Gaussian Jumps Research Articles

Fitting Rwanda’s Currency Market Returns with the Poisson Compound Model with Normal Inverse Gaussian Jumps

Fitting Rwanda’s Currency Market Returns with the Poisson Compound Model with Normal Inverse Gaussian Jumps

Merton's Optimal Investment Problem with Jump Signals

This paper presents a new framework for Merton's optimal investment problem which uses the theory of Meyer $\sigma$-fields to allow for more flexibility in the information structure by adding a signal that possibly warns the investor about impending jumps. By means of dynamic programming, we solve the problem explicitly. In a case study with Gaussian jumps, we find, for instance, that an investor may prefer to disinvest completely even after a mildly positive signal. Our setting allows us also to address questions such as whether it is better to improve signal quality or quantity and how much extra value can be generated from either choice.

The Value of Insight

An investor may invest in a riskless bank account and in a stock that is a standard Black–Scholes asset with occasional Gaussian jumps of the log price, as proposed by Merton [Merton RC ( 1976 ) Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3(1):125–144.]. It is well known how to solve the standard running consumption problem for this investor, which we take as a benchmark for comparing the performance of two different insiders, one who knows in advance of each jump exactly when the jump will happen, and the other who has information in advance of each jump about the size of the jump but no information about the time. These considerations give rise to two novel and concrete stochastic control problems. For each problem, rigorous verification proofs for optimality are presented.

A mathematical model for preventing HIV virus from proliferating inside CD4 T brownian cell using Gaussian jump nanorobot

We present a statistical distribution of a nanorobot motion inside the blood. This distribution is like the distribution of A and B particles in continuous time random walk scheme inside the fluid reactive anomalous transport with stochastic waiting time depending on the Gaussian distribution and a Gaussian jump length which is detailed in Zhang and Li [J. Stat. Phys., Published Online with https://doi.org/10.1007/s10955-018-2185-8 , 2018]. Rather than estimating the length parameter of the jumping distance of the nanorobot, we normalize the Probability Density Function (PDF) and present some reliability properties for this distribution. In addition, we discuss the truncated version of this distribution and its statistical properties, and estimate its length parameter. We use the estimated distance to study the conditions that give a finite expected value of the first meeting time between this nanorobot in the case of nonlinear flow with independent [Formula: see text]-dimensional Gaussian jumps and an independent [Formula: see text]-dimensional CD4 T Brownian cell in the blood ([Formula: see text]-space) to prevent the HIV virus from proliferating within this cell.

Mean perimeter and mean area of the convex hull over planar random walks

We investigate the geometric properties of the convex hull over n successive positions of a planar random walk, with a symmetric continuous jump distribution. We derive the large n asymptotic behavior of the mean perimeter. In addition, we compute the mean area for the particular case of isotropic Gaussian jumps. While the leading terms of these asymptotics are universal, the subleading (correction) terms depend on the finer details of the jump distribution and describe a ‘finite size effect’ of discrete-time jump processes, allowing one to accurately compute the mean perimeter and the mean area even for small n, as verified by Monte Carlo simulations. This is particularly valuable for applications dealing with discrete-time jumps processes and ranging from the statistical analysis of single-particle tracking experiments in microbiology to home range estimations in ecology.

Optimal search in interacting populations: Gaussian jumps versus Lévy flights.

We investigated the relationships between search efficiency, movement strategy, and nonlocal communication in the biological context of animal foraging. We considered situations where the members of a population of foragers perform either Gaussian jumps or Lévy flights, and show that the search time is minimized when communication among individuals occurs at intermediate ranges, independently of the type of movement. Additionally, while Brownian strategies are more strongly influenced by the communication mechanism, Lévy flights still result in shorter overall search durations.

Adjoint Expansions in Local Lévy Models

We propose a novel method for the analytical approximation in local volatility models with Levy jumps. The main result is an expansion of the characteristic function in a local Levy model, which is worked out in the Fourier space by considering the adjoint formulation of the pricing problem. Combined with standard Fourier methods, our result provides efficient and accurate pricing formulae. In the case of Gaussian jumps, we also derive an explicit approximation of the transition density of the underlying process by a heat kernel expansion: the approximation is obtained in two ways, using partial integro-differential equation techniques and working in the Fourier space. Numerical tests confirm the effectiveness of the method.

Competitive Brownian and Lévy walkers

Population dynamics of individuals undergoing birth and death and diffusing by short- or long-range two-dimensional spatial excursions (Gaussian jumps or Lévy flights) is studied. Competitive interactions are considered in a global case, in which birth and death rates are influenced by all individuals in the system, and in a nonlocal but finite-range case in which interaction affects individuals in a neighborhood (we also address the noninteracting case). In the global case one single or few-cluster configurations are achieved with the spatial distribution of the bugs tied to the type of diffusion. In the Lévy case long tails appear for some properties characterizing the shape and dynamics of clusters. Under nonlocal finite-range interactions periodic patterns appear with periodicity set by the interaction range. This length acts as a cutoff limiting the influence of the long Lévy jumps, so that spatial configurations under the two types of diffusion become more similar. By dividing initially everyone into different families and following their descent it is possible to show that mixing of families and their competition is greatly influenced by the spatial dynamics.

Adjoint Expansions in Local Lévy Models

We propose a novel method for the analytical approximation in local volatility models with Levy jumps. The main result is an expansion of the characteristic function in a local Levy model, which is worked out in the Fourier space by considering the adjoint formulation of the pricing problem. Combined with standard Fourier methods, our result provides efficient and accurate pricing formulae. In the case of Gaussian jumps, we also derive an explicit approximation of the transition density of the underlying process by a heat kernel expansion: the approximation is obtained in two ways, using PIDE techniques and working in the Fourier space. Numerical tests confirm the effectiveness of the method.

Mass transport with sorption in porous media

Abstract Small-scale models in the form of random walks, combining Gaussian jumps, advection by mean flow field and possibly very long sorbing durations, correspond to experimental data in many porous media, in the laboratory and in the field. Within this frame-work, solutes are observed in two phases, which are mobile and immobile. For such random walks, in the hydrodynamic limit, the densities of that phases are linked by a relationship involving a fractional integral. This implies that the total density of tracer evolves according to a fractional variant of Fourier’s law.