This paper is a natural continuation of [8], 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 higher summability of Green's functions, boundedness of resolvent operators in Lebesgue spaces, establish Itô's formula, and so on.
In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributionsof a continuous function.
We consider a stochastic flow on $\mathds{R}$ generated by an SDE with its drift being a function of bounded variation. We show that the flow is differentiable with respect to the initial conditions. Asymptotic properties of the flow are studied.
A new class of random partial differential equations of parabolic type is considered, where the stochastic term consists of an irregular noisy drift, not necessarily Gaussian, for which a suitable interpretation is provided. After freezing a realization of the drift (stochastic process), we study existence and uniqueness (in some appropriate sense) of the associated parabolic equation and a probabilistic interpretation is investigated.
In this paper we study the drifted Brownian meander, that is a Brownian motion starting from $ u $ and subject to the condition that $ \min_{ 0\leq z \leq t} B(z)> v $ with $ u > v $. The limiting process for $ u \downarrow v $ is analyzed and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribu…
We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.
We study the convergence of a drift implicit scheme for one-dimensional SDEs that was considered by Alfonsi for the Cox-Ingersoll-Ross (CIR) process. Under general conditions, we obtain a strong convergence of order 1. In the CIR case, Dereich, Neuenkirch and Szpruch have shown recently a strong convergence of order 1/2 for this scheme. Here, we obtain a strong convergence of order 1 under more restrictive assumptions on the CIR parameters.
We study the dynamical properties of the Brownian diffusions having $σ{\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We characterize this class through the equality $D^2_+=D^2_-$, where $D_{+}$ (resp. $D_-$) denotes the forward (resp. backward) stochastic derivative of Nelson's type. Our proof is based on a remarkable identity for $D_+^2-D_-^2$ and on the use of the martingale problem. We also give a new formulation of a famous theorem of Kolmogorov concerning reversible diffusions. We finally relate our characterization to some questions about the complex stochast…
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…
In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well …
Existence and uniqueness are proved for Mckean-Vlasov type distribution dependent SDEs with singular drifts satisfying an integrability condition in space variable and the Lipschitz condition in distribution variable with respect to $W_0$ or $W_0+W_θ$ for some $θ\ge 1$, where $W_0$ is the total variation distance and $W_θ$ is the $L^θ$-Wasserstein distance. This improves some existing results where the drift is either locally bounded in the space variable or continuous in the distribution variable with respect to the Wasserstein distance.
We consider two reflecting diffusion processes $(X_t)_{t \ge 0}$ with a moving reflection boundary given by a non-decreasing pure jump Markov process $(R_t)_{t \ge 0}$. Between the jumps of the reflection boundary the diffusion part behaves as a reflecting Brownian motion with negative drift or as a reflecting Ornstein-Uhlenbeck process. In both cases at rate $γ(X_t -R_t)$ for some $γ\ge 0$ the reflection boundary jumps to a new value chosen uniformly in $[R_{t-},X_t]$. Since after each jump of the reflection boundary the diffusions are reflected at a higher level we call the processes Brownia…
We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of an arbitrary order. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some additional assumptions on the noise, we prove that the solution has moments of all orders. We complete the study providing a numerical scheme for the solution. As an illustration of our results and motivation for applications, we suggest two stochastic volatili…
In this paper, we are interested in the following one dimensional forward stochastic differential equation (SDE) \[ d X_{t}=b(t,X_{t},ω)d t +σd B_{t},\quad 0\leq t\leq T,\quad X_{0}=\,x\in \mathbb{R}, \] where the driving noise $B_{t}$ is a $d$-dimensional Brownian motion. The drift coefficient $b:[0,T] \timesΩ\times \mathbb{R}\longrightarrow \mathbb{R}$ is Borel measurable and can be decomposed into a deterministic and a random part, i.e., $b(t,x,ω) = b_1(t,x) + b_2(t,x,ω)$. Assuming that $b_1$ is of spacial linear growth and $b_2$ satisfies some integrability conditions, we obtain the existe…
We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get…
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.
Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by rand…
In this paper we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points, and the Malliavin differentiability with respect to sample paths. We also establish Bismut-Elworthy-Li's formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier-Stokes equations given by Constantin-Iyer \cite{Co-Iy} to prove the local well-posedness of NSEs in $\m…
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…
In this paper, the coupling by change of measure is constructed for a class of SDEs with integrable drift and additive noise, from which the Harnack and shift Harnack inequalities are derived. Finally, as applications, the gradient estimate, the regularity of the heat kernel and the distribution properties of the associated transition probability are also obtained. The important tool is Krylov's estimate.
We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein--Uhlenbeck process driven by mixed fractional Brownian motion.
A careful look at rough path topology applied to Brownian motion reveals new possible properties of the well-known Lévy area, in particular the presence of an intrinsic drift of this area. Using renormalization limit of Markov chains on periodic graphs, we present a construction of such a non-trivial drift and give an explicit formula for it. Several examples for which explicit computations are made are included.
We investigate some smoothness properties for a transport-diffusion equation involving a class of non-degerate L{é}vy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some H{ö}lder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey-Campanato space for which the regularizing properties of the L{é}vy operator suffice to obtain global H{ö}lder continuity.
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 …
We consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates fo…
This paper deals with the ergodicity and the existence of a strong law of large numbers for adaptive Markov Chain Monte Carlo. We show that a diminishing adaptation assumption together with a drift condition for positive recurrence is enough to imply ergodicity. Strengthening the drift condition to a polynomial drift condition yields a strong law of large numbers for possibly unbounded functions. These results broaden considerably the class of adaptive MCMC algorithms for which rigorous analysis is now possible. As an example, we give a detailed analysis of the Adaptive Metropolis Algorithm of…
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.
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.
In this paper, the distribution dependent stochastic differential equation in a separable Hilbert space with a Dini continuous drift is investigated. The existence and uniqueness of weak and strong solutions are obtained. Moreover, some regularity results as well as gradient estimates and log-Harnack inequality are derived for the associated semigroup. In addition, dimensional free Harnack inequality with power and shift Harnack inequality are also proved when the noise is additive. All of the results extend the ones in the distribution independent situation.
The study of records in the Linear Drift Model (LDM) has attracted much attention recently due to applications in several fields. In the present paper we study $δ$-records in the LDM, defined as observations which are greater than all previous observations, plus a fixed real quantity $δ$. We give analytical properties of the probability of $δ$-records and study the correlation between $δ$-record events. We also analyse the asymptotic behaviour of the number of $δ$-records among the first $n$ observations and give conditions for convergence to the Gaussian distribution. As a consequence of our …
In this paper we consider the drift estimation problem for a general differential equation driven by an additive multidimensional fractional Brownian motion, under ergodic assumptions on the drift coefficient. Our estimation procedure is based on the identification of the invariant measure, and we provide consistency results as well as some information about the convergence rate. We also give some examples of coefficients for which the identifiability assumption for the invariant measure is satisfied.
Many applications in networked control require intermittent access of a controller to a system, as in event-triggered systems or information constrained control applications. Motivated by such applications and extending previous work on Lyapunov-theoretic drift criteria, we establish both subgeometric and geometric rates of convergence for Markov chains under state dependent random time drift criteria. We quantify how the rate of ergodicity, nature of Lyapunov functions, their drift properties, and the distributions of stopping times are related. We finally study an application in networked co…
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorov's backward equations in some second order Sobolev spaces, which is analytically interesting in itself.
We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the $L^d$ class, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$ for any $\varepsilon>0$.
We prove unique weak solvability and Feller property for stochastic differential equations with drift in a large class of time-dependent vector fields. This class contains, in particular, the critical Ladyzhenskaya-Prodi-Serrin class, the weak $L^d$ class as well as some vector fields that are not even in $L^{2+\varepsilon}_{\rm loc}$, $\varepsilon>0$.
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some deli…
This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices, and the stability of a corresponding random transition matrix (or exponential semigroup). We consider a class of random matrix drift coefficients that involves random perturbations of an exponentially stable flow of deterministic (time-varying) drift matrices. In contrast with more conventional studies, our analysis is not based on the existence of Lyapunov functions, and it does not rely on any ergodic properties. These approaches are often d…
A parameter estimation problem for a class of semilinear stochastic evolution equations is considered. Conditions for consistency and asymptotic normality are given in terms of growth and continuity properties of the nonlinear part. Emphasis is put on the case of stochastic reaction-diffusion systems. Robustness results for statistical inference under model uncertainty are provided.
In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequenci…
