Limits Of Random Walks Research Articles

Brownian motion in a vector space over a local field is a scaling limit

For any natural number d, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a d-dimensional vector space V over a local field K. Just as the Laplace operator on L2(Rd) is the infinitesimal generator of Brownian motion with state space Rd, the Vladimirov–Taibleson operator on L2(V) is the infinitesimal generator of real-time Brownian motion with state space V. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted V to be the p-adic numbers.

Interpreting the modern history of finance theory from Henri Poincaré’s perspective

Henri Poincaré is well known for his work in mathematics, physics, and the philosophy of science. From 1900 onwards, Poincaré had the opportunity to become a forerunner of the Efficient Market Hypothesis (EMH), a concept that was formally developed 65 years later, but he never seised it. We show that Poincaré’s view on randomness was incompatible with the EMH framework. Although he did not work directly on financial topics, his work has helped to understand the limits of random walks in asset pricing; a way to reconcile the mathematician with the field of finance.

Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains

Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations.

Fractional diffusion and Fokker-Planck equationsare widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equationsfrom the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equationsbased on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equationsfor the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equationsare unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

On the Causality Problem in Focused Particle Transport

When perpendicular diffusion is included into the focused transport equation, certain assumptions of the perpendicular diffusion coefficient (including the well-known fieldline random walk limit) clearly lead to faster-than-light motion for relativistic particles, and transport that does not preserve causality in general. In this work we show that this discrepancy can be solved by including the effect of turbulence on particle drifts, especially the streaming term present in the focused transport equation, leading to a so-called drift reduction factor. We discuss this drift reduction factor, how it solves the above-mentioned causality problem, and its general implications for particle transport. Under certain simplifying assumptions we show the interplay between particle streaming (as a drift process) and perpendicular diffusion, and how these processes can be consistently coupled through the turbulent magnetic field.

Discrete approximation to Brownian motion with varying dimension in bounded domains

In this paper, we study a discrete approximation to Brownian motion with varying dimension (BMVD) introduced by Chen and Lou in their 2019 paper with continuous time random walks on square lattices. The state space of BMVD contains a 2-dimensional component, a 3-dimensional component, and a “darning point” which joins these two components. Such a state space is equipped with the geodesic distance under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous-time reversible random walks with exponential holding times.

Scaling limits of branching random walks and branching-stable processes

AbstractBranching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanisms for point processes. Here we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results obtained during the past decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.

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.

Scaling limits of tree-valued branching random walks

We consider a branching random walk (BRW) taking its values in the b-ary rooted tree Wb (i.e. the set of finite words written in the alphabet {1,…,b}, with b≥2). The BRW is indexed by a critical Galton–Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a γ-stable law, γ∈(1,2]. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on Wb (reflection at the root of Wb and otherwise: probability 1∕2 to move closer to the root of Wb and probability 1∕(2b) to move away from it to one of the b sites above). We denote by Rb(n) the range of the BRW in Wb which is the set of all sites in Wb visited by the BRW. We first prove a law of large numbers for #Rb(n) and we also prove that if we equip Rb(n) (which is a random subtree of Wb) with its graph-distance dgr, then there exists a scaling sequence (an)n∈N satisfying an→∞ such that the metric space (Rb(n),an−1dgr), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

Pricing and Hedging Options Conditional on Market Activity

We replicate and price European options on stocks modeled by time-changed geometric Brownian motion. The time change is obtained as the integrated intensity of random arrival times of price changes of the underlier over the life of the option. For European call options we obtain explicit hedging and pricing formulas. This approach is motivated by the need to connect option prices directly to the microstructure properties of the limit order book that determines tick-by-tick stock price changes. The continuous time model is obtained as an appropriate limit of discrete time random walks with random jump times, in the limit of infinitely many independent representative agents.

Perpendicular Diffusion of Energetic Particles: A Complete Analytical Theory

Abstract Over the past two decades scientists have significantly improved our understanding of the transport of energetic particles across a mean magnetic field. Due to test-particle simulations, as well as powerful nonlinear analytical tools, our understanding of this type of transport is almost complete. However, previously developed nonlinear analytical theories do not always agree perfectly with simulations. Therefore, a correction factor a 2 was incorporated into such theories with the aim to balance out inaccuracies. In this paper a new analytical theory for perpendicular transport is presented. This theory contains the previously developed unified nonlinear transport theory, the most advanced theory to date, in the limit of small Kubo number turbulence. New results have been obtained for two-dimensional turbulence. In this case, the new theory describes perpendicular diffusion as a process that is sub-diffusive while particles follow magnetic field lines. Diffusion is restored as soon as the turbulence transverse complexity becomes important. For long parallel mean-free paths, one finds that the perpendicular diffusion coefficient is a reduced field line random walk limit. For short parallel mean-free paths, on the other hand, one gets a hybrid diffusion coefficient that is a mixture of collisionless Rechester & Rosenbluth and fluid limits. Overall, the new analytical theory developed in the current paper is in agreement with heuristic arguments. Furthermore, the new theory agrees almost perfectly with previously performed test-particle simulations without the need of the aforementioned correction factor a 2 or any other free parameter.

Large deviations of currents in diffusions with reflective boundaries

We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman–Kac equation underlying the evolution of generating functions. Our results generalize recent works on density-type observables, and are illustrated for an N-particle single-file diffusion on a ring, which can be mapped to a reflected N-dimensional diffusion.

Brownian motion on stable looptrees

In this article, we introduce Brownian motion on α-stable looptrees using resistance techniques, where α∈(1,2). We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of rα and t−αα+1 respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

Relation between generalized diffusion equations and subordination schemes.

Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology, and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.

Limits of random walks with distributionally robust transition probabilities

We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Levy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial Levy process.

Heuristic Description of Perpendicular Transport

The problem of the transport of energetic particles across a mean magnetic field is known since more than 50 years. Previous attempts to describe perpendicular transport theoretically were either based on complicated non-linear theories or computationally expensive simulations. In either case it remained unclear how particles really experience perpendicular transport. In this paper I will present a heuristic approach to solve this problem. Simple arguments will lead to several formulas for the perpendicular diffusion coefficient. These formulas include well-known cases such as compound sub-diffusion and the field line random walk limit but also newer cases such as the collisionless Rechester and Rosenbluth limit. Furthermore, analytical theories such as NLGC and UNLT theories contain a correction factor a2 which is often assumed to be 1/3. The heuristic approach discussed in this article explains this value as well.

Mixture of Gaussians in the open quantum random walks

We discuss the Gaussian and the mixture of Gaussians in the limit of open quantum random walks. The central limit theorems for the open quantum random walks under certain conditions were proven by Attal et al (Ann Henri Poincare 16(1):15–43, 2015) on the integer lattices and by Ko et al (Quantum Inf Process 17(7):167, 2018) on the crystal lattices. The purpose of this paper is to investigate the general situation. We see that the Gaussian and the mixture of Gaussians in the limit depend on the structure of the invariant states of the intrinsic quantum Markov semigroup whose generator is given by the Kraus operators which generate the open quantum random walks. Some concrete models are considered for the open quantum random walks on the crystal lattices. Due to the intrinsic structure of the crystal lattices, we can conveniently construct the dynamics as we like. Here, we consider the crystal lattices of $$\mathbb {Z}^2$$ with intrinsic two points, hexagonal, triangular, and Kagome lattices. We also discuss Fourier analysis on the crystal lattices which gives another method to get the limit theorems.

Random time-change with inverses of multivariate subordinators: Governing equations and fractional dynamics

It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wide interest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of Rd-valued Markov processes with the components of an independent multivariate inverse subordinator. As a possible application, we present a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of suitable continuous-time random walks.

Anomalous Diffusions in Option Prices: Connecting Trade Duration and the Volatility Term Structure

Anomalous diffusions arise as scaling limits of continuous-time random walks (CTRWs) whose innovation times are distributed according to a power law. The impact of a non-exponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modelling this has been used to accommodate for random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behaviour of the implied volatility more consistently than usual Levy or stochastic volatility models.We focus on two distinct classes of underlying asset models, one with independent price innovations and waiting times, and one allowing dependence between these two components. These two models capture the well-known paradigm according to which shorter trade duration is associated with higher return impact of individual trades. We fully describe these processes in a semimartingale setting leading no-arbitrage pricing formulae, and study their statistical properties. We observe that skewness and kurtosis of the asset returns do not tend to zero as time goes by. We also characterize the large-maturity asymptotics of Call option prices, and find that the convergence rate is slower than in standard Levy regimes, which in turn yields a declining implied volatility term structure and a slower decay of the skew.

Anomalous Diffusions in Option Prices: Connecting Trade Duration and the Volatility Term Structure

Anomalous diffusions arise as scaling limits of continuous-time random walks whose innovation times are distributed according to a power law. The impact of a nonexponential waiting time does not va...