I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to stochastic volatility and jump models, Wilmott, 2002.
Is it correct to write the general stock price process in a SVJ model as follows: $$ S_T = \tilde S_T X_T Y_T $$ with $$ \tilde S_T = S_t \exp \left\{ -\frac{1-\rho^2}{2} \int_t^T \sigma^2_u du + \sqrt{1-\rho^2} \int_t^T \sigma_u dZ_u \right\} $$ $$ X_T = \exp \left\{ -\frac{\rho^2}{2} \int_t^T \sigma^2_u du + \rho\int_t^T \sigma_u dW_u \right\} $$ where $\sigma_t$ is the stochastic instantaneous volatility and adapted $W$, $dW dZ = 0$, $\rho$ the correlation between the vol and the stock price, and $$ Y_T = \exp \left \{ \int_t^T dJ_u \right\} $$ is the (compensated) jump process (eg a compensated compound Poisson process)? Also, for simplicity I assume that $J$ is independent of the standard Brownian motions $W$ and $Z$ and the instantaneous volatility $\sigma$
If so (if what I wrote above is correct), then I can write the price of a vanilla option as follows, correct? $$ C^{SVJ}(S_t,K,T) = E_t \left[ C^{BS}\left(S_tX_TY_T,K,T; \bar\sigma \sqrt{1-\rho^2} \right) \right] $$ Here $C^{SVJ}$ is the SVJ model option price, and $C^{BS}$ the Black-Scholes formula, which in the expectation has implied volatility $\bar\sigma \sqrt{1-\rho^2}$ and $$ \bar\sigma = \left( \frac{1}{T-t} \int_t^T \sigma^2_u du \right)^{1/2} $$
