Consider the following boundary value problem in the domain $[0,T]$ x $R$ for an unknown function F.
$\frac{\partial F}{\partial t}(t,x) + \mu(t,x)\frac{\partial F}{\partial x}(t,x) + \frac {1}{2}\sigma^2(t,x)\frac{\partial^2 F}{\partial t^2}(t,x) = 0$
$F(T,x) = \Phi(x)$
$\Phi, \mu, \sigma$ are assumed to be known functions.
Derive a stochastic representation formula for this problem. Make sure it is clear at which points the functions should be evaluated.
So this is how I think you do this, but I need some help understanding the steps.
We first assume that it actually exists such stochastic representation that is the solution to the SDE
$dX_s = \mu(t,X_s)ds + \sigma(t,X_s)dB_s$
$X_t = x$
And the infinitesimal generator $\mathcal{A}$ of X is
$\mu(t,x)\frac{\partial F}{\partial x}(t,x) + \frac {1}{2}\sigma^2(t,x)\frac{\partial^2 F}{\partial t^2}(t,x)$
so we can rewrite the the PDE as
$\frac{\partial F}{\partial t} + \mathcal{A}F(t,x) = 0$ (or should it be a minus sign)
$F(T,x) = \Phi(x)$ $(\star)$
And now we apply the Itô formula on $F(s,X_s)$ and this step I don't understand (if someone could explain it I would be very pleased, I know the Itô formula but on this problem I don't get it), but if I'm correct it we get
$F(T,X_T) = F(t,X_t) + \int^T_t \big(\frac{\partial F}{\partial t}(s,X_s) + \mathcal{A}F(s, X_s)\big)ds + \int^T_t\sigma(s, X_s)\frac{\partial F}{\partial x}(s,X_s)dB_s$ (where the ds-integral is $0$ and $F(T,X_t) = \Phi(X_T)$ by assumption)
So now we take expectations on both sides and we get:
$E_{t,x}[\Phi(X_T)] = F(t, x) + E_{t,x}[\int^T_t \sigma(s,X_s)\frac{\partial F}{\partial x}(s,X_s)dB_s]$
Where the integral is $0$ if $\sigma(s,X_s)\frac{\partial F}{\partial x}(s,X_s)$ is sufficiently nice.
So we then have our stochastic representation of $F(t,x) = E_{t,x}[\Phi(X_T)]$
So, how do you apply the Itô formula on $(\star)$? Also I'm a bit confused if it should be a minus sign at $(\star)$ aswell, I think it should? Is this the Kolmogorovs backward equation? And if you instead have a initial condition in the PDE you get the foward equation? I think we should also be able to this if you add some nice function to the PDE, say $r(x)$. How would that change the derivation of this stochastic representation?