I'm trying to derive an approximation for the zero-rebate barrier option under the Heston model: $$dS_t=\mu S_tdt+\sqrt{v_t}S_tdW^S_t$$ $$dv_t=\kappa(\bar{v}-v_t)dt+\eta\sqrt{v_t}dW^v_t,\quad d\langle W^S,W^v\rangle_t=\rho dt$$ The payoff of a down-and-out option is: $$\mathcal{C}_T=(S_T-k)\mathbb{I}_{\{S_T\geq K\}}\mathbb{I}_{\{(\max_{0\leq t\leq T}S_t=:)m_T\geq H\}}$$
Under the Black-Scholes dynamics we have the closed-form solution: $$\mathcal{C}_t=S_te^{r\tau}\left(\Phi\left(\frac{\ln(S_t/K)+\nu\tau}{\sigma\sqrt{\tau}}\right)-\left(\frac{H}{S_t}\right)^{1+2r/\sigma^2}\Phi\left(\frac{\ln(H^2/(S_tK))+\nu\tau}{\sigma\sqrt{\tau}}\right)\right) - K\left(\Phi\left(\frac{\ln(S_t/K)+(\nu-\sigma^2)\tau}{\sigma\sqrt{\tau}}\right)-\left(\frac{H}{S_t}\right)^{-1+2r/\sigma^2}\Phi\left(\frac{\ln(H^2/(S_tK))+(\nu-\sigma^2)\tau}{\sigma\sqrt{\tau}}\right)\right)$$ where $\nu=r+\frac{\sigma^2}{2}$ Obviously, under the Heston dynamics we don't have a closed-form solution. However, I would like to know whether there exists approximations allowing to price the down-and-out call option in a similar fashion.
As a starting point, I can use the result of the hitting probability of Heston for $\rho$ and $\mu$ equal to 0, i.e. the probability that during the time interval $[0,t]$ the process $S_t$ was positive, with $x$ and $y$ are initial values, solved via the Fourier transform in $x$: $$\frac{2}{\pi}\int^\infty_0 \frac{\sin (\omega x)}{\omega}\bigg(\frac{\Delta(\omega)e^{-m_-(\omega)\kappa t}}{m_+(\omega)+m_-(\omega)e^{-\Delta(\omega)\kappa t}}\bigg)^{\frac{2\theta\kappa}{\xi^2}}e^{-\frac{2y\kappa}{\xi^2}B(\omega,\kappa t)}d\omega$$ where $B(\omega,\tau)$ is a solution of the Riccati equation $$\frac{\partial B}{\partial \tau}(\omega,\tau)=-B(\omega,\tau)-B(\omega,\tau)^2+\frac{\xi^2 \omega^2}{4\kappa^2}, B(\omega,0)=0;$$ $$\Delta(\omega)=\sqrt{1+(\xi\omega/\kappa)^2}, m_{\pm}(\omega)=\frac{\Delta(\omega)\pm 1}{2}$$
Any hints in this direction as much as completely different solutions are much appreciated.
