# Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector of unknown parameters. I need to esimate vector of unknown parameters $\theta$ of characteristic function.

I tried to find a PDF using the inverse Fourier transform to use the maximum likelihood method, but the characteristic function is too complicated for that. I also thought about building the empirical characteristic function using the time series of assets and to estimate parameters using the least square method, but I do not know how to build the empirical characteristic function, because time series is not just a sample of random variables, it is a random process that depends on time.

• you seems crossposted this question to several conferences on stackexchange... it it really so wide? would it be possible to choose the most appropriate conference? so that if people answered it in one conf, then others don't spend time answering in other conf. – Denis Dec 12 '13 at 2:16
• Is the process p-Markovian? Is there a log P over which the process does not depend on it s past? – Drmanifold Dec 16 '13 at 18:55
• No, this process is non Markovian. Basically, I have stochastic model for an assets with stochastic volatility (I've tryied to estimate Heston model and BNS model using theirs CF). For example in Heston model it is proved, that process $\{A_1 - A_0, A_2 - A_1,...,A_n - A_{n-1}\}$ is Markovian. – Dmitry Pavliv Dec 16 '13 at 19:32