Bruno Rémillard

HEC Montréal

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Bouchra Nasri

Université de Montréal

Christian Genest

McGill University

Kilani Ghoudi

United Arab Emirates University

Nicolas Papageorgiou

HEC Montréal

Jean Vaillancourt

HEC Montréal

Anatoliy Swishchuk

University of Calgary

Alexandre Hocquard

Fio Corporation (Canada)

Jean‐François Quessy

Université du Québec à Trois-Rivières

Jean‐François Renaud

Université du Québec à Montréal

Hatem Ben‐Ameur

HEC Montréal

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HEC Montréal

Université de Montréal

Université Laval

McGill University

Centre National de la Recherche Scientifique

Université du Québec à Montréal

McMaster University

Carleton University

Natural Sciences and Engineering Research Council

Statistics Canada

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On one-dimensional Riccati diffusions

The Annals of Applied Probability (2019)

Adrian N. Bishop Pierre Del Moral Kengo Kamatani Bruno Rémillard

This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional 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.

10.1214/18-aap1431

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Citations (17)

Testing For Equality Between Two Copulas

Swiss Finance Institute Research Paper Series (2006)

Bruno Rémillard Olivier Scaillet

We develop a test of equality between two dependence structures estimated through empirical copulas. We provide inference for independent or paired samples. The multiplier central limit theorem is used for calculating p-values of the Cram´er-von Mises test statistic. Finite sample properties are assessed with Monte Carlo experiments. We apply the testing procedure on empirical examples in finance, psychology, insurance and medicine.

Statistic

Central limit theorem

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Multivariate General Compound Point Processes in Limit Order Books

SSRN Electronic Journal (2020)

Qi Guo Bruno Rémillard Anatoliy Swishchuk

In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. Law of large numbers (LLN) and two functional central limit theorems (FCLTs) for the MGCPP were proved in this work. Applications of the MGCPP in the limit order market were also considered. We provided numerical simulations and comparisons for the MGCPP and MGCHP by applying Google, Apple, Microsoft, Amazon, and Intel trading data.

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MULTIVARIATE HAWKES-BASED MODELS IN LIMIT ORDER BOOK: EUROPEAN AND SPREAD OPTION PRICING

International Journal of Theoretical and Applied Finance (2024)

Qi Guo Anatoliy Swishchuk Bruno Rémillard

In this paper, we consider the pricing problem of European options and spread options for the Hawkes-based model in the limit order book (LOB). We introduce a variant of Hawkes process and consider its limit theorems, namely the exponential multivariate general compound Hawkes process (EMGCHP). We also consider a special case of one-dimensional EMGCHP and its limit theorems. Option pricing with one-dimensional EMGCHP in LOB and numerical examples are presented. We also discuss implied volatility and implied order flow. It reveals the relationship between stock volatility and the order flow in the LOB system. In this way, the Hawkes-based model can provide more market forecast information than the classical Black–Scholes model. Margrabe’s spread options valuations with two one-dimensional and one two-dimensional Hawkes-based models for two assets are presented.

10.1142/s0219024923500280

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Sur les structures cristallines de deux diesters apparentés du trans-bis(hydroxyméthyl)-1,4 cyclohexane

Acta Crystallographica Section B (1982)

F. Brisse Bruno Rémillard

10.1107/s0567740882004154

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Compound Hawkes Processes in Limit Order Books

RePEc: Research Papers in Economics (2017)

Anatoliy Swishchuk Bruno Rémillard Robert J. Elliott Jonathan A. Chávez-Casillas

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.

Central limit theorem

Order book

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Option Pricing and Hedging for Discrete Time Autoregressive Hidden Markov Model

Les Cahiers du GERAD (2017)

Massimo Caccia Bruno Rémillard

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 proposed methodology, 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. Secondly, we present out-of-sample hedging results on S\&P 500 vanilla options as well as a trading strategy based on theoretical prices, which we compare to simpler models including the classical Black-Scholes delta-hedging approach.

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A Stochastic Analysis of a Siqr Epidemic Model With Short and Long-term Prophylaxis

Research Square (Research Square) (2022)

Idriss Sekkak Bouchra Nasri Bruno Rémillard Jude Dzevela Kong Mohamed El Fatini

Abstract This paper aims to incorporate a high order stochastic perturbation into a SIQR epidemic model with transient prophylaxis and lasting prophylaxis. The existence and uniqueness of the global positive solution is proven and a stochastic condition in order to study the extinction of an infectious disease is established. The existence of a stationary distribution for the stochastic epidemic model is investigated as well. Numerical simulations are conducted to support our theoretical results and an example of application with COVID-19 data from Canada is used to estimate the transmission rate and basic reproduction number while constructing a model fitting the data.

Epidemic model

Basic reproduction number

Extinction (optical mineralogy)

Stochastic modelling

Stationary distribution

10.21203/rs.3.rs-1762043/v1

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On the joint asymptotic behavior of two rank-based estimators of the association parameter in the gamma frailty model

Statistics & Probability Letters (2005)

Christian Genest Jean‐François Quessy Bruno Rémillard

Goodness of fit

Rank (graph theory)

10.1016/j.spl.2005.03.016

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Citations (19)

Une introduction aux probabilités.

HAL (Le Centre pour la Communication Scientifique Directe) (2006)

Sylvain Rubenthaler Pierre Del Moral Bruno Rémillard

Cet ouvrage presente un panorama complet des probabilites de la theorie de la mesure aux diverses applications de cette branche des mathematiques. La redaction souligne divers points de contacts entre les probabilites, la combinatoire, l'algebre, la theorie de graphes, et l'ingenierie stochastique moderne. La lecture de ce livre ne necessite aucune connaissance prealable sur la theorie de la mesure, et tous les resultats enonces sont demontres avec soin. Une centaine d'exercices et problemes corriges ainsi qu'une cinquantaine de figures faciliteront la comprehension du lecteur. La completude de ce livre et ses nombreux exemples en font le compagnon ideal de l'etudiant interesse par les applications des mathematiques, de la licence au master, des classes preparatoires, ainsi que du candidat a l'agregation.

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