Independent Brownian Motions Research Articles

A law of large numbers for interacting diffusions via a mild formulation

Consider a system of n weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called McKean-Vlasov or Fokker-Planck equation, as n tends to infinity. We propose a relatively new approach to show this convergence by directly studying the stochastic partial differential equation that the empirical measure satisfies for each fixed n. Under a suitable control on the noise term, which appears due to the finiteness of the system, we are able to prove that the stochastic perturbation goes to zero, showing that the limiting measure is a solution to the classical McKean-Vlasov equation. In contrast with known results, we do not require any independence or finite moment assumption on the initial condition, but the only weak convergence. The evolution of the empirical measure is studied in a suitable class of Hilbert spaces where the noise term is controlled using two distinct but complementary techniques: rough paths theory and maximal inequalities for self-normalized processes.

On the limiting law of the length of the longest common and increasing subsequences in random words with arbitrary distribution

Let (Xk)k≥1 and (Yk)k≥1 be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet Am:={1,…,m}, m≥2, having respective probability mass function p1X,…,pmX and p1Y,…,pmY. Let LCIn be the length of the longest common and weakly increasing subsequences in X1,...,Xn and Y1,...,Yn. Once properly centered and normalized, LCIn is shown to have a limiting distribution which is expressed as a functional of two independent multidimensional Brownian motions.

Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

We first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb R^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability $1$, t

Rank dependent branching-selection particle systems

We consider a large family of branching-selection particle systems. The branching rate of each particle depends on its rank and is given by a function $b$ defined on the unit interval. There is also a killing measure $D$ supported on the unit interval as well. At branching times, a particle is chosen among all particles to the left of the branching one by sampling its rank according to $D$. The measure $D$ is allowed to have total mass less than one, which corresponds to a positive probability of no killing. Between branching times, particles perform independent Brownian Motions in the real line. This setting includes several well known models like Branching Brownian Motion (BBM), $N$-BBM, rank dependent BBM, and many others. We conjecture a scaling limit for this class of processes and prove such a limit for a related class of branching-selection particle system. This family is rich enough to allow us to use the behavior of solutions of the limiting equation to prove the asymptotic velocity of the rightmost particle under minimal conditions on $b$ and $D$. The behavior turns out to be universal and depends only on $b(1)$ and the total mass of $D$. If the total mass is one, the number of particles in the system $N$ is conserved and the velocities $v_N$ converge to $\sqrt{2 b(1)}$. When the total mass of $D$ is less than one, the number of particles in the system grows up in time exponentially fast and the asymptotic velocity of the rightmost one is $\sqrt{2 b(1)}$ independently of the number of initial particles.

Model of Continuous Random Cascade Processes in Financial Markets

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

Process convergence of fluctuations of linear eigenvalue statistics of random circulant matrices

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to [Formula: see text]. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.

Partial Information Stochastic Differential Games for Backward Stochastic Systems Driven by Lévy Processes

In this paper, we consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by a backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. A sufficient condition and a necessary one for the existence of the saddle point for the game are proved. As an application, a linear quadratic stochastic differential game problem is discussed.

A Dynamic Network Model of Interbank Lending—Systemic Risk and Liquidity Provisioning

We develop a dynamic model of interbank borrowing and lending activities in which banks are organized into clusters, and adjust their monetary reserve levels to meet prescribed capital requirements. Each bank has its own initial monetary reserve level and faces idiosyncratic risks characterized by an independent Brownian motion, whereas system wide, the banks form a hierarchical structure of clusters. We model the interbank transactional dynamics through a set of interacting measure-valued processes. Each individual process describes the intracluster borrowing/lending activities, and the interactions among the processes capture the intercluster financial transactions. We establish the weak limit of the interacting measure-valued processes as the number of banks in the system grows large. We then use the weak limit to develop asymptotic approximations of two proposed macromeasures (the liquidity stress index and the concentration index), both capturing the dynamics of systemic risk. We use numerical examples to illustrate the applications of the asymptotics and conduct-related sensitivity analysis with respect to various indicators of financial activity.

Process convergence of fluctuations of linear eigenvalue statistics of band Toeplitz matrices

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of band Toeplitz matrices with independent Brownian motion entries, as the dimension of matrix goes to infinity. Here the band width bn of the n×n Toeplitz matrix goes to infinity with n at the rate of o(n).

Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion

We obtain the high-order asymptotic expansion for the distribution of the quadratic variation of the mixed fractional Brownian motion, which is defined as the sum of a Brownian motion and an independent fractional Brownian motion. Our approach is based on the analysis of the cumulants of this sequence. We show that both the Brownian and fractional Brownian parts contribute to the asymptotic expansion.

Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions

<p style='text-indent:20px;'>This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, it is proved that if <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>, the disease is permanent and there is a stationary distribution. And when <inline-formula><tex-math id="M3">\begin{document}$ \lambda&lt;0 $\end{document}</tex-math></inline-formula>, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction.

Millstone Hill ISR Measurements of Small Aspect Angle Spectra

AbstractThe Millstone Hill incoherent scatter (IS) radar is used to measure spectra close to perpendicular to the Earth's magnetic field, and the data are fit to three different forward models to estimate ionospheric temperatures. IS spectra measured close to perpendicular to the magnetic field are heavily influenced by Coulomb collisions, and the temperature estimates are sensitive to the collision operator used in the forward model. The standard theoretical model for IS radar spectra treats Coulomb collisions as a velocity independent Brownian motion process. This gives estimates of Te/Ti &lt; 1 when fitting the measured spectra for aspect angles up to 3.6°, which is a physically unrealistic result. The numerical forward model from Milla and Kudeki (2011, https://doi.org/10.1109/TGRS.2010.2057253) incorporates single‐particle simulations of velocity‐dependent Coulomb collisions into a linear framework, and when applied to the Millstone data, it predicts the same Te/Ti ratios as the Brownian theory. The new approach is a nonlinear particle‐in‐cell (PIC) code that includes velocity‐dependent Coulomb collisions which produce significantly more collisional and nonlinear Landau damping of the measured ion‐acoustic wave than the other forward models. When applied to the radar data, the increased damping in the PIC simulations will result in more physically realistic estimates of Te/Ti. This new approach has the greatest impact for the largest measured ionospheric densities and the lowest radar frequencies. The new approach should enable IS radars to obtain accurate measurements of plasma temperatures at times and locations where they currently cannot.

Maximum likelihood estimators from discrete data modeled by mixed fractional Brownian motion with application to the Nordic stock markets

Mixed fractional Brownian motion is a linear combination of Brownian motion and independent Fractional Brownian motion that is extensively used for option pricing. The consideration of the mixed process is able to capture the long–range dependence property that financial time series exhibit. This paper examines the problem of deriving simultaneously the estimators of all the unknown parameters for a model driven by the mixed fractional Brownian motion using the maximum likelihood estimation method. The consistency and asymptotic normality properties of these estimators are provided. The performance of the methodology is tested on simulated data sets, and the outcomes illustrate that the maximum likelihood technique is efficient and reliable. An empirical application of the proposed method is also made to the real financial data from four Nordic stock market indices.

Infinite horizon optimal control for mean‐field stochastic delay systems driven by Teugels martingales under partial information

SummaryIn this article, we discuss an infinite horizon optimal control of the stochastic system with partial information, where the state is governed by a mean‐field stochastic differential delay equation driven by Teugels martingales associated with Lévy processes and an independent Brownian motion. First, we show the existence and uniqueness theorem for an infinite horizon mean‐field anticipated backward stochastic differential equation driven by Teugels martingales. Then applying different approaches for the underlying system, we establish two classes of stochastic maximum principles, which include two necessary conditions and two sufficient conditions for optimality, under a convex control domain. Moreover, compared with the finite horizon optimal control, we add the transversality conditions to the two kinds of stochastic maximum principles. Finally, using the stochastic maximum principle II, we settle an infinite horizon optimal consumption problem driven by Teugels martingales associated with Gamma processes.

A Limit Law for Functionals of Multiple Independent Fractional Brownian Motions

A Limit Law for Functionals of Multiple Independent Fractional Brownian Motions

Stochastic Characteristics and Optimal Control for a Stochastic Chemostat Model with Variable Yield

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

Fractional smoothness of derivative of self-intersection local times with respect to bi-fractional Brownian motion

Let BHi,Ki={BtHi,Ki,t≥0},i=1,2 be two independent bifractional Brownian motions with respective indices Hi∈(0,1) and Ki∈(0,1]. In this paper, we study the fractional smoothness of derivative of the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motion.

CREDIT DEFAULT SWAPS IN TWO-DIMENSIONAL MODELS WITH VARIOUS INFORMATIONS FLOWS

We study a credit risk model of a financial market in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the no-arbitrage prices of both risk-free and risky credit default swaps given the reference filtration initially and progressively enlarged by the two default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

Impulsive stochastic fractional differential equations driven by fractional Brownian motion

In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index 1/2< H<1 under a non-Lipschitz condition with the Lipschitz one as a particular case. Our analysis depends on an approximation scheme of Carathéodory type. Some previous results are improved and extended.

Sharp Threshold for the Dynamics of a SIRS Epidemic Model With General Awareness- Induced Incidence and Four Independent Brownian Motions

In this paper, a stochastic SIRS epidemic model with general awareness-induced and four independent Brownian Motions is established. We verify the global existence of a unique positive solution and find out the noise modified reproduction number R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> which is a sharp threshold for the dynamics: If R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> <; 1, the disease will die out; if R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> > 1, the disease persists and there exists a global asymptotically stable stationary distribution under parameter restrictive conditions. Numerical simulations are presented to illustrate the theoretical results.