Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying:
$dB(t)=r(t)B(t)dt$ and
$dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$
where $r,b,\sigma$ deterministic, positive and bounded.
I know that the price of european call option with strike price $K$ is given by
$\tilde{V}(t)=U(t,S(t))$ and
$U(t,S(t)=\tilde{E}[max[S(0)e^{\int_t^T \sigma(s)d\tilde{W}(s)-\dfrac{1}{2}\int_t^T \sigma^2(s)ds}-Ke^{-\int_t^T r(s)ds}],0]$
where $\tilde{E}$ the expected value under the new probability measure $\tilde{P}$ and $\tilde{W}$ is a wiener process. $S(t)$ here is the discounted price of the stock.
Can someone help me how to find the PDE and the replicating strategy of the european call option?
