SPDE with generalized drift and fractional-type noise

We consider a stochastic partial differential equation involving a second order differential operator whose drift is discontinuous. The equation is driven by a Gaussian noise which behaves as a Wiener process in space and the time covariance generates a signed measure. This class includes the Brownian motion, fractional Brownian motion and other related processes. We give a necessary and sufficient condition for the existence of the solution and we study the path regularity of this solution.     

Singular fractional differential equations

Fractional differential equations have received increasing attention during recent years since the behavior of many physical systems can be properly described using the fractional order system theory. By fractional analog for Duhamel principle we give the existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions. In order to find the explicit solutions we use integral representation of the solution obtained via Laplace and Fourier transforms in succession and their inverses. We deal with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes, extending the results in viscoelasticity, continuum random walk, seismology, continuum mechanics and many other branches of life and science. The main task is finding existence-uniqueness results like in the case of evolution equations with entire derivatives. By examining the fractional evolution equations it turns out that they lead to till now known results from the evolution equations with entire derivatives in limiting case. They give more, behavior of the solution when order of derivatives are inside the intervals of entire points. In this way we can follow the influence of the operators generated by entire derivative in many fractional time evolution PDEs especially with singular initial data, and non-Lipschitz's nonlinear term. Apart from evolution equations we prove also an existence-uniqueness result for an initial value problem with singularities for linear and nonlinear fractional elliptic equation of Helmholtz type and fractional order α, where 1 < Re(α) ≤ 2, with respect to the one variable from R ₊. As a framework, we employ also Colombeau algebra of generalized functions containing fractional derivatives and operations among them in order to deal with the fractional equations with singularities. We apply the same techniques to the fractional Laplace and Poisson equation linear and nonlinear ones. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large scale behavior of wavelet coefficients of non-linear subordinated processes with long memory

We study the asymptotic behavior of wavelet coefficients of random processes with long memory. These processes may be stationary or not and are obtained as the output of non-linear filter with Gaussian input. The wavelet coefficients that appear in the limit are random, typically non-Gaussian and belong to a Wiener chaos. They can be interpreted as wavelet coefficients of a generalized self-similar process.     

Integration with respect to Fractal Functions and Stochastic Calculus II

The link between fractional and stochastic calculus established in part I of this paper is investigated in more detail.We study a fractional integral operator extending the Lebesgue – Stieltjes integral and introduce a related concept of stochastic integral which is similar to the so – called forward integral in stochastic integration theory.The results are applied to ODE driven by fractal functions and to anticipative SDE whose noise processes possess absolutely continuous generalized covariation processes.A survey on this approach may be found in [21].     

Asymptotic stability of nonlinear fractional neutral singular systems

In this paper, fractional calculus has been introduced into neutral singular systems. The (asymptotical) stability and (generalized) Mittag-Leffler stability of nonlinear fractional neutral singular systems with Caputo and Riemann-Liouville derivatives are studied, respectively. Several sufficient conditions guaranteeing stability of such systems are established by using the Lyapunov direct method.     

Diffusions with singular drift related to wave functions

Summary Schrödinger equations are equivalent to pairs of mutually time-reversed non-linear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space-time Sobolev space.By means of Nagasawa's multiplicative functionalN _s ^t , a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived.The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solutions of a Feynman-Kac integral equation with a locally integrable potential function.     

A note on arbitrage‐free pricing of forward contracts in energy markets

Arbitrage theory is used to price forward (futures) contracts in energy markets, where the underlying assets are non‐tradeable. The method is based on the so‐called ‘fitting of the yield curve’ technique from interest rate theory. The spot price dynamics of Schwartz is generalized to multidimensional correlated stochastic processes with Wiener and Lévy noise. Findings are illustrated with examples from oil and electricity markets.     

Uniqueness for solutions of Fokker–Planck equations related to singular SPDE driven by Lévy and cylindrical Wiener noise

We generalize recent results concerning uniqueness of solutions to Fokker–Planck equations (FPE) related to singular Hilbert space-valued SPDE from the (cylindrical) Wiener noise case to the case of SPDE driven by noise with jumps. Using a different space of test functions, we can relax the usual integrability assumptions and obtain more general uniqueness results for FPE, even in the case of SPDE driven by Wiener noise.     

Multi-valued,singular stochastic evolution inclusions

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast-diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-⁎ mean ergodic invariant measure.     

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.     

A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force

This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang-Gao-Tenreiro Machado-Baleanu and Yang-Abdel-Aty-Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional-exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang-Abdel-Aty-Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang-Abdel-Aty-Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.     

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.     

Generalized variational calculus in terms of multi-parameters fractional derivatives

In this paper, we briefly introduce two generalizations of work presented a few years ago on fractional variational formulations. In the first generalization, we consider the Hilfer’s generalized fractional derivative that in some sense interpolates between Riemann–Liouville and Caputo fractional derivatives. In the second generalization, we develop a fractional variational formulation in terms of a three parameter fractional derivative. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed.     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

Absolute continuity of Wiener processes

By a Wiener process we mean a countably additive random measure taking independent values on disjoint sets. Given two continuous Wiener processes we give their decomposition into weakly equivalent and mutually singular parts.     

Generalized solution of some parabolic equations with a random drift

Some stochastic partial differential equations arising from a turbulent transport model are studied using Hida's theory of Brownian functionals. For the spatially homogeneous case, the solutions are constructed as a regular or generalized Brownian functional, depending on a small parameter. The regularity property of such solutions is also determined. However, for the spatially nonhomogeneous equations, only generalized solutions in a series form involving iterated singular Wiener integrals are found.This work was supported in part by NSF Grant DMS-87-02236.     

Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method

Physical processes with memory and hereditary properties can be best described by fractional differential equations due to the memory effect of fractional derivatives. For that reason reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation.     

Extending the D’alembert solution to space–time Modified Riemann–Liouville fractional wave equations

In the realm of complexity, it is argued that adequate modeling of TeV-physics demands an approach based on fractal operators and fractional calculus (FC). Non-local theories and memory effects are connected to complexity and the FC. The non-differentiable nature of the microscopic dynamics may be connected with time scales. Based on the Modified Riemann–Liouville definition of fractional derivatives, we have worked out explicit solutions to a fractional wave equation with suitable initial conditions to carefully understand the time evolution of classical fields with a fractional dynamics. First, by considering space–time partial fractional derivatives of the same order in time and space, a generalized fractional D’alembertian is introduced and by means of a transformation of variables to light-cone coordinates, an explicit analytical solution is obtained. To address the situation of different orders in the time and space derivatives, we adopt different approaches, as it will become clear throughout this paper. Aspects connected to Lorentz symmetry are analyzed in both approaches.