Case Of Brownian Motion Research Articles

Sample Paths Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian Motion

We analyze the effect of additive fractional noise with Hurst parameter H > {1}/{2} on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.

Enlarged Filtrations and Indistinguishable Processes

We study modification properties of stochastic processes under different probability measures in an initially enlarged filtration setup. For this purpose, we consider several pure-jump Levy processes under two equivalent probability measures and derive the associated martingale compensators with respect to different enlarged filtrations. As our main result, we prove that the obtained martingale processes under different probability measures in our enlarged filtration approach are indistinguishable. In addition, we provide a condition under which the pure-jump result can be carried over to the Brownian motion case. In this context, we show how indistinguishable Brownian motions under different probability measures can be constructed in an enlarged filtration framework. We finally apply our theoretical results to precipitation derivatives pricing under weather forecasts and optimal portfolio selection under future information.

On the potential in non‐Gaussian chain polymer models

In this paper, we investigate the potential for a class of non‐Gaussian processes so‐called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M‐Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.

An inverse first-passage problem revisited: the case of fractional Brownian motion, and time-changed Brownian motion

We revisit an inverse first-passage time (IFPT) problem, in the cases of fractional Brownian motion, and time-changed Brownian motion. Let X(t) be a one dimensional continuous stochastic process starting from a random position and let S(t) be an assigned continuous boundary, such that and F an assigned distribution function. The IFPT problem here considered consists in finding the distribution of η such that the first-passage time of X(t) below S(t) has distribution F. We study this IFPT problem for fractional Brownian motion and a constant boundary we also obtain some extension to other Gaussian processes, for one, or two, time-dependent boundaries.

Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda _{1}^{(n)}(t),\dots , \lambda _{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu ^{(n)}=(\mu _{t}^{(n)}; t\geq 0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu ^{(n)}$ converges in probability to a deterministic limit $\mu $, in the topology of uniform convergence over compact sets. We show that the process $\mu $ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers’ equation. Our results extend those of Rogers and Shi [14] for the free Brownian motion and Pardo et al. [12] for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

Statistical inference for Vasicek-type model driven by self-similar Gaussian processes

In this paper, we consider the drift parameters estimation problem for the Vasicek-type model defined aswhere a < 0 and are considered as unknown drift parameters and Gt is a self-similar Gaussian process with index . We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distributions of our estimators of a and of b based on the observation as . Our approach extend the result of Xiao and Yu (2017) for the case when G is a fractional Brownian motion with Hurst parameter . We also discuss the cases of sub-fractional Browian motion and bi-fractional Brownian motion. The conclusion can also be extended to more general self-similarity processes, such as Hermite processes.

Fine properties of fractional Brownian motions on Wiener space

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter H under the (p,r)-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and Üstünel, and show non-differentiability, modulus of continuity, law of the iterated logarithm (LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.

State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions

The abrupt changes that are ubiquitous in physical and natural systems are often well characterized by shot noise with a state dependent recurrence frequency and jump amplitude. For such state dependent behavior, we derive the transition probability for both the It\^o and Stratonovich jump interpretations, and subsequently use the transition probability to pose a master equation for the jump process. For exponentially distributed inputs, we present a novel class of transient solutions, as well as a generic steady state solution in terms of a potential function and the Pope-Ching formula. These new results allow us to describe state dependent jumps in a double well potential for steady state particle dynamics, as well as transient salinity dynamics forced by state dependent jumps. Both examples showcase a stochastic description that is more general than the limiting case of Brownian motion to which the jump process defaults in the limit of infinitely frequent and small jumps. Accordingly, our analysis may be used to explore a continuum of stochastic behavior from infrequent, large jumps to frequent, small jumps approaching a diffusion process.

Bayesian model selection with fractional Brownian motion

We implement Bayesian model selection and parameter estimation for the case of fractional Brownian motion with measurement noise and a constant drift. The approach is tested on artificial trajectories and shown to make estimates that match well with the underlying true parameters, while for model selection the approach has a preference for simple models when the trajectories are finite. The approach is applied to observed trajectories of vesicles diffusing in Chinese hamster ovary cells. Here it is supplemented with a goodness-of-fit test, which is able to reveal statistical discrepancies between the observed trajectories and model predictions.

An efficient and stable method for short maturity Asian options

In this paper, we develop a Markov chain‐based approximation method to price arithmetic Asian options for short maturities under the case of geometric Brownian motion. It has the advantage of being a closed‐form approximation involving only matrices. It is an accurate, efficient, and stable method for the pricing and hedging of short maturity arithmetic Asian options for which previous methods in the literature have shown either slower convergence or instabilities in hedging parameters. We demonstrate that this method is as good as and sometimes better than existing approximation methods in the literature.

Time since maximum of Brownian motion and asymmetric Lévy processes

Motivated by recent studies of record statistics in relation to strongly correlated time series, we consider explicitly the drawdown time of a Lévy process, which is defined as the time since it last achieved its running maximum when observed over a fixed time period . We show that the density function of this drawdown time, in the case of a completely asymmetric jump process, may be factored as a function of t multiplied by a function of T − t. This extends a known result for the case of pure Brownian motion. We state the factors explicitly for the cases of exponential down-jumps with drift, and for the downward inverse Gaussian Lévy process with drift.

Convergence of a finite volume scheme for a stochastic conservation law involving a &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$Q$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-brownian motion

We study a time explicit finite volume method for a first order conservation law with a multiplicative source term involving a \begin{document}$Q$\end{document} -Wiener process. After having presented the definition of a measure-valued weak entropy solution of the stochastic conservation law, we apply a finite volume method together with Godunov scheme for the space discretization, and we denote by \begin{document}$\{u_{\mathcal{T}, k}\}$\end{document} its discrete solution. We present some a priori estimates including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities for the discrete solution. We show that the discrete solution \begin{document}$\{u_{\mathcal{T}, k}\}$\end{document} converges along a subsequence to a measure-valued entropy solution of the conservation law in the sense of Young measures as the maximum diameter of the volume elements and the time step tend to zero. Some numerical simulations are presented in the case of the stochastic Burgers equation. The empirical average turns out to be a regularization of the deterministic solution; moreover, the variance in the case of the \begin{document}$Q$\end{document} -Brownian motion converges to a constant while that in the Brownian motion case keeps increasing as time tends to infinity.

An Efficient and Stable Method for Short Maturity Asian Options

In this paper, we develop a Markov chain-based approximation method to price arithmetic Asian options for short maturities under the case of geometric Brownian motion. It has the advantage of being a closed-form approximation involving only matrices. It is an accurate, efficient, and stable method for the pricing and hedging of short maturity arithmetic Asian options for which previous methods in the literature have shown either slower convergence or instabilities in hedging parameters. We also consider the pricing and hedging of floating-strike Asian options and fixed-strike in-progress Asian options, and demonstrate that our method is as good as and sometimes better than existing approximation methods in the literature.

A generalization of Strassen's law and L&amp;eacute;vy's modulus of continuity for $\boldsymbol{G}$-Brownian motion

The Strassen theorem and Levy's modulus of continuity for Brownian motion case play important roles in the probability theory. Muller gave the unification form on them in 1981. In this paper, the Strassen's law and Levy's modulus of continuity for $G$-Brownian motion are extended to the general case. It can be seen as the unification of Strassen's law and Levy's modulus of continuity for $G$-Brownian motion.

Maximal Exponential Inequalities for Certain Diffusion Processes

In this paper, we prove maximal exponential inequalities for a class of diffusion processes under certain conditions. The present results extend the well-known maximal inequalities of the power-type proved in the case of a Brownian motion, and in the geometric Brownian motion case. The method of proof is essentially based on explicit forms of optimal stopping problems related to those treated by Shepp and Shiryaev [Ann. Probab. Appl., 3 (1993), pp. 631--640], but considered here with a multiplicative reward functional. The key tool in the proof of one of our main results is a comparison principle for solutions of a system of first-order nonlinear differential equations. This is one of the novel features in this paper.

On divergence of expectations of the Feynman–Kac type with singular potentials

Motivated by the work of Baras-Goldstein (1984), we discuss when expectations of the Feynman-Kac type with singular potentials are divergent. Underlying processes are Brownian motion and $\alpha$-stable process. In connection with the work of Ishige-Ishiwata (2012) concerned with the heat equation in the half-space with a singular potential on the boundary, we also discuss the same problem in the half-space for the case of Brownian motion.

Mean first-passage times of non-Markovian random walkers in confinement.

The first-passage time, defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes. Its importance comes from its crucial role in quantifying the efficiency of processes as varied as diffusion-limited reactions, target search processes or the spread of diseases. Most methods of determining the properties of first-passage time in confined domains have been limited to Markovian (memoryless) processes. However, as soon as the random walker interacts with its environment, memory effects cannot be neglected: that is, the future motion of the random walker does not depend only on its current position, but also on its past trajectory. Examples of non-Markovian dynamics include single-file diffusion in narrow channels, or the motion of a tracer particle either attached to a polymeric chain or diffusing in simple or complex fluids such as nematics, dense soft colloids or viscoelastic solutions. Here we introduce an analytical approach to calculate, in the limit of a large confining volume, the mean first-passage time of a Gaussian non-Markovian random walker to a target. The non-Markovian features of the dynamics are encompassed by determining the statistical properties of the fictitious trajectory that the random walker would follow after the first-passage event takes place, which are shown to govern the first-passage time kinetics. This analysis is applicable to a broad range of stochastic processes, which may be correlated at long times. Our theoretical predictions are confirmed by numerical simulations for several examples of non-Markovian processes, including the case of fractional Brownian motion in one and higher dimensions. These results reveal, on the basis of Gaussian processes, the importance of memory effects in first-passage statistics of non-Markovian random walkers in confinement.

Bounds for expected maxima of Gaussian processes and their discrete approximations

The paper deals with the expected maxima of continuous Gaussian processes that are Hölder continuous in -norm and/or satisfy the opposite inequality for the -norms of their increments. Examples of such processes include the fractional Brownian motion and some of its “relatives” (of which several examples are given in the paper). We establish upper and lower bounds for and investigate the rate of convergence to that quantity of its discrete approximation . Some further properties of these two maxima are established in the special case of the fractional Brownian motion.

Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt=θXtdt+dGt,t≥0 with an unknown parameter θ>0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimator θ˜t of θ based on the observation {Xs,s∈[0,t]} as t→∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H∈(0,1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.

Dimension of fractional Brownian motion with variable drift

Let X be a fractional Brownian motion in \({\mathbb {R}}^d\). For any Borel function \(f:[0,1] \rightarrow {\mathbb {R}}^d\), we express the Hausdorff dimension of the image and the graph of \(X+f\) in terms of f. This is new even for the case of Brownian motion and continuous f, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of f is a self-affine McMullen–Bedford carpet, we obtain an explicit formula for the dimension of the graph of \(X+f\) in terms of the generating pattern. In particular, we show that it can be strictly bigger than the maximum of the Hausdorff dimension of the graph of f and that of X. Despite the random perturbation, the Minkowski and Hausdorff dimension of the graph of \(X+f\) can disagree.