Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition between cosecutive partition times of distance dt, and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tighness results which together yield convergence in law of such measures as the partition gets finer. In particular let L be a closed smooth submanifold of a Riemannian manifold M. We prove convergence of Brownian motion on M, conditioned to visit L at all partition times, to a process on L whose law has a Radon-Nikodym density with repect to Brownian motion on L which contains scalar, mean and sectional curvature terms. Various approximation schemes for Brownian motion are also given. These results substantially extend earlier work by the authors and by Andersson and Driver.
The classical Chernoff's theorem is a statement about discrete-time approximations of semigroups, where the approximations are consturcted as products of time-dependent contraction operators strongly differentiable at zero. We generalize the version of Chernoff's theorem for semigroups proved in a paper by Smolyanov et al., and obtain a theorem about descrete-time approximations of backward propagators.
A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many non-regular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical non-regular statistical estimation problem of isotonic (or monotone) regression. The …
Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial v…
We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter 3/4<H<1. Using results from infinite ergodic theory it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.
In this paper we study the convergence of dynamical discrete web (DyDW) to the dynamical Brownian web (DyBW) in the path space topology. We show that almost surely the DyBW has RCLL paths taking values in an appropriate metric space and as a sequence of RCLL paths, the scaled dynamical discrete web converges to the DyBW. This proves weak convergence of the DyDW process to the DyBW process.
The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.
In a previous work, we have defined a Tanaka SDE related to Walsh Brownian motion which depends on kernels. It was shown that there are only one Wiener solution and only one flow of mappings solving this equation. In the terminology of Le Jan and Raimond, these are respectively the stronger and the weaker among all solutions. In this paper, we obtain these solutions as limits of discrete models.
We show that the distribution of the maximum of the fractional Brownian motion $B^H$ with Hurst parameter $H\to 0$ over an $n$-point set $τ\subset [0,1]$ can be approximated by the normal law with mean $\sqrt{\ln n}$ and variance $1/2$ provided that $n\to \infty$ slowly enough and the points in $τ$ are not too close to each other.
In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains $D$ in $\mathbb{R}^n$ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on $D\cap2^{-k}\mathbb{Z}^n$ moving at the rate $2^{-2k}$ with stationary initial distribution converge weakly in the space $\mathbf{D}([0,1],\mathbb{R}^n)$, equipped with the Skorokhod topology, to the law of the stationary reflected Brownian mo…
We propose discrete random-field models that are based on random partitions of $\mathbb{N}^2$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.
In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\to\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total…
We analyze the classical Brownian risk models discussing the approximation of ruin probabilities (classical, γ-reflected, Parisian and cumulative Parisian) for the case that ruin can occur only on specific discrete grids. A practical and natural grid of points is for instance G(1) = {0,1,2,...}, which allows us to study the probability of the ruin on the first day, second day, and so one. For such a discrete setting, there are no explicit formulas for the ruin probabilities mentioned above. In this contribution we derive accurate approximations of ruin probabilities for uniform grids by lettin…
This paper derives the asymptotic behavior of the following ruin probability $$P\{\exists t \in G(δ):B_H(t)-c_1t>q_1u,B_H(t)-c_2t>q_2u\}, \ \ \ u \rightarrow \infty,$$ where $B_H$ is a standard fractional Brownian motion, $c_1,q_1,c_2,q_2>0$ and $G(δ)$ denotes a regular grid $\{0,δ, 2δ,...\}$ for some $δ>0$. The approximation depends on $H$, $δ$ (only when $H\leq 1/2$) and the relations between parameters $c_1,q_1,c_2,q_2$.
Brownian motion with darning (BMD in abbreviation) is introduced and studied in [4] and [5, Chapter 7]. Roughly speaking, BMD travels across the "darning area" at infinite speed, while it behaves like a regular BM outside of this area. In this paper we show that starting from a single point in its state space, BMD is the weak limit of a family of continuous-time simple random walks on square lattices with diminishing mesh sizes. From any vertex in their state spaces, the approximating random walks jump to its nearest neighbors with equal probability after an exponential holding time.
The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. …
We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the restrictions in [11] (or [3]) that the underlying state space has to be bounded, and that the initial distribution of the limiting continuous process has to be its invariant distribution. The approach in this paper is that we first obtain heat kernel upper bounds for the approximating random walks that are uniform in their mesh size, by establishing a Nash-type inequal…
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…
For large $n$, take a random $n \times n$ permutation matrix and its associated discrete copula $X_n$. For $a, b = 0, 1, \ldots, n$, let $y_n(\frac{a}{n},\frac{b}{n}) = \frac{1}{n} ( X_{a,b} - \frac{ab}{n} )$; define $y_n: [0,1]^2 \to R$ by interpolating quadratically on squares of side $\frac{1}{n}$. We prove a Donsker type central limit theorem: $\sqrt{n} y_n$ approaches a bridged Brownian sheet on the unit square.
The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black-Scholes model, with …
In this paper we study the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) introduced in [4] by continuous time random walks on square lattices. The state space of BMVD contains a $2$-dimensional component, a $3$-dimensional component, and a "darning point" which joins these two components. Such a state space is equipped with the geodesic distance, under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous time reversible random walks with exponent…
We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly con…
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
We study the asymptotic behaviour of the maximum local time L*(t) of the Brox's process, the diffusion in Brownian environment. Shi proved that the maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in the discrete case it is t. We show here that t log(log(log t)) is the proper rate and we prove that for the minimum speed the rate is the same as in the discrete case namely t/log(log(log t)).
We propose some class of statistics suitable for estimation of the Hurst index of the fractional Brownian motion based on the second order increments of an observed discrete trajectory.
We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimators for both the Hurst index as well as a vector of unknown parameters in the model based on a single time series of observations from the slow process only. We prove that these estimators are both consistent and asymptotically normal as the amplitude of the perturbation and the time-scale separation parameter go to zero. Numerical simulations illustrate the theoretical results.
This note extends the results in [8] to construct the rough super-Brownian motion on finite volume with Dirichlet boundary conditions. The backbone of this study is the convergence of discrete approximations of the parabolic Anderson model (PAM) on a box.
We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.
The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(δ^{2H-1})$ ($δ$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(δ^H)$.
Strongly consistent and asymptotically normal estimators of the Hurst parameter of solutions of stochastic differential equations are proposed. The estimators are based on discrete observations of the underlying processes.
We prove that the Airy process, A(t), locally fluctuates like a Brownian motion. In the same spirit we also show that in a certain scaling limit, the so called discrete polynuclear growth (PNG) process behaves like a Brownian motion.
We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes.
We show that for all positive beta the semigroups of beta-Dyson Brownian motions of different dimensions are intertwined. The proof relates beta-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption beta>1 in [GS]. The corresponding results for beta-Dyson Ornstein-Uhlenbeck processes are also presented.
Strongly consistent and asymptotically normal estimators of the Hurst index and volatility parameters of solutions of stochastic differential equations with polynomial drift are proposed. The estimators are based on discrete observations of the underlying processes.
In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the est…
We study the scenery reconstruction problem on the $d$-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces. In particular, with the right drift, Brownian motion can be used to reconstruct any scenery. To this end, we prove an injectivity property of an infinite Vandermonde matrix.
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
We consider branching random walks built on Galton-Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of "globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, "globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the "lineage'' of the nodes in a Galton-Watson tree conditioned b…
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam and van Moerbeke that the process of eigenvalues of two consecutive minors of an Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.
For stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H>1/2$, Harnack type inequalities are established by constructing a coupling with unbounded time-dependent drift. These inequalities are applied to the study of existence and uniqueness of invariant measure for a discrete Markov semigroup constructed in terms of the distribution of the solution. Furthermore, we show that entropy-cost inequality holds for the invariant measure.
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is the fractional Brownian motion. We have to use two radically different models for both cases ${1\over2}\leq H<1$ and $0<H<{1\over2}$. This result provides an algorithm for the simulation of the fractional Brownian motion, which appears to be quite efficient.
In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call $D$th-random feuilletages (denoted by ${\bf r}[D]$), indexed by a parameter $D\geq 0$ and which are candidate to play the role of the Brownian map in dimension $D$. The construction relies on some obj…
In this monograph, we construct and study a sigma-finite measure on continuous functions from R_+ to R, strongly related to many probability measures obtained by penalisation of Brownian motion, i.e. as limits of probabilities which are absolutely continuous with respect to Wiener measure. This remarkable sigma-finite measure can be generalized in three other cases: one can start from a two-dimensional Brownian motion, from a recurrent diffusion with values in R_+, and from a discrete, recurrent Markov chain.
The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for coupling the two processes, as well as solutions to the discrete Dirichlet problem in various domains which, to the author's knowledge, have not been published anywhere. The main focus is on the two-dimensional processes.
We consider a Brownian motion with drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.
Strongly consistent and asymptotic normal estimators of the Hurst index of a stochastic differential equation driven by a fractional Brownian motion are proposed. The estimators are based on discrete observations of the underlying process.
We consider super-Brownian motion whose historical paths reflect from each other, unlike those of the usual historical super-Brownian motion. We prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open. We prove a few results about path behavior for processes under any limit distribution. In particular, we show that for any $γ>0$, a "typical" increment of a reflecting historical path over a small time interval $Δt$ is not greater than $(Δt)^{3/4 - γ}$.
The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form $X_t = θG(t) + B_t$, where $B$ is a Gaussian process, $G(t)$ is a known function, and $θ$ is an unknown drift parameter. The estimation techniques for the cases of discrete-time and continuous-time observations are presented. As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered. Secondly, we study in detail the model with two independent fractional Brownian motio…
