Papers by Bruno Remillard
Social Science Research Network, 2017
Parrondo's paradox is extended to regime switching random walks in random environments. The parad... more Parrondo's paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec's theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.
Statistical Science, 2021
Donald Andrew Dawson (Don Dawson) was born in 1937. He received a bachelor's degree in 1958 and a... more Donald Andrew Dawson (Don Dawson) was born in 1937. He received a bachelor's degree in 1958 and a master's degree in 1959 from McGill University and a Ph.D. in 1963 from M.I.T. under the supervision of Henry P. McKean, Jr. Following an appointment at McGill University as professor for 7 years, he joined Carleton University in 1970 where he remained for the rest of his career. Among his many contributions to the theory of stochastic processes, his work leading to the creation of the Dawson-Watanabe superprocess and the analysis of its remarkable properties in describing the evolution in space and time of populations, stand out as milestones of modern probability theory. His numerous papers span the whole gamut of contemporary hot areas, notably the study of stochastic evolution equations, measure-valued processes, McKean-Vlasov limits, hierarchical structures, super-Brownian motion, as well as branching, catalytic and historical processes. He has over 200 refereed publications and 8 monographs, with an impressive number of citations, more than 7000. He is elected Fellow of the Royal Society and of the Royal Society of Canada, as well as Gold medalist of the Statistical Society of Canada and elected Fellow of the Institute of Mathematical Statistics. We realized this interview to celebrate the outstanding contribution of Don Dawson to 50 years of Stochastics at Carleton University.
The Annals of Applied Probability, 2019
This article is concerned with the fluctuation analysis and the stability properties of a class o... more This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These onedimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.
SSRN Electronic Journal, 2019
We consider several time series and for each of them, we fit an appropriate dynamic parametric mo... more We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of- fit procedure based on Cramer-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.
SSRN Electronic Journal, 2017
In this paper we solve the discrete time mean-variance hedging problem when asset returns follow ... more In this paper we solve the discrete time mean-variance hedging problem when asset returns follow a multivariate autoregressive hidden Markov model. Time dependent volatility and serial dependence are well established properties of financial time series and our model covers both. To illustrate the relevance of our recommended approach, we first compare the proposed model with the well-known hidden Markov model via likelihood ratio tests and a novel goodness-of-fit test on the S&P 500 daily returns. In addition, we present out-of-sample hedging results on S&P 500 vanilla options as well as a trading strategy based on the difference between theoretical and market prices. This strategy is compared to simpler models including the classical Black-Scholes delta-hedging approach.
Journal of Multivariate Analysis, 2019
La collection Les Cahiers du GERAD est constituée des travaux de recherche menés par nos membres.... more La collection Les Cahiers du GERAD est constituée des travaux de recherche menés par nos membres. La plupart de ces documents de travail aété soumisà des revues avec comité de révision. Lorsqu'un document est accepté et publié, le pdf original est retiré si c'est nécessaire et un lien vers l'article publié est ajouté.
SSRN Electronic Journal, 2017
In this paper we introduce two new Hawkes processes, namely, compound and regime-switching compou... more In this paper we introduce two new Hawkes processes, namely, compound and regime-switching compound Hawkes processes, to model the price processes in limit order books. We prove Law of Large Numbers and Functional Central Limit Theorems (FCLT) for both processes. The two FCLTs are applied to limit order books where we use these asymptotic methods to study the link between price volatility and order flow in our two models by using the diffusion limits of these price processes. The volatilities of price changes are expressed in terms of parameters describing the arrival rates and price changes. We also present some numerical examples.
SSRN Electronic Journal, 2015
We find the variance-optimal equivalent martingale measure when multivariate assets are modeled b... more We find the variance-optimal equivalent martingale measure when multivariate assets are modeled by a regime-switching geometric Brownian motion, and the regimes are represented by a homogeneous continuous time Markov chain. Under this new measure, the Markov chain driving the regimes is no longer homogeneous, which differs from the equivalent martingale measures usually proposed in the literature. We show the solution minimizes the mean-variance hedging error under the objective measure. As argued by Schweizer (1996), the variance-optimal equivalent measure naturally extends canonical option pricing results to the case of an incomplete market and the expectation under the proposed measure may be interpreted as an option price. Solutions for the option value and the optimal hedging strategy are easily obtained from Monte Carlo simulations. Two applications are considered.
The aim of this paper is to present efficient algorithms for the detection of multiple targets in... more The aim of this paper is to present efficient algorithms for the detection of multiple targets in noisy images. The algorithms are based on the optimal filter of a multidimensional Markov chain signal. We also present some simulations, in the case of one, two and three targets, showing the efficiency of the method for detecting the positions of the targets.
It can be shown that when the payoff function is convex and decreasing (respectively increasing) ... more It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not preserve the convexity or monotonicity properties. In this paper, we propose a method of approximation for American options which can preserve both convexity and monotonicity. The resulting values can then be used to define exercise times and can also be used in combination with primal-dual methods to get sharper bounds. Other application of the algorithm include finding optimal hedging strategies. Recently, two-stage Monte Carlo methods were developed by Rogers (2002), Andersen and Broadie (2004) and Haugh and Kogan (2004). For an interesting review of Monte Carlo methods, see Fu et al. (2001).
SSRN Electronic Journal, 2013
In this paper we present a forecasting method for time series using copula-based models for multi... more In this paper we present a forecasting method for time series using copula-based models for multivariate time series. We study how the performance of the predictions evolves when changing the strength of the di erent possible dependencies, as well as the structure of the dependence. We also look at the impact of the marginal distributions. The impact of estimation errors on the performance of the predictions is also considered. In all the experiments, we compare predictions from our multivariate method with predictions from the univariate version which has been introduced in the literature recently. To simplify implementation, a test of independence between univariate Markovian time series is proposed. Finally, we illustrate the methodology by a practical implementation with nancial data.
Asymptotic Methods in Probability and Statistics, 1998
ABSTRACT . Usually, empirical distribution functions are used to estimate the theoretical distrib... more ABSTRACT . Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions `(X) of the observable random variable X. In practice, many researchers are using empirical distribution functions constructed from residuals, which are estimations of a non-observable error terms in linear models. This falls under a class of more general problems in which one is interested in the estimation of the distribution function of a non-observable random variable `(Q; X) depending on an observable random variable X together with its unknown law Q. When Q is estimated by some Qn , the quantities `(Qn ; X i ) are called pseudo-observations. Some work has been done recently when the pseudo-observations are the so-called residuals of linear models. The aim of this paper is to provide some tools to study the asymptotic behavior of empirical processes constructed from general pseudo-observations. Examples of pseudoobservations will be given together with applica...
SSRN Electronic Journal, 2011
This paper aims at developing a robust and omnibus procedure for checking the independence of two... more This paper aims at developing a robust and omnibus procedure for checking the independence of two time series. Li and Hui (1994) proposed a robustified version of Haugh's (1976) classic portmanteau statistic which is based on a fixed number of lagged residual cross-correlations. In order to obtain a consistent test for independence against an alternative of serial crosscorrelation of an arbitrary form between the two series, Hong's (1996a) introduced a class of statistics that take into account all possible lags. The test statistic is a weighted sum of residual cross-correlations and the weighting is determined by a kernel function. With the truncated uniform kernel, we retrieve a normalized version of Haugh's statistic. However, several kernels lead to a greater power. Here, we introduce a robustified version of Hong's statistic. We suppose that for each series, the true ARMA model is estimated by a n 1/2-consistent robust method and the robust cross-correlation is so obtained. Under the null hypothesis of independence, we show that the robust statistic asymptotically follows a N (0, 1) distribution. Using a result of Li and Hui, we also propose a robust procedure for checking independence at individual lags and a descriptive causality analysis in the Granger's sense is discussed. The level and power of the robust version of Hong's statistic are studied by simulation in finite samples. Finally, the proposed robust procedures are applied to a set of financial data.
SSRN Electronic Journal, 2007
discuss the technical challenges of implementing a multi-variate extension of Dybvig (1988) model... more discuss the technical challenges of implementing a multi-variate extension of Dybvig (1988) model and discuss the possible solutions.
Journal of Multivariate Analysis, 1996
Let Z 1 , ..., Z n be a random sample of size n 2 from a d-variate continuous distribution functi... more Let Z 1 , ..., Z n be a random sample of size n 2 from a d-variate continuous distribution function H, and let V i, n stand for the proportion of observations Z j , j{i, such that Z j Z i componentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution function K n derived from the (dependent) pseudo-observations V i, n. This random quantity is a natural nonparametric estimator of K, the distribution function of the random variable V=H(Z), whose expectation is an affine transformation of the population version of Kendall's tau in the case d=2. Since the sample version of { is related in the same way to the mean of K n , Genest and Rivest (1993, J. Amer. Statist. Assoc.) suggested thatn[K n (t)&K(t)] be referred to as Kendall's process. Weak regularity conditions on K and H are found under which this centered process is asymptotically Gaussian, and an explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions.
Canadian Journal of Statistics, 2003
ABSTRACT Weighted symmetry is an extension of the classical notion of symmetry in which the tails... more ABSTRACT Weighted symmetry is an extension of the classical notion of symmetry in which the tails of a distribution are similar, up to a scaling factor. The authors develop test statistics of weighted symmetry based on empirical processes. The finite-dimensional distributions of the proposed statistics are either non-parametric or conditionally nonparametric, according as the parameters of weighted symmetry are known or estimated. Asymptotically, the distributions of the processes behave like Brownian bridges or motions, leading to familiar distributions for the proposed test statistics. The authors also establish the asymptotic normality of Hodges-Lehmann type estimators based on a generalization of the Wilcoxon signed rank test. Furthermore, they propose density estimators in mat setting.La symétrie pondérée est une généralisation de la notion classique de symétrie dans laquelle les queues d'une distribution sont similaires, à un facteur près. Les auteurs déduisent de processus empiriques des statistiques permettant de tester la symétrie pondérée. À taille finie, les lois des statistiques étudiées sont soit non paramétriques, soit conditionnellement non paramétriques, selon que les paramètres régissant la symétrie pondérée sont connus ou estimés. Asymptotiquement, les processus empiriques se comportent comme le mouvement ou le pont brownien, ce qui conduit à des lois limites familières pour les statistiques proposées. Les auteurs établissent aussi la normalité asymptotique d'estimateurs de type Hodges-Lehmann obtenus par généralisation de la statistique des rangs signés de Wilcoxon. Ils proposent de plus des estimateurs de la densité adaptés à ce contexte.
Bernoulli, 1998
Let fL(t): t > 0g be Le Âvy's area process, let ã: R U 3 R, and let f Z t : t > 3g be the stochas... more Let fL(t): t > 0g be Le Âvy's area process, let ã: R U 3 R, and let f Z t : t > 3g be the stochastic process de®ned by Z t (s) L(ts)a(2t log log t), 0 < s < 1. Conditions on ã are given such that the set of all limit points of fã(t) Z t : t > 3g as t 3 I is a.s. equal to the set of all continuous functions de®ned on [0, 1] which vanish at 0.
Advances in Applied Probability, 2008
While the convergence properties of many sampling selection methods can be proven, there is one p... more While the convergence properties of many sampling selection methods can be proven, there is one particular sampling selection method introduced in Baker (1987), closely related to ‘systematic sampling’ in statistics, that has been exclusively treated on an empirical basis. The main motivation of the paper is to start to study formally its convergence properties, since in practice it is by far the fastest selection method available. We will show that convergence results for the systematic sampling selection method are related to properties of peculiar Markov chains.
Stochastic Processes and their Applications, 2014
We study existence, uniqueness and mass conservation of signed measure valued solutions of a clas... more We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized two-dimensional Navier-Stokes equations in vorticity form introduced by Kotelenez.
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Papers by Bruno Remillard