Case Of Brownian Motion Research Articles

On the ergodicity of interacting particle systems under number rigidity

In this paper, we provide relations among the following properties: the tail triviality of a probability measure mu on the configuration space {varvec{Upsilon }};the finiteness of a suitable L^2-transportation-type distance bar{textsf {d} }_{varvec{Upsilon }};the irreducibility of local {mu }-symmetric Dirichlet forms on {varvec{Upsilon }}. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including text {sine}_{2}, text {Airy}_{2}, text {Bessel}_{alpha , 2} (alpha ge 1), and text {Ginibre} point processes. In particular, the case of the unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role.

Ergodic properties of Brownian motion under stochastic resetting.

We study the ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin- or fat-tailed distributions the normalized or non-normalized invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equationdescribing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this nonequilibrium process which should hold far beyond the case of Brownian motion analyzed here.

Effusion of stochastic processes on a line

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.

Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme

Abstract One of the open problems in the study of stochastic differential equations is regularity analysis and approximations to stochastic partial differential equations driven by multiplicative fractional Brownian motions (fBms), especially for the case $H\in (0,\frac {1}{2})$. In this paper, we address this problem by considering a class of stochastic evolution equations (SEEs) driven by multiplicative fBms. We analyze the well-posedness and regularity of mild solutions to such equations with $H\in (0,\frac {1}{2})$ and $H\in (\frac {1}{2},1)$ under the Lipschitz conditions and linear growth conditions. The two cases are treated separately. Compared with the standard Brownian motion case, the main difficulty is that fBm is neither a Markov process nor a semimartingale such that some classical stochastic calculus theories are unavailable. As a consequence, we need to explore some new strategies to complete the existence and uniqueness and regularity analysis of the solutions. Especially for the case $H\in (0,\frac {1}{2})$, we utilize some delicate techniques to overcome the difficulties from the singularity of the covariance of fBms. In addition, we give a fully discrete scheme for such equations, carried out by the spectral Galerkin method in space and a time-stepping method in time. The obtained regularity results of the equations help us to examine the strong convergence of the discrete scheme. In final, several numerical examples are done to substantiate the theoretical findings.

Predicting the last zero before an exponential time of a spectrally negative Lévy process

AbstractGiven a spectrally negative Lévy process, we predict, in an $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises [2], where the infinite-horizon problem is solved. Using a similar argument as that in [24], we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite-horizon setting. Surprisingly (unlike the infinite-horizon problem), an optimal stopping time is based on a curve that is killed at the moment the mean of the exponential time is reached. That is, an optimal stopping time is the first time the process crosses above a non-negative, continuous, and non-increasing curve depending on time. This curve and the value function are characterised as a solution of a system of nonlinear integral equations which can be understood as a generalisation of the free boundary equations (see e.g. [21, Chapter IV.14.1]) in the presence of jumps. As an example, we numerically calculate this curve in the Brownian motion case and for a compound Poisson process with exponential-sized jumps perturbed by a Brownian motion.

Supermartingale Brenier's Theorem with full-marginals constraint

We explicitly construct the supermartingale version of the Fréchet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordère et al. [41] obtained in the martingale setting. Our construction is based on the Markovian iteration of one-period optimal supermartingale couplings. In the limit, as the number of iterations goes to infinity, we obtain a pure jump process that belongs to a family of local Lévy models introduced by Carr et al. [22]. We show that the constructed processes solve the continuous-time supermartingale optimal transport problem for a particular family of (path-dependent) cost functions. The explicit computations are provided in the following three cases: the uniform case, the Bachelier model and the Geometric Brownian Motion case.

Wiener–Hopf factorization for arithmetic Brownian motion with time-dependent drift and volatility

In this paper we obtain a Wiener–Hopf type factorization for a real-valued arithmetic Brownian motion with time-dependent drift and volatility. To the best of our knowledge, this paper is the very first step towards realizing the objective of deriving Wiener–Hopf type factorizations for (real-valued) time-inhomogeneous Lévy processes. In order to prove our main theorem, we derive some new results regarding time-inhomogeneous noisy Wiener–Hopf factorization. We demonstrate that in the special case of the arithmetic Brownian motion with constant drift and volatility our main result agrees with classical Wiener–Hopf factorization for this particular time-homogenous Lévy process.

Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomization

BackgroundClassical Brownian motion (BM) has been commonly used in monitoring clinical trials including those with covariate adaptive randomization (CAR). Independent increment property is commonly assumed in the sequential monitoring process of the clinical trials with CAR designs. However, in reality, correlation may exist in the error terms of the underlying model, resulting in dependent increment in the sequential monitoring process. MethodsWe conducted simulations for estimating the Hurst exponent to evaluate the stochastic property in the covariate adaptive randomized clinical trials under two scenarios: 1. CAR designs with independent and identically distributed error terms. 2. CAR designs with correlated error terms. The theoretical properties of covariate adaptive randomized clinical trials with correlated error structure were investigated. A test statistic including the covariance pattern of the error terms was proposed. ConclusionIn our study, the sequential test statistics under CAR procedure is shown to be asymptotically Brownian motion when the error structure is correctly specified. Further, Brownian motion is a special case of fractional Brownian motion when Hurst exponent equals to 0.5. Our simulations are consistent with the theoretical asymptotic results.

Pathwise large deviations for white noise chaos expansions

We consider a family of continuous processes $\{X^\varepsilon\}_{\varepsilon>0}$ which are measurable with respect to a white noise measure, take values in the space of continuous functions $C([0,1]^d:\mathbb{R})$, and have the Wiener chaos expansion \[ X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). \] We provide sufficient conditions for the large deviations principle of $\{X^\varepsilon\}_{\varepsilon>0}$ to hold in $C([0,1]^d:\mathbb{R})$, thereby refreshing a problem left open by P\'erez-Abreu (1993) in the Brownian motion case. The proof is based on the weak convergence approach to large deviations: it involves demonstrating the convergence in distribution of certain perturbations of the original process, and thus the main difficulties lie in analysing and controlling the perturbed multiple stochastic integrals. Moreover, adopting this representation offers a new perspective on pathwise large deviations and induces a variety of applications thereof.

Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

We study the fluctuations of the area under a self-similar Gaussian process x(τ) with Hurst exponent H > 0 (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate r. Typical fluctuations of A(t) scale as for large t and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of A(t). In the long-time limit t → ∞, we find that the full distribution of the area takes the form with anomalous exponents α = 1/(2H + 2) and β = (2H + 3)/(4H + 4) in the regime of moderately large fluctuations, and a different anomalous scaling form in the regime of very large fluctuations. The associated rate functions Φ(y) and Ψ(w) depend on H and are found exactly. Remarkably, Φ(y) has a singularity that we interpret as a first-order dynamical condensation transition, while Ψ(w) exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of Φ(y) around the origin y = 0 correctly describes the typical, Gaussian fluctuations of A(t). Despite these anomalous scalings, we find that all of the cumulants of the distribution grow linearly in time, , in the long-time limit. For the case of reset Brownian motion (corresponding to H = 1/2), we develop a recursive scheme to calculate the coefficients c n exactly and use it to calculate the first six nonvanishing cumulants.

A generative model for fBm with deep ReLU neural networks

We provide a large probability bound on the uniform approximation of fractional Brownian motion with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a N-dimensional Gaussian vector, with bounds on the network design (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of O(N−H) is possible with log⁡(N) hidden layers and O(Nlog⁡N) parameters. Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction and in the general fractional Brownian motion case (H≠1/2), on the Lemarié-Meyer wavelet representation. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.

Mean arc theorem for exploring domains with randomly distributed arbitrary closed trajectories

A remarkable result from integral geometry is Cauchy’s formula, which relates the mean path length of ballistic trajectories randomly crossing a convex 2D domain, to the ratio between the region area and its perimeter. This theorem has been generalized for non-convex domains and extended to the case of Brownian motion to find many applications in various fields including biological locomotion and wave physics. Here, we generalize the theorem to arbitrary closed trajectories exploring arbitrary domains. We demonstrate that, regardless of the complexity of this trajectory, the mean arc length still satisfies Cauchy’s formula provided that no closed trajectory is entirely contained in the domain. Below this threshold, the mean arc length decreases with the size of the closed trajectory. In this case, an approximate analytical formula can still be given for convex closed trajectories intersecting convex domains provided they are small in comparison. To validate our analysis, we performed numerical simulations of different types of trajectories exploring arbitrary 2D domains. Our results could be applied to retrieve geometric information of bounded domains from the mean first entrance–exit length.

Rate of strong convergence to Markov-modulated Brownian motion

AbstractLatouche and Nguyen (2015b) constructed a sequence of stochastic fluid processes and showed that it converges weakly to a Markov-modulated Brownian motion (MMBM). Here, we construct a different sequence of stochastic fluid processes and show that it converges strongly to an MMBM. To the best of our knowledge, this is the first result on strong convergence to a Markov-modulated Brownian motion. Besides implying weak convergence, such a strong approximation constitutes a powerful tool for developing deep results for sophisticated models. Additionally, we prove that the rate of this almost sure convergence is $o(n^{-1/2} \log n)$ . When reduced to the special case of standard Brownian motion, our convergence rate is an improvement over that obtained by a different approximation in Gorostiza and Griego (1980), which is $o(n^{-1/2}(\log n)^{5/2})$ .

Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<frac{1}{2}. Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps

We study existence of densities for solutions to stochastic differential equations with H\"older continuous coefficients and driven by a $d$-dimensional L\'evy process $Z=(Z_{t})_{t\geq 0}$, where, for $t>0$, the density function $f_{t}$ of $Z_{t}$ exists and satisfies, for some $(\alpha_{i})_{i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*} \limsup\limits _{t \to 0}t^{1/\alpha_{i}}\int\limits _{\mathbb{R}^{d}}|f_{t}(z+e_{i}h)-f_{t}(z)|dz\leq C|h|,\ \ h\in \mathbb{R},\ \ i=1,\dots,d. \end{align*} Here $e_{1},\dots,e_{d}$ denote the canonical basis vectors in $\mathbb{R}^{d}$. The latter condition covers anisotropic $(\alpha_{1},\dots,\alpha_{d})$-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citep{DF13}.

A financial market with singular drift and no arbitrage

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay theta > 0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as theta > 0. This implies that there is no arbitrage in the market in that case. However, when theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.

On local linearization method for stochastic differential equations driven by fractional Brownian motion

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., H = 1/2), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method.

Optimising dividends and consumption under an exponential CIR as a discount factor

We consider an economic agent (a household or an insurance company) modelling its surplus process by a deterministic process or by a Brownian motion with drift. The goal is to maximise the expected discounted spending/dividend payments under a discounting factor given by an exponential CIR process. In the deterministic case, we are able to find explicit expressions for the optimal strategy and the value function. For the Brownian motion case, we are able to show that for a special parameter choice the optimal strategy is a constant-barrier strategy.

Williamson nanofluid flow through porous medium in the presence of melting heat transfer boundary condition: semi-analytical approach

PurposePresent investigation based on the flow of electrically conducting Williamson nanofluid embedded in a porous medium past a linearly horizontal stretching sheet. In addition to that, the combined effect of thermophoresis, Brownian motion, thermal radiation and chemical reaction is considered in both energy and solutal transfer equation, respectively.Design/methodology/approachWith suitable choice of nondimensional variables the governing equations for the velocity, temperature, species concentration fields, as well as rate shear stress at the plate, rate of heat and mass transfer are expressed in the nondimensional form. These transformed coupled nonlinear differential equations are solved semi-analytically using variation parameter method.FindingsThe behavior of characterizing parameters such as magnetic parameter, melting parameter, porous matrix, Brownian motion, thermophoretic parameter, radiation, Lewis number and chemical particular case present result validates with earlier established results and found to be in good agreement. Finally reaction parameter is demonstrated via graphs and numerical results are presented in tabular form.Originality/valueThe said work is an original work of the authors.

Non-linear filtering for bilinear stochastic differential systems: A Stratonovich perspective

Bilinear stochastic differential equations have found applications to model turbulence in autonomous systems as well as switching uncertainty in non-linear dynamic circuits. In signal processing and control literature, bilinear stochastic differential equations are ubiquitous, since they capture non-linear qualitative characteristics of dynamic systems as well as offer closed-form solutions. The novelties of the paper are two: we weave bilinear filtering for the Stratonovich stochasticity. Then this paper unfolds the usefulness of bilinear filtering for switched dynamic systems. First, the Stratonovich stochasticity is embedded into a vector ‘bilinear’ time-varying stochastic differential equations. Then, coupled non-linear filtering equations are achieved. Finally, the non-linear filtering results are applied to an appealing bilinear stochastic Ćuk converter circuit. This paper also encompasses a system of coupled bilinear filtering equations for the vector input Brownian motion case. This paper brings the notions of systems theory, that is, bilinearity, Stratonovich stochasticity, non-linear filtering techniques and switched electrical networks together. Numerical simulation results are presented to demonstrate that the proposed bilinear filter can achieve much better and accurate filtering performance than the conventional Extended Kalman Filter (EKF).