Volume 19, Issue 6

We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.

This paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes.

In this paper, a new model for financial processes in form of a space-time fractional diffusion equation of varying order is introduced, analyzed, and applied for some financial data. While the orders of the spatial and temporal derivatives of this equation can vary on different time intervals, their ratio remains constant and thus the global scaling properties of its solutions are conserved. In this way, the model covers both a possible complex short-term behavior of the financial processes and their long-term dynamics determined by its characteristic time-independent scaling exponent. As an application, we consider the option pricing and describe how it can be modeled by the space-time fractional diffusion equation of varying order. In particular, the real option prices of index S&P 500 traded in November 2008 are analyzed in the framework of our model and the results are compared with the predictions made by other option pricing models.

This paper derives the weak-sense Gaussian solution to a family of fractional-in-time and multifractional-in-space stochastic partial differential equations, driven by fractional-integrated-in-time spatiotemporal white noise. Some fundamental results on the theory of pseudodifferential operators of variable order, and on the Mittag-Leffler function are applied to obtain the temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the derived solution.

An identity is presented, in which an expression involving Riemann–Liouville fractional derivative is integrated with respect to the derivative’s order. As a by-product an integral analogue of an identity for a sum of binomial coefficients is derived.

In 1962 Lamperti introduced a transformation that associates to every non-trivial self-similar process a strictly stationary process. This transform has been widely studied for Gaussian processes and in particular for fractional Brownian motion. Our aim is to analyze various properties of the Lamperti transform of the fractional Brownian sheet. We give the stochastic differential equation satisfied by this transform and we represent it as a series of independent Ornstein-Uhlenbeck sheets.

In this paper we derive explicit formulas for the densities of Lévy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer G -function. This allows to efficiently evaluate them numerically. We also compute two-point distribution of wait-first model. Our results show perfect agreement with the Monte Carlo simulations.

In this article, we are concerned with a class of fractional stochastic evolution equations with nonlocal initial conditions in Hilbert spaces. The existence of mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using fractional calculations, Schauder fixed point theorem, stochastic analysis theory, α -order fractional solution operator theory and α -resolvent family theory. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[λσ(u)F⋅(t,x)] $$\partial^\beta_tu_t(x) = -\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u) \stackrel{\cdot}{F}(t,x)] $$ in ( d +1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂tβ $\partial^\beta_t$ is the Caputo fractional derivative, −(−Δ) α /2 is the generator of an isotropic stable process and I 1 − β t is the fractional integral operator. The forcing noise denoted by F⋅(t,x)$\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. The multiplicative non-linearity σ : ℝ → ℝ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane [32]. We first study the existence and uniqueness of the solution of these equations and under suitable conditions on the initial function, we also study the asymptotic behavior of the solution with respect to the parameter λ . In particular, our results are significant extensions of those in [14], [16], [32], and [33].

Time-dependent fractional-derivative problems Dtα u+Au=f $D_t^\alpha u + Au = f$ are considered, where Dtα $D_t^\alpha$ is a Caputo fractional derivative of order α ∈ (0, 1)∪(1, 2) and A is a classical elliptic operator, and appropriate boundary and initial conditions are applied. The regularity of solutions to this class of problems is discussed, and it is shown that assuming more regularity than is generally true—as many researchers do—places a surprisingly severe restriction on the problem.