Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u s_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the process $S$ at time $t$, i.e.: $$ \mathbb{P}(S_T > x) = \frac{1}{2} + \frac{1}{\pi}\int_{0}^{\infty} Re\bigg(\frac{e^{-i u \log(x)} \Phi_{T,0}(u)}{i u }\bigg) du $$ (can be found in Carr Madan FFT approach to option pricing for example).

So, once we know the characteristic function of the process, we have all the information on the distribution and on the density of the process at a given time $t$, since we need only the information of the initial price $S_0$, and the initial volatility $V_0$ (here I am using an easy possible example, the Heston model). $$ \frac{dS_t}{S_t} = r dt + \sqrt{V_t} dW_t$$$$ dV_t = \kappa ( \theta - V_t) + \sqrt{V_t} dB_t$$ $$d<W,B>_t = \rho $$ Is it possible, using the characteristic function, to compute the following transition probability? $$ \mathbb{P}(S_T \leq x | S_t = y) $$ Of course, in the condition where I know $S_t$ and $V_t$ it's easy using the characteristic function, but I know only $S_t$ and I don't have any information on $V_t$.



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