## Project Details

### Description

The main part of the project is devoted to complete the non adapted stochastic calculus for Levy processes, the statistical tools to fit models and the practical fit of financial time series. Specifically,

a) Stochastic calculus: Strook formula in order to get the kernel of some chaotic decomopositions. Aplications to local time and functionals of the Levy process. Study of a Kabanov formula for the product of Teugels' martingales. Definition an properties of the Skorohod integral in this setup. Ito's formula. Existence and uniqueness of solucions of linear stochastic differential equations driven by a simple Levy process.

b) Statistical analysis of Levy processes. Estimation of the number of jumps using wavelets techniques. Estimation of the parameters of a simple Levy process (drift and diffusion parameters of the Brownian part, and of the parameters of the Poisson processes).

c) Fit of financial time series which likely have discontinuities using simple Levy processe.

A second set of problem that we will deal are related to stochastic differential-algeraic equations (existence and uniqueness of solutions) beginning with the case where the equation is driven by a white noise (formal derivative of a Brownian motion) and extending the results to more generals situations (Levy processes, semimartingales,) Further the applications of this equations will be studied.

Finally the third block is devoted to the study of multiple Stratonovich integrals with respect to the fractionary Brownian motion (which for H=1/2 is an ordinary Brownian motion, hence, a Levy process). The concrete objective are: first to investigate is it is possible to find a nice space for the multiple integral using a measure which gives mass to the diagonals. Second, to build the multiple integral for H

a) Stochastic calculus: Strook formula in order to get the kernel of some chaotic decomopositions. Aplications to local time and functionals of the Levy process. Study of a Kabanov formula for the product of Teugels' martingales. Definition an properties of the Skorohod integral in this setup. Ito's formula. Existence and uniqueness of solucions of linear stochastic differential equations driven by a simple Levy process.

b) Statistical analysis of Levy processes. Estimation of the number of jumps using wavelets techniques. Estimation of the parameters of a simple Levy process (drift and diffusion parameters of the Brownian part, and of the parameters of the Poisson processes).

c) Fit of financial time series which likely have discontinuities using simple Levy processe.

A second set of problem that we will deal are related to stochastic differential-algeraic equations (existence and uniqueness of solutions) beginning with the case where the equation is driven by a white noise (formal derivative of a Brownian motion) and extending the results to more generals situations (Levy processes, semimartingales,) Further the applications of this equations will be studied.

Finally the third block is devoted to the study of multiple Stratonovich integrals with respect to the fractionary Brownian motion (which for H=1/2 is an ordinary Brownian motion, hence, a Levy process). The concrete objective are: first to investigate is it is possible to find a nice space for the multiple integral using a measure which gives mass to the diagonals. Second, to build the multiple integral for H

Status | Finished |
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Effective start/end date | 1/12/03 → 30/11/06 |

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