Stochastic processes with statistical applications and fractional stochastic calculus

A time continuous statistical model of multiple chirp signals observed against the background of strongly-dependent stationary Gaussian noise is considered. Strong consistency of the least squares estimate for such trigonometric regression model parameters is proved.

We are interested in fractional stochastic differential equations (FSDEs) with a soft wall, i.e. we are considering FSDEs with a permeable wall. The process defined by FSDE with a soft wall may cross the wall, but it is affected by the force of the selected quantity in the opposite direction. When the process is far from the wall, the force acts weakly. As it approaches or crosses the wall, the force acts stronger. When the process crosses the wall, the current force does not allow him to get away from it. We find conditions under which SDE with a soft wall has a unique solution and construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behaviour of the solution on the repulsion force.

In option pricing we often deal with the options whose payoffs depend on multiple factors such as foreign exchange rates, stocks etc. Usually it leads to a knowledge of the joint distributions and complicated integration procedures. We use an alternative approach which converts the option pricing problem to a dual one. In the talk we give a solution of the problem in the optional semimartingale setting. It is based on joint research with Andrey Pak.

In this talk we will present some recent results on first and second order fluctuations of a class of additive functionals of a fractional Brownian motion. Two different behaviors arise depending on the value of the Hurst parameter. When the Hurst parameter is larger than or equal to 1/3, the limit in distribution turns out to be an independent Brownian motion subordinated to the local time. When the Hurst parameter is less than 1/3 the limit is a constant multiple of the derivative of the local time.

At first we consider a fractional Ornstein-Uhlenbeck process as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. Under specified hypotheses on the forcing process involved in the drift, we can show a sort of short- or long-range dependence useful in some applications. Then, we define a time-changed fractional Ornstein-Uhlenbeck process by composing the fractional Ornstein-Uhlenbeck process with the inverse of a subordinator and we show some convergence results and properties.

We investigate the mixed fractional Brownian motion with trend, that is, a linear model driven by a standard Brownian motion and a fractional Brownian motion with Hurst index H. We develop and compare two approaches to estimation of all unknown model parameters by discrete-time observations. The first algorithm is more traditional: we estimate the parameters of noise using the quadratic variations, while the estimator of trend is obtained as a discretization of a continuous-time estimator of maximum likelihood type. This approach has several limitations, in particular, it assumes that H < 3/4, moreover, some estimators have too low rate of convergence. Therefore, we propose a new method for simultaneous estimation of all parameters, which is based on the ergodic theorem. We also consider the problem of drift parameter estimation in a similar model with two independent fractional Brownian motions. The performance of all estimators is studied numerically.

We review a few facts and proofs on the classical and the non-local versions of Liouville's classical theorem on harmonic functions.

This is joint work with David Berger (TU Dresden) and Eugene Shargorodsky (King's College, London).

We introduce the long-range dependent completely correlated mixed fractional Brownian motion (ccmfBm). This is a process that is driven by a mixture of Brownian motion (Bm) and a long-range dependent completely correlated fractional Brownian motion (fBm, ccfBm) that is constructed from the Brownian motion via the Molchan-Golosov representation. Thus, there is a single Bm driving the mixed process. In the short time-scales the ccmfBm behaves like the Bm (it has Brownian Hölder index and quadratic variation). However, in the long time-scales it behaves like the fBm (it has long-range dependence governed by the fBms Hurst index). We provide a transfer principle for the ccmfBm and use it to construct the Cameron-Martin-Girsanov-Hitsuda theorem and prediction formulas. Finally, we illustrate the ccmfBm by simulations.