  # Multi-parameter Multi-fractional Brownian Motion

## Final Report Summary - MULTIFRACTIONALITY (Multi-parameter Multi-fractional Brownian Motion)

The main objectives studied within the project are complex stochastic systems nonlinear in space and in time. Important properties of these systems, like long-range dependence, are usually modelled by fractional, or fractal, processes. The notion of multi-fractionality generalizes that of fractionality by allowing the characteristics of these phenomena to be dependent on time and state. This leads to a wider class of models, which imitate complex behaviour in a more appropriate way. The notion of multi-fractionality requires a development and an implementation of new mathematical techniques. The project aimed at development of these techniques, as well as statistical and numerical tools to simulate and predict the systems behaviour.
To achieve the goals of the project the following problems were solved.
First, stochastic calculus of fractional and multifractional Brownian motion was advanced to the level needed, including definition of stochastic integral, path regularity, change-of-variable formulas etc. In addition, a theory of spatiotemporal random fields governed by a multi-fractional evolution equation was developed, and limit theorems for the renormalized solutions for the equations with random data were obtained. To develop models further, a harmonizable multifractional stable process and multifractional Rosenblatt process were defined and there properties were studied.
In order for the models to be treated numerically, various approximation results were obtained,
including rate of discrete-time approximations for a mixed Brownian-fractional Brownian stochastic differential equation (SDE), smooth approximations of SDEs with fractional and multifractional noises, including a two-dimensional case, which is particularly important for geophysical applications.
In view of statistical applications, a theory of estimation of scaling functions was developed for
number of mono-fractal and multi-fractal models, and this theory is tested on the real data from
finance. Fractional versions of non-linear, linear and sublinear death processes were introduced and studied. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities. Explicit formulas were derived for the state probabilities of the three death processes and the related probability generating functions and mean values were examined. A useful subordination relation is also proved, allowing to express the death processes as compositions of their classical counterparts with the random time process, which has one-dimensional distribution related to a solution of the fractional diffusion equation.
Different types of processes obtained by composing Brownian motion, fractional Brownian motion and Cauchy processes were studied via derivation partial differential equations satisfied by the distributions of these processes. It is shown that many important partial differential equations, like wave equation, equation of vibration of rods, fractional diffusion equations, higher-order heat equation are satisfied by the laws of some iterated processes.
For mixed stochastic differential equations involving fractional and standard Brownian motions, new existence and uniqueness results were proved. A limit theorem for solutions of mixed stochastic differential equation is established. Approximations by solutions of ordinary stochastic differential equations with random drift are constructed and their convergence is proved. With the help of such approximations, stochastic viability and comparison theorems are established. Equations with unknown drift parameter were studied. We investigated the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein-Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behavior of the fractional derivative of the fractional Brownian motion is established. The conditions of the convergence of the solutions of stochastic differential equations involving fractional and standard Brownian motions with respect to a parameter were established. For a linear combination of fractional Brownian motion and Wiener process, asymptotic behavior of power variations is completely described. Using this description, strongly consistent estimators and asymptotic confidence intervals for Hurst parameter are constructed.
Absolutely continuous stochastic processes that converge to the multifractional Brownian motion in Besov-type spaces are defined. The convergence of solutions of stochastic differential equations with these processes to that of the equation with the multifractional Brownian motion is proved. Absolute continuous random fields that converge to anisotropic fractional and multifractional Brownian sheets in Besov-type spaces are constructed. The rate of convergence of Euler scheme applied to the solution of stochastic differential equation with nonhomogeneous coefficients and non-Lipschitz diffusion was established. A substantial progress in the problem of optimal approximation of a fractional Brownian motion by martingales is made. Existence a unique martingale closest to fractional Brownian motion in a specific sense is proved, and the form of this martingale is found. Numerical results concerning the approximation problem are given.
Existence of strong arbitrage in a fractional Black-Scholes market is proved. All contingent claims in this market model are shown to be weakly hedgeable. A sufficient condition for a contingent claim to be hedgeable is given. The mixed fractional financial models were studied. The problems of quantile and efficient hedging were solved explicitly and asymptotically as well. In both fractional and mixed Cox-Ingersoll-Ross models, bounds for interest rate option price are established with the help of comparison theorems. In Levy financial market model with jumps the conditions which guarantee non-emptiness of stopping domain are found, and it is shown that for pure jump processes the stopping domain is non-empty for a very wide class of payoffs. Sufficient condition are found, when the stopping domain has a threshold structure, i.e. it is an epigraph of some function.
Representations of tensor random fields on the sphere are studied basing on the theory of representations of the rotation group. Specific components of a tensor field are introduced for which spectral decompositions are derived in terms of generalized spherical functions, under the conditions of weak isotropy and mean square continuity of these conponents. The properties of random coeffcients of the decompositions are characterized, including such an important question as conditions of Gaussianity. Limit theorems for the integrals of random fields are stated under integrability conditions on higher order spectral densities.
A class of minimum contrast estimators based on the objective function which is composed with the use of the squared periodogram is introduced, consistency and asymptotic normality of the proposed estimators are stated. The problem of parameter estimation for a process of Ornstein-Uhlenbeck type with reciprocal gamma marginal distribution is studied. Minimum contrast estimators of unknown parameters are derived based on both the discrete and the continuous observations from the process as well as moments based estimators based on discrete observations. It is proved that proposed estimators are consistent and asymptotically normal. The explicit forms of the asymptotic covariance matrices are determined by using the higher order spectral density and cumulants of the RGOU process.
The multi-fractal scenarios for a multi-fractal product of stochastic processes and fields related to the logarithms of the moment generating functions of hyperbolic distributions have been obtained and tested for both real and simulated data. The Renyi functions related to multi-fractal spectrum are obtained in explicit form for both multi-fractal stochastic processes and random fields. The Renyi function for spherical random fields also was developed and studied. The general scaling limit theorem where developed for stochastic integrals contained the so-called Green function and random initial conditions (assuming that Green functions itself satisfied the scaling laws, which is typical for fractional equations, such as equation for Riez-Bessel motion). The spatial scaling for randomly initialized Riesz-Bessel and Burgers equation with quadratic potential is established. The resulting Gaussian and non-Gaussian scenarios are obtained and described.
Four workshops were organized covering the topics developed in the project, short course on rough paths was given, the results were published in 37 papers and 33 presentations were made at the conferences and scientific seminars. Five Ph.D. students involved in the projects have made a considerable progress and are about to finish their theses.

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