Processes with inert drift

The Inert Drift Atlas Model

Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)). We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z…

On nondegenerate Itô processes with moderated drift

In this paper we present an approach to proving parabolic Aleksandrov estimates with mixed norms for stochastic integrals with singular ``moderated'' drift.

On stochastic Itô processes with drift in $L_{d}$

For Itô stochastic processes in $\mathbb{R}^{d}$ with drift in $L_{d}$ Aleksandrov's type estimates are established in the elliptic and parabolic settings. They are applied to estimating the resolvent operators of the corresponding elliptic and parabolic operators in $L_{p}$ and $L_{p+1}$, respectively, where $p\geq d$.

Independent factorization of the last zero arcsine law for Bessel processes with drift

We show that the last zero before time $t$ of a recurrent Bessel process with drift starting at $0$ has the same distribution as the product of an independent right censored exponential random variable and a beta random variable. This extends a recent result of Schulte-Geers and Stadje (2017) from Brownian motion with drift to recurrent Bessel processes with drift. Our proof is intuitive and direct while avoiding heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.

The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift is derived through a system…

Weak uniqueness for SDEs driven by supercritical stable processes with Holder drifts

In this paper, we investigate stochastic differential equations(SDEs) driven by a class of supercritical $α$-stable process(including the rotational symmetric $α-$stable process) with drift $b$. The weak well-posedness is proved, provided that the $(1-α)$-Hölder semi-norm of $b$ is sufficient small.

Cramér's estimate for stable processes with power drift

We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.

On potentials of Itô's processes with drift in $L_{d+1}$

This paper is a natural continuation of \cite{Kr_20_2}, where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Here we study some properties of these processes such as the probability to pass through narrow tubes, higher summability of Green's functions, and so on. The results seem to be new even if the diffusion is constant.

An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.

On diffusion processes with drift in $L_{d+1}$

This paper is a natural continuation of \cite{Kr_20_2} and \cite{Kr_21_1} where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$ and some properties of their Green's functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Hölder continuity of potentials, Fanghua Lin estimates and so on.

Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary $λ$-Hölder continuous process, $λ\in(0,1)$. We prove that, under some mild moment assumptions on the Hölder constant of the noise, the $L^r(Ω;L^\infty([0,T]))$-rate of convergence is equal to $λ$. To exemplify, we consider numerical schemes for the generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models. The results are illustrated by simulations.

On invariant probability measures of regime-switching diffusion processes with singular drifts

For regime-switching diffusions processes with singular drifts, we introduce integrability conditions involving a nice reference probability measure and the $Q$-matrix of the jump part to study the existence of the invariant probability measures. Consequently, the generator of the regime-switching diffusions process has an extension in the $L^1$-space w.r.t the invariant probability measure to be a generator of $C_0$-semigroup. Moreover, we prove the uniqueness of the extension. Regularities and the uniqueness of the invariant probability density w.r.t the nice reference probability measure ar…

Escape probabilities of compound renewal processes with drift

We consider the problem of determining escape probabilities from an interval of a general compound renewal process with drift. This problem is reduced to the solution of a certain integral equation. In an actuarial situation where only negative jumps arise we give a general solution for escape and survival probabilities under Erlang$(n)$ and hypo-exponential arrivals. These ideas are generalized to the class of arrival distributions having rational Laplace transforms. In a general situation with two-sided jumps we also identify important families of solvable cases. A parallelism with the "scal…

具有哈密顿噪声的惯性凝聚粒子的质量指数衰减

Exponential decay of mass for inertial coalescing particles with Hamiltonian noise

研究二维环面上惯性粒子在哈密顿噪声下凝聚过程的长期行为，证明在特定条件下期望总质量随时间指数衰减，并给出显式衰减率。

关于漂移项属于包含 $L_{d+2}$ 的 Morrey 类的扩散过程

On diffusion processes with drift in a Morrey class containing $L_{d+2}$

本文为混合范数 Morrey 类漂移项建立新条件，从而在混合范数空间及 $L_{d_{0}+1}$（$d_{0}<d$）中获得时间非齐次扩散过程势的 Aleksandrov 型估计。

带漂移相互作用的连续状态分支过程从无穷远处下降现象研究

On the coming down from infinity of continuous-state branching processes with drift-interaction

本文研究连续状态分支过程从无穷远处下降的条件，给出充分条件确保不爆炸及下降，并证明在漂移满足单侧Lipschitz条件时过程收敛于随机微分方程的强解，主要运用比较原理和Lyapunov函数方法。

具有不规则漂移的Itô过程的显式局部密度界

Explicit local density bounds for Itô-processes with irregular drift

本文研究了具有不规则漂移的连续扩散过程，假设扩散系数为常数或局部Lipschitz连续函数，通过将其与双重反射Brown运动进行比较，推导出了边际密度的显式上界。

带漂移的稳定Lévy过程的抛物分形几何

Parabolic Fractal Geometry of Stable Lévy Processes with Drift

该研究引入抛物Hausdorff维数，利用抛物柱体覆盖的时空非线性标度匹配自相似性，显式求出带确定性漂移的各向同性稳定Lévy过程的图像与值域的Hausdorff维数，证明漂移项的抛物维数的贡献，并推导相关公式与界。

Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive Lévy process with non-zero Lévy measure. In this paper we study the asymptotic behavior of the local time process, in the spatial variable, of these processes killed at two different random times: either at the time of the first visit of the Lévy process to 0, in which case we prove results at the excursion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under gener…

具有有界不规则漂移和扩散系数的Itô过程期望占用密度的精确上界

A sharp upper bound for the expected occupation density of Itô processes with bounded irregular drift and diffusion coefficients

本文在系数有界和椭圆性条件下，找到了漂移和扩散系数未知的Itô过程期望占用密度的显式最优上界，并据此得到了路径积分均值的显式上界。

Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise

We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $ρ$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $ρ\ge d$, the order of convergence at…

Uniform Asymptotics for Compound Poisson Processes with Regularly Varying Jumps and Vanishing Drift

This paper addresses heavy-tailed large deviation estimates for the distribution tail of functionals of a class of spectrally one-sided Lévy process. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting $[0,\infty)$, the maximum jump until that time, and the time it takes until exiting $[0,\infty)$. The proofs rely, among other things, on pr…

由具有均场漂移相互作用的粒子系统诱导的点过程的混沌传播

Propagation of chaos for point processes induced by particle systems with mean-field drift interaction

研究由具有均场漂移相互作用的粒子系统诱导的点过程的渐进行为，证明在合适条件下该点过程的混沌传播，其弱极限与独立同分布的麦肯维拉索方程解的点过程相同，弱极限为强度测度与经典极值分布相关的泊松点过程，特别地，得到归一化上序统计量的极限分布。

Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations

Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process…

Boundary crossing probabilities for diffusions with piecewise linear drifts

We propose an approach to approximate the boundary crossing probabilities for general one-dimensional diffusion processes, and derive the convergence rate for this approximation scheme. There results are based on the explicit expression of the Laplace transforms of the first passage densities for diffusions with piecewise linear drifts. The proposed method is applied to a reliability problem where the standard degradation model based on Wiener process is extended to diffusion processes with piecewise linear drifts.

Fractional Poisson process with random drift

We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in …

Stochastic equations with time-dependent drift driven by Levy processes

Using the method of Krylov's estimates, we prove the existence of weak solutions of stochastic differential equations driven by purely discontinuous Levy processes satisfying an additional assumption. The diffusion coefficient is assumed to be one and the time-dependent drift is measurable and bounded.

带漂移的Bessel与Dunkl过程的中心极限定理

A central limit theorem for Bessel and Dunkl processes with drift

本文针对正则漂移向量及任意根系和重数，证明了带漂移的Bessel与Dunkl过程的中心极限定理，改进了作者近期结果。

Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions

We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction makes use of transformations of diffusion processes to eliminate the drift which combine a measure change given by Doob's h-transform and a diffeomorphism. Such transformations have the important property of preserving analytic solvability of the process: the transition probability density for the driftless process can be expressed through the transition prob…

Exit times in non-Markovian drifting continuous-time random walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

贝塞尔和杜克尔过程与漂移

Bessel and Dunkl processes with drift

本文研究了带有漂移的贝塞尔和杜克尔过程的构造及其特性，包括径向部分、吉拉斯定理、矩、鞅、大数定律和中心极限定理。

Strong Feller processes with measure-valued drifts

We construct a strong Feller process associated with $-Δ+ σ\cdot \nabla$, with drift $σ$ in a wide class of measures (weakly form-bounded measures, e.g. combining weak $L^d$ and Kato class measure singularities), by exploiting a quantitative dependence of the smoothness of the domain of an operator realization of $-Δ+ σ\cdot \nabla$ generating a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the value of the form-bound of $σ$. Our method admits extension to other types of perturbations of $-Δ$ or $(-Δ)^{\fracα{2}}$, e.g. to yield new $L^p$-regularity results for Schrödinger ope…

基于漂移理论的直观分析

Intuitive Analyses via Drift Theory

本文系统介绍漂移理论的核心定理及其在优惠券收集、顶点覆盖、2-SAT、Moran过程等随机过程分析中的应用，旨在推广该工具以简化首达时间相关问题的证明。

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow …

A Limit Theorem for Financial Markets with Inert Investors

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modelling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long range dependence and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated market parti…

Stochastic analysis on Gaussian space applied to drift estimation

In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Loève expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of J…

Explicit moments of decision times for single- and double-threshold drift-diffusion processes

We derive expressions for the first three moments of the decision time (DT) distribution produced via first threshold crossings by sample paths of a drift-diffusion equation. The "pure" and "extended" diffusion processes are widely used to model two-alternative forced choice decisions, and, while simple formulae for accuracy, mean DT and coefficient of variation are readily available, third and higher moments and conditioned moments are not generally available. We provide explicit formulae for these, describe their behaviors as drift rates and starting points approach interesting limits, and, …

具有时间非齐次漂移的高频率数据的高斯过程M估计

M-estimation for Gaussian processes with time-inhomogeneous drifts from high-frequency data

本文提出了一种基于对比的估计方法，用于估计具有时间非齐次漂移的高斯过程，该方法在高频采样下具有高效计算能力，并证明了估计量的一致性和渐近正态性，同时展示了非标准收敛速度和对协方差核参数的修正方法。

高维莱维驱动的奥本-乌尔申贝克过程的低秩与稀疏漂移估计

Low-Rank and Sparse Drift Estimation for High-Dimensional Lévy-Driven Ornstein--Uhlenbeck Processes

本文研究高维莱维驱动的奥本-乌尔申贝克过程，提出低秩加稀疏结构的漂移矩阵估计方法，通过核范数与ℓ1惩罚结合的凸估计器，在受限强凸性条件下推导出估计器的Frobenius风险非渐近Oracle不等式。