## Abstract

This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process S = (S_{t})_{t≥o} is given by dS_{t} =m(θ_{t})S_{t} dt + ν(θ _{t})S_{t} dB_{t}, where B = (B_{t}) _{t≥0} is a Brownian motion, ν is a positive function and θ = (θ_{t})_{t≥0} is a cádlág strong Markov process. The random process θ is unobservable. We assume also that the asset price S_{t} is observed only at random times 0 < τ_{1} < τ_{2} < ⋯. This is an appropriate assumption when modeling high frequency financial data (e.g., tick-by-tick stock prices). In the above setting the problem of estimation of θ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process (τ_{k}, log S_{τk}). While quite natural, this problem does not fit into the "standard" diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for θ_{t}, based on the observations of (τ_{k}, log S _{τk})_{k≥1}. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy.

Original language | English |
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Pages (from-to) | 1633-1652 |

Number of pages | 20 |

Journal | Annals of Applied Probability |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2006 |

## Keywords

- Discrete observations
- Nonlinear filtering
- Volatility estimation