Time-changed Brownian Motion Research Articles

η-Stability for stochastic functional differential equation driven by time-changed Brownian motion

This manuscript focuses on a class of stochastic functional differential equations driven by time-changed Brownian motion. By utilizing the Lyapunov method, we capture some sufficient conditions to ensure that the solution for the considered equation is η-stable in the pth moment sense. Subsequently, we present some new criteria of the η-stability in mean square by using time-changed Itô formula and proof by contradiction. Finally, we provide some examples to demonstrate the effectiveness of our main results.

Transportation Inequalities for Stochastic Differential Equations Driven by the Time-Changed Brownian Motion

Transportation Inequalities for Stochastic Differential Equations Driven by the Time-Changed Brownian Motion

On a Class of Distribution Dependent Stochastic Differential Equations Driven by Time-Changed Brownian Motions

On a Class of Distribution Dependent Stochastic Differential Equations Driven by Time-Changed Brownian Motions

McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion

In this paper, we study a class of McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion and impulsive McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion. We will first discuss questions of existence and uniqueness of solutions under some appropriate conditions by some fixed point theorems. Subsequently, we consider various stability properties with respect to initial data and coefficients by some time-changed Gronwall-like inequalities and time-changed Gronwall-like inequalities with jumps, generalizing some known results.

Variance Gamma (nonlocal) equations

Some equations are provided for the Variance Gamma process using the definition other than that based on a time-changed Brownian motion. A new nonlocal equation is obtained involving generalized Weyl derivatives, which is true even in the drifted case. The connection to special functions is in focus, and a space equation for the process is studied. In conclusion, the convergence in distribution of a compound Poisson process to the Variance Gamma process is observed.

The tempered space-fractional Cattaneo equation

We consider the time-fractional Cattaneo equation involving the tempered Caputo space-fractional derivative. There is an increasing interest in the recent literature for the applications of the fractional-type Cattaneo equations to heat transfer models. Our main aim is to discuss the role played by a fractional tempered operator in this framework. We show that the fundamental solution coincides with the probability law of a time-changed Brownian motion, obtained by means of a tempered stable subordinator. We find the characteristic function of this process and we explain the main differences with previous stochastic treatments of the time-fractional Cattaneo equation. We also provide the solution of a Dirichlet problem for the tempered fractional Cattaneo equation by means of the H-Fox function.

On the practical stability with regard to a part of the variables for distribution-dependent SDEs driven by time-changed Brownian motion

Distribution-dependent stochastic differential equations (DDSDEs, in short), also called as McKean–Vlasov SDEs, are a special class of SDEs and have some concrete applications in mean-field control, mean-field games and complex networks. Stability plays fundamental role among the theory and applications for DDSDEs. In this paper, we discuss a class of distribution-dependent stochastic differential equations driven by time-changed Brownian motions (TDDSDEs, in short). The main aim of this paper is to establish the stability for TDDSDEs. Sufficient conditions are provided to guarantee the solutions to be stable in different senses with regard to a part of the variables in terms of a Lyapunov function. Two examples are given to show the usefulness of the obtained theoretical results.

ARMA–GARCH model with fractional generalized hyperbolic innovations

In this study, a multivariate ARMA–GARCH model with fractional generalized hyperbolic innovations exhibiting fat-tail, volatility clustering, and long-range dependence properties is introduced. To define the fractional generalized hyperbolic process, the non-fractional variant is derived by subordinating time-changed Brownian motion to the generalized inverse Gaussian process, and thereafter, the fractional generalized hyperbolic process is obtained using the Volterra kernel. Based on the ARMA–GARCH model with standard normal innovations, the parameters are estimated for the high-frequency returns of six U.S. stocks. Subsequently, the residuals extracted from the estimated ARMA–GARCH parameters are fitted to the fractional and non-fractional generalized hyperbolic processes. The results show that the fractional generalized hyperbolic process performs better in describing the behavior of the residual process of high-frequency returns than the non-fractional processes considered in this study.

Global attracting sets and exponential stability of stochastic functional differential equations driven by the time-changed Brownian motion

In this paper, we study a class of stochastic functional differential equations driven by the time-changed Brownian motion. Some sufficient conditions ensuring the existence of the global attracting sets of considered equations are presented. As applications, some new criteria for the exponential stability in mean square of the considered equations are obtained. Lastly, some examples are investigated to illustrate the theory.

Stochastic solutions for time-fractional heat equations with complex spatial variables

We deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is analyzed and, in particular, some results on its moments, as well as its construction as weak limit of continuous-time random walks, are obtained. The extension of our approach to the higher dimensional case is also provided.

Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching

Abstract The aim of this work is to study the asymptotic stability of the time-changed stochastic delay differential equations (SDDEs) with Markovian switching. Some sufficient conditions for the asymptotic stability of solutions to the time-changed SDDEs are presented. In contrast to the asymptotic stability in existing articles, we present the new results on the stability of solutions to time-changed SDDEs, which is driven by time-changed Brownian motion. Finally, an example is given to demonstrate the effectiveness of the main results.

First passage times for some classes of fractional time-changed diffusions

We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given.

Fractional Cauchy problem on random snowflakes

We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.

Fast pricing of American options under variance gamma

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utilized for the variance gamma model. We develop a new fast method inspired by the quadratic approximation method, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our proposed method with those of the existing methods and show that this method is efficient and accurate for practical use.

A new discretization scheme for one dimensional stochastic differential equations using time change method

We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of an SDE using a time-changed Brownian motion, which dates back to Doeblin (1940). In cases where the diffusion coefficient is bounded and is β-Holder continuous with 0<β≤1, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat SDEs with no strong solution. Our scheme is the first to achieve strong convergence for the case of 0<β<1∕2.

Gaussian fluctuations and a law of the iterated logarithm for Nerman’s martingale in the supercritical general branching process

In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale (Wt)t≥0 to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (also known as Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value W figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman’s martingale, converges to its limit W. More precisely, assuming the existence of a Malthusian parameter α>0 and W0∈L2, we prove a functional central limit theorem for (W−Wt+s)s∈R, properly normalized, as t→∞. The weak limit is a randomly scaled time-changed Brownian motion. Under an additional technical assumption, we prove a law of the iterated logarithm for W−Wt.

On absolute continuity and singularity of multidimensional diffusions

Consider two laws $P$ and $Q$ of multidimensional possibly explosive diffusions with common diffusion coefficient $\mathfrak {a}$ and drift coefficients $\mathfrak {b}$ and $\mathfrak {b}+ \mathfrak {a}\mathfrak {c}$, respectively, and the law $P^{\circ }$ of an auxiliary diffusion with diffusion coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {a}$ and drift coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {b}$. We show that $P \ll Q$ if and only if the auxiliary diffusion $P^{\circ }$ explodes almost surely and that $P\perp Q$ if and only if the auxiliary diffusion $P^{\circ }$ almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.

Hypothesis testing for high-dimensional time series via self-normalization

Self-normalization has attracted considerable attention in the recent literature of time series analysis, but its scope of applicability has been limited to low-/fixed-dimensional parameters for low-dimensional time series. In this article, we propose a new formulation of self-normalization for inference about the mean of high-dimensional stationary processes. Our original test statistic is a U-statistic with a trimming parameter to remove the bias caused by weak dependence. Under the framework of nonlinear causal processes, we show the asymptotic normality of our U-statistic with the convergence rate dependent upon the order of the Frobenius norm of the long-run covariance matrix. The self-normalized test statistic is then constructed on the basis of recursive subsampled U-statistics and its limiting null distribution is shown to be a functional of time-changed Brownian motion, which differs from the pivotal limit used in the low-dimensional setting. An interesting phenomenon associated with self-normalization is that it works in the high-dimensional context even if the convergence rate of original test statistic is unknown. We also present applications to testing for bandedness of the covariance matrix and testing for white noise for high-dimensional stationary time series and compare the finite sample performance with existing methods in simulation studies. At the root of our theoretical arguments, we extend the martingale approximation to the high-dimensional setting, which could be of independent theoretical interest.

Modeling stochastic mortality for joint lives through subordinators

There is a burgeoning literature on mortality models for joint lives. In this paper, we propose a new model in which we use time-changed Brownian motion with dependent subordinators to describe the mortality of joint lives. We then employ this model to estimate the mortality rate of joint lives in a well-known Canadian insurance data set. Specifically, we first depict an individual’s death time as the stopping time when the value of the hazard rate process first reaches or exceeds an exponential random variable, and then introduce the dependence through dependent subordinators. Compared with existing mortality models, this model better interprets the correlation of death between joint lives, and allows more flexibility in the evolution of the hazard rate process. Empirical results show that this model yields highly accurate estimations of mortality compared to the baseline non-parametric (Dabrowska) estimation.

On the Fractional Wave Equation

In this paper we study the time-fractional wave equation of order 1 &lt; ν &lt; 2 and give a probabilistic interpretation of its solution. In the case 0 &lt; ν &lt; 1 , d = 1 , the solution can be interpreted as a time-changed Brownian motion, while for 1 &lt; ν &lt; 2 it coincides with the density of a symmetric stable process of order 2 / ν . We give here an interpretation of the fractional wave equation for d &gt; 1 in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for ν &gt; 2 and also at space-time fractional wave equations.