Discrete-time Random Walk Research Articles

Exceptional points of discrete-time random walks in planar domains

Exceptional points of discrete-time random walks in planar domains

Random walk with multiple memory channels.

A class of one-dimensional, discrete-time random walk models with memory, termed "random walk with n memory channels" (RWnMC), is proposed. In these models the information of n (n∈Z) previous steps from the walker's entire history is needed to decide a future step. Exact calculation of the mean and variance of position of the RW2MC (n=2) has been done, which shows that it can lead to asymptotic diffusive and superdiffusive behavior in different parameter regimes. A connection between RWnMC and a Pólya-type urn model evolving by drawing n balls at a time has also been reported. This connection for the RW2MC is discussed in detail and suggests the applicability of RW2MC in many population dynamics models with multiple competing species.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Random walks on complex networks under time-dependent stochastic resetting.

We study discrete-time random walks on networks subject to a time-dependent stochastic resetting, where the walker either hops randomly between neighboring nodes with a probability 1-ϕ(a) or is reset to a given node with a complementary probability ϕ(a). The resetting probability ϕ(a) depends on the time a since the last reset event (also called a, the age of the walker). Using the renewal approach and spectral decomposition of the transition matrix, we formulate the stationary occupation probability of the walker at each node and the mean first passage time between two arbitrary nodes. Concretely, we consider two different time-dependent resetting protocols that are both exactly solvable. One is that ϕ(a) is a step-shaped function of a and the other one is that ϕ(a) is a rational function of a. We demonstrate the theoretical results on several different networks, also validated by numerical simulations, and find that the time-modulated resetting protocols can be more advantageous than the constant-probability resetting in accelerating the completion of a target search process.

How to estimate the memory of the elephant random walk

We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show the almost sure convergence of our estimate in the diffusive, critical and superdiffusive regimes. The local asymptotic normality of our statistical procedure is established in the diffusive regime, while the local asymptotic mixed normality is proven in the superdiffusive regime. Asymptotic and exact confidence intervals as well as statistical tests are also provided. All our analysis relies on asymptotic results for martingales and the quadratic variations associated.

Compound Option Pricing and the Roll-Geske-Whaley Formula under the Conjugate-Power Dagum Distribution

We explore the pricing of compound derivatives under the newly introduced conjugate-power Dagum distribution. Assuming a discrete-time multiplicative conjugate-power Dagum random walk, we first provide an alternative derivation of the price of a married put based on a change of measure, which is helpful for the pricing of compound options. Then, we apply these results to obtain the equivalent of the Roll-Geske-Whaley formula for the pricing of American options in presence of one known discrete dividend under this alternative distribution.

Further results on the minimal random walk

The aim of this paper is to go further in the analysis of the asymptotic behavior of the so-called minimal random walk (MRW) using a new martingale approach. The MRW is a discrete-time random walk with infinite memory that has three regimes depending on the location of its two parameters. In the diffusive and critical regimes, we establish new results on the asymptotic behavior of the MRW such as quadratic strong laws and functional central limit theorems. In the superdiffusive regime, we prove the almost sure convergence of the MRW, properly normalized, to a nondegenerate random variable. Moreover, we show that the fluctuation of the MRW around its limiting random variable is still Gaussian. Finally, several results on the center of mass of the MRW are also provided.

Non-equilibrium random walks on multiplex networks

We introduce a non-equilibrium discrete-time random walk model on multiplex networks, in which at each time step the walker first undergoes a random jump between neighboring nodes in the same layer, and then tries to hop from one node to one of its replicas in another layer. We derive the so-called supra-Markov matrix that governs the evolution of the occupation probability of the walker. The occupation probability at stationarity is different from the weighted average over the counterparts on each layer, unless the transition probabilities between layers vanish. However, they are approximately equal when the transition probabilities between layers are very small, which is given by the first-order degenerate perturbation theory. Moreover, we compute the mean first passage time (MFPT) and the graph MFPT (GrMFPT) that is the average of the MFPT over all pairs of distinct nodes. Interestingly, we find that the GrMFPT can be smaller than that of any layer taken in isolation. The result embodies the advantage of global search on multiplex networks.

The enhanced strong invariance principle for the elephant random walk

The elephant random walk (ERW) is a discrete-time random walk on Z with a memory about the whole past. It has been shown that the asymptotic behavior of the ERW depends heavily on a memory parameter 0≤p≤1. In this work, we establish the strong invariance principle for the ERW in the diffusive regime 0≤p<3/4 and the critical regime p = 3/4, which enhances the known results of Coletti, Gava and Schütz.

Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criterion for finite networks that establishes when there exists a non-zero resetting probability that minimizes the mean first passage time (MFPT) at a target node. Right at optimality, the coefficient of variation of the first passage time is not unity, unlike in continuous time processes with instantaneous resetting, but above 1 and depends on the minimal MFPT. The approach is general and applicable to the study of different discrete-time ergodic Markov processes such as Lévy flights, where the long-range dynamics is introduced in terms of the fractional Laplacian of the graph. We apply these results to the study of optimal transport on rings and Cayley trees.

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.

Survival probability of random walks leaping over traps

We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length ℓ over which the RWs can jump. We study the survival probability of such RWs when the traps are periodically distributed and separated by a distance L. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. The decay rate of the survival probability depends in a non-trivial way on the trap length ℓ and exhibits an interesting regime when ℓ → 0 as it tends to the ratio ℓ/L, which is reminiscent of strongly chaotic deterministic systems. We generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump. In this case, we find that the survival probability decays algebraically with an exponent that is independent of the trap length. Finally, we derive the diffusive limit of our model and show that, depending on the chosen scaling, we obtain either diffusion with uniform absorption, or diffusion with periodically distributed point absorbers.

Generating stochastic trajectories with global dynamical constraints

We propose a method to exactly generate Brownian paths x c (t) that are constrained to return to the origin at some future time t f , with a given fixed area under their trajectory. We derive an exact effective Langevin equation with an effective force that accounts for the constraint. In addition, we develop the corresponding approach for discrete-time random walks, with arbitrary jump distributions including Lévy flights, for which we obtain an effective jump distribution that encodes the constraint. Finally, we generalise our method to other types of dynamical constraints such as a fixed occupation time on the positive axis or a fixed generalised quadratic area .

Asymmetric random walks with bias generated by discrete-time counting processes

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the ‘generator process’ of the walk. We refer the so defined walk to as ‘asymmetric discrete-time random walk’ (ADTRW). We highlight connections of the waiting time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and analyze recurrent/transient features of simple ADTRWs (walks with unit jumps in both directions). We explore the connections of the recurrence/transience with the bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple ADTRWs are either unbiased in the large time limit or ‘strictly unbiased’ at all times with symmetric Bernoulli generator process. In this analysis we highlight the connections of bias and light-tailed/fat-tailed features of the waiting time density in the generator process. As a prototypical example with fat-tailed feature we consider the ADTRW with Sibuya distributed waiting times.We also introduce time-changed versions: We subordinate the ADTRW to a continuous-time renewal process which is independent from the generator process and the jumps to define the new class of ‘asymmetric continuous-time random walk’ (ACTRW). This new class – apart of some special cases – is not a Montroll–Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may open large interdisciplinary fields in anomalous transport, birth–death models and others.

Local time of random walks on graphs.

We investigate the local (or occupation) time of a discrete-time random walk on a generic graph, and present a general method for calculating sample-path averages of local time functionals in terms of the resolvent of the transition matrix.

Random walks on complex networks with multiple resetting nodes: A renewal approach.

Due to wide applications in diverse fields, random walks subject to stochastic resetting have attracted considerable attention in the last decade. In this paper, we study discrete-time random walks on complex networks with multiple resetting nodes. Using a renewal approach, we derive exact expressions of the occupation probability of the walker in each node and mean first-passage time between arbitrary two nodes. All the results can be expressed in terms of the spectral properties of the transition matrix in the absence of resetting. We demonstrate our results on circular networks, stochastic block models, and Barabási-Albert scale-free networks and find the advantage of the resetting processes to multiple resetting nodes in a global search on such networks. Finally, the distribution of resetting probabilities is optimized via a simulated annealing algorithm, so as to minimize the mean first-passage time averaged over arbitrary two distinct nodes.

Functional limit theorems for the multi-dimensional elephant random walk

In this article we shall derive functional limit theorems for the multi-dimensional elephant random walk (MERW) and thus extend the results provided for the one-dimensional marginal by Bercu and Laulin. The MERW is a non-Markovian discrete-time random walk on which has a complete memory of its whole past, in allusion to the traditional saying that an elephant never forgets. As the name suggests, the MERW is a d-dimensional generalization of the elephant random walk (ERW), the latter was first introduced by Schütz and Trimper in 2004. We measure the influence of the elephant’s memory by a so-called memory parameter p between zero and one. A striking feature that has been observed in Schütz and Trimper is that the long-time behavior of the ERW exhibits a phase transition at some critical memory parameter pc . We investigate the asymptotic behavior of the MERW in all memory regimes by exploiting a connection between the MERW and Pólya urns, following similar ideas as in the work by Baur and Bertoin for the ERW.

Generating discrete-time constrained random walks and Lévy flights.

We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.

Expected maximum of bridge random walks & Lévy flights

We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions f(η), including the case of Lévy flights. We study the expected maximum of bridge RWs, i.e. RWs starting and ending at the origin after n steps. We obtain an exact analytical expression for valid for any n and jump distribution f(η), which we then analyze in the large n limit up to second leading order term. For jump distributions whose Fourier transform behaves, for small k, as with a Lévy index 0 < μ ⩽ 2 and an arbitrary length scale a > 0, we find that, at leading order for large n, . We obtain an explicit expression for the amplitude h 1(μ) and find that it carries the signature of the bridge condition, being different from its counterpart for the free RW. For μ = 2, we find that the second leading order term is a constant, which, quite remarkably, is the same as its counterpart for the free RW. For generic 0 < μ < 2, this second leading order term is a growing function of n, which depends non-trivially on further details of , beyond the Lévy index μ. Finally, we apply our results to compute the mean perimeter of the convex hull of the 2d Rouse polymer chain and of the 2d run-and-tumble particle, as well as to the computation of the survival probability in a bridge version of the well-known ‘lamb–lion’ capture problem.

Universal record statistics for random walks and Lévy flights with a nonzero staying probability

We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability 0 ⩽ p ⩽ 1, while with the complementary probability 1 − p, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution f 0(η). We have shown that, for arbitrary p, the statistics of records up to step N is completely universal, i.e. independent of f 0(η) for any N. We also compute the connected two-time correlation function C p (m 1, m 2) of the record-breaking events at times m 1 and m 2 and show it is also universal for all p. Moreover, we demonstrate that C p (m 1, m 2) < C 0(m 1, m 2) for all p > 0, indicating that a nonzero p induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing p. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when p → 1, N → ∞ with the product t = (1 − p)N fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit.