Using Ito formula on $f(x_T)$ we get :
$$ f(x_T) = f(\alpha) + \int^T_t Df(x_s)ds + \int^T_t \sum_{i=1}^n \frac{\partial f(x)}{\partial x^i}\sigma^i(t,x)dW^i_t, $$
with (using the Einstein convention)
- $ D=b^i(t,x)\frac{\partial}{\partial x^i}+\frac{1}{2}\sigma^i(t,x)\sigma^j(t,x)\rho_{ij}(t)\frac{\partial^2}{\partial x^i\partial x^j} $
Injecting it in the first formula we obtain :
$$ C(\alpha, t, T)=f(\alpha)+\mathbb{E}\left(\int^T_t Df(x_s)ds|\mathcal{F}_t\right). $$
Differentiating over $T$ :
$$ \frac{\partial C(\alpha, t, T)}{\partial T}=\mathbb{E}\left(Df(x_T)|\mathcal{F}_t\right)=\int \prod\limits_{i=1}^{n} dx^iDf(x)p(T,x|t,\alpha), $$
with
- $p(.|.)$ the conditional probability.
Till there everything is ok. But then, he says he does an integration by parts and "discarding the surface term" (i don't even know what he wants to say here), he finally gets :
$$ \frac{\partial C(\alpha, t, T)}{\partial T}=\int \prod\limits_{i=1}^{n} dx^if(x)D_2 p(T,x|t,\alpha), $$
with
- $ D_2=-\frac{\partial}{\partial x^i}b^i(t,x)+\frac{1}{2}\rho_{ij}(t)\frac{\partial^2}{\partial x^i\partial x^j}\sigma^i(t,x)\sigma^j(t,x). $
It seems so simple but I can't find how he find the last equality, if someone can help ! Thank you very much !