Lauri Viitasaari

Lauri Viitasaari
Aalto University

PhD

About

74
Publications
6,430
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454
Citations
Additional affiliations
September 2015 - December 2017
Aalto University
Position
  • PostDoc Position
September 2014 - present
Universität des Saarlandes
Position
  • PostDoc Position
March 2014 - present
Aalto University
Position
  • PostDoc Position

Publications

Publications (74)
Article
In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H∈(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provi...
Article
Full-text available
In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space–time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial averag...
Article
Full-text available
We prove a result on the fractional Sobolev regularity of composition of paths of low fractional Sobolev regularity with functions of bounded variation. The result relies on the notion of variability, proposed by us in the previous article [43, https://arxiv.org/abs/2003.11698]. Here we work under relaxed hypotheses, formulated in terms of Sobolev...
Article
A complex-valued linear mixture model is considered for discrete weakly stationary processes. Latent components of interest are recovered, which underwent a linear mixing. Asymptotic properties are studied of a classical unmixing estimator which is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag...
Article
In this article, we study the asymptotic behaviour of the least-square estimator in a linear regression model based on random observation instances. We provide mild assumptions on the moments and dependence structure on the randomly spaced observations and the residuals under which the estimator is strongly consistent. In particular, we consider ob...
Preprint
In this article we study the asymptotic behaviour of the least square estimator in a linear regression model based on random observation instances. We provide mild assumptions on the moments and dependence structure on the randomly spaced observations and the residuals under which the estimator is strongly consistent. In particular, we consider obs...
Article
This paper develops a new integrated ball (pseudo)metric which provides an intermediary between a chosen starting (pseudo)metric d and the Lp distance in general function spaces. Selecting d as the Hausdorff or Fréchet distances, we introduce integrated shape-sensitive versions of these supremum-based metrics. The new metrics allow for finer analys...
Article
Full-text available
Background: Respiratory infection is the 4th most common reason for absence from work in Finland. There is limited knowledge of how social distancing affects the spread of respiratory infections during respiratory epidemics. We assessed the effect of nationwide infection control strategies against coronavirus disease in 2020 on various respiratory...
Article
Full-text available
Generalisations of the Ornstein‐Uhlenbeck process defined through Langevin equations, such as fractional Ornstein‐Uhlenbeck processes, have recently received a lot of attention. However, most of the literature focuses on the one dimensional case with Gaussian noise. In particular, estimation of the unknown parameter is widely studied under Gaussian...
Preprint
Full-text available
This paper develops a new integrated ball (pseudo)metric which provides an intermediary between a chosen starting (pseudo)metric d and the L_p distance in general function spaces. Selecting d as the Hausdorff or Fr\'echet distances, we introduce integrated shape-sensitive versions of these supremum-based metrics. The new metrics allow for finer ana...
Preprint
Full-text available
We prove a result on the fractional Sobolev regularity of composition of paths of low fractional Sobolev regularity with functions of bounded variation. The result relies on the notion of variability, proposed by us in the previous article [43, https://arXiv.org/abs/2003.11698]. Here we work under relaxed hypotheses, formulated in terms of Sobolev...
Preprint
Full-text available
In this article we consider existence and uniqueness of the solutions to a large class of stochastic partial differential of form $\partial_t u = L_x u + b(t,u)+\sigma(t,u)\dot{W}$, driven by a Gaussian noise $\dot{W}$, white in time and spatial correlations given by a generic covariance $\gamma$. We provide natural conditions under which classical...
Preprint
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In this paper we introduce the long-range dependent completely correlated mixed fractional Brownian motion (ccmfBm). This is a process that is driven by a mixture of Brownian motion (Bm) and a long-range dependent completely correlated fractional Brownian motion (fBm, ccfBm) that is constructed from the Brownian motion via the Molchan--Golosov repr...
Article
Full-text available
The classical ARCH model together with its extensions have been widely applied in the modeling of financial time series. We study a variant of the ARCH model that takes account of liquidity given by a positive stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution for this model. Moreo...
Article
This paper develops a new class of functional depths. A generic member of this class is coined Jth order kth moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show tha...
Article
Let Z=(Zt)t≥0 be the Rosenblatt process with Hurst index H∈(1∕2,1). We prove joint continuity for the local time of Z, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of Z. Based on analogy with similar known results in the case of fractional Brownian motion, we believe o...
Article
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In this article we introduce and study oscillating Gaussian processes defined by Xt=α+Yt1Yt>0+α-Yt1Yt<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_t = \alpha _+ Y...
Preprint
In this article we present a {\it quantitative} central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space-time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial...
Article
Full-text available
The complexity of biological systems is encoded in gene regulatory networks. Unravelling this intricate web is a fundamental step in understanding the mechanisms of life and eventually developing efficient therapies to treat and cure diseases. The major obstacle in inferring gene regulatory networks is the lack of data. While time series data are n...
Article
Full-text available
We consider the one-dimensional stochastic heat equation driven by a multiplicative space–time white noise. We show that the spatial integral of the solution from −R to R converges in total variance distance to a standard normal distribution as R tends to infinity, after renormalization. We also show a functional version of this central limit theor...
Article
Full-text available
In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the...
Preprint
Let $Z = (Z_t)_{t \geq 0}$ be the Rosenblatt process with Hurst index $H \in (1/2, 1)$. We prove joint continuity for the local time of $Z$, and establish H\"older conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Bro...
Article
We develop a unified framework for analyzing and optimizing costs for systems of FCFS queues with batch arrivals, setup delays and a general nonlinear cost structure that includes costs associated with energy used, setup times and Quality of Service (QoS) measures. We focus on the MX/G/1 and GeoX/G/1 queues with i.i.d. service times, but our result...
Preprint
Full-text available
We define compositions $\varphi(X)$ of H\"older paths $X$ in $\mathbb{R}^n$ and functions of bounded variation $\varphi$ under a relative condition involving the path and the gradient measure of $\varphi$. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions $\varphi(X)$ with respect to a given H\"older p...
Preprint
We propose a novel strategy for multivariate extreme value index estimation. In applications such as finance, volatility and risk present in the components of a multivariate time series are often driven by the same underlying factors, such as the subprime crisis in the US. To estimate the latent risk, we apply a two-stage procedure. First, a set of...
Preprint
We consider a complex-valued linear mixture model, under discrete weakly stationary processes. We recover latent components of interest, which have undergone a linear mixing. We study asymptotic properties of a classical unmixing estimator, that is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag...
Article
Full-text available
The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.
Article
Full-text available
In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X = ∑ k ≥ 1 λ k ( ξ k 2 - 1 ) , where ξ k ∼ N ( 0 , 1 ) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In p...
Article
Full-text available
In this article we study the asymptotic behaviour of the realized quadratic variation of a process $\int_{0}^{t}u_{s}dY_{s}^{(1)}$, where $u$ is a $\beta$-H\"older continuous process with $\beta > 1-H$ and $% Y_{t}^{(1)}=\int_{0}^{t}e^{-s}dB^{H}_{a_s}$, where $a_{t}=He^{\frac{t}{H}} $ and $B^H$ is a fractional Brownian motion with Hurst index $H\in...
Preprint
Full-text available
Generalisations of the Ornstein-Uhlenbeck process defined through Langevin equation $dU_t = - \Theta U_t dt + dG_t,$ such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention in the literature. In particular, estimation of the unknown parameter $\Theta$ is widely studied under Gaussian stationary increment noise $G$...
Preprint
We study one-dimensional stochastic differential equations of form $dX_t = \sigma(X_t)dY_t$, where $Y$ is a suitable H\"older continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $\sigma$ for which we assume very mild conditions....
Article
Full-text available
We study the regular conditional law of mixed Gaussian Volterra processes under the influence of model disturbances. More precisely, we study prediction of Gaussian Volterra processes driven by a Brownian motion in a case where the Brownian motion is not observable, but only a noisy version is observed. As an application, we discuss how our result...
Preprint
Denote by $\mu_\beta="\exp(\beta X)"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\subset\mathbb{R}^d$. The case $\beta\in\mathbb{R}$ has been studied quite intensively, and then $\mu_\beta$ is a random measure on $U$. It is known that $\mu_\beta$ can also be defined for complex values...
Preprint
In this article we introduce and study oscillating Gaussian processes defined by $X_t = \alpha_+ Y_t {\bf 1}_{Y_t >0} + \alpha_- Y_t{\bf 1}_{Y_t<0}$, where $\alpha_+,\alpha_->0$ are free parameters and $Y$ is either stationary or self-similar Gaussian process. We study the basic properties of $X$ and we consider estimation of the model parameters....
Preprint
In this article we study effects that small perturbations in the noise have to the solution of differential equations driven by H\"older continuous functions of order $H>\frac12$. As an application, we consider stochastic differential equations driven by a fractional Brownian motion. We introduce a Wong--Zakai type stationary approximation to the f...
Preprint
Full-text available
We study the regular conditional law of mixed Gaussian Volterra processes under the influence of model disturbances. More precisely, we study prediction of Gaussian Volterra processes driven by a Brownian motion in a case where the Brownian motion is not observable, but only a noisy version is observed. As an application, we discuss how our result...
Preprint
Full-text available
The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.
Preprint
Full-text available
In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the...
Article
Full-text available
It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this e...
Chapter
Let X be a Lei-Nualart process with Hurst index H ∈ (0, 1). We study the asymptotic behavior of quadratic variations of process X in case when the Hurst parameter H greater or equal than 3/4 using the total variation distance.
Preprint
Full-text available
We consider the one-dimensional stochastic heat equation driven by a multiplicative space-time white noise. We show that the spatial integral of the solution from $-R$ to $R$ converges in total variance distance to a standard normal distribution as $R$ tends to infinity, after renormalization. We also show a functional version of this central limit...
Article
Full-text available
We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry--Esseen bound of the so-called alternative estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of conti...
Preprint
One of the focus areas of modern scientific research is to reveal mysteries related to genes and their interactions. The dynamic interactions between genes can be encoded into a gene regulatory network (GRN), which can be used to gain understanding on the genetic mechanisms behind observable phenotypes. GRN inference from time series data has recen...
Preprint
Full-text available
We study a generalized ARCH model with liquidity given by a general stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution. In addition, we provide consistent estimators for the model parameters by using AR(1) type characterisation. We illustrate our results with several examples and s...
Preprint
Full-text available
It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this e...
Preprint
Structural change detection problems are often encountered in analytics and econometrics, where the performance of a model can be significantly affected by unforeseen changes in the underlying relationships. Although these problems have a comparatively long history in statistics, the number of studies done in the context of multivariate data under...
Article
Full-text available
The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion fr...
Article
We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every $\frac{1}{2}<H<1$. More precisely, we extend the consistency proved in \cite{DFW} for $\frac{1}{2}<H<\frac{3}{4}$ to the strong consistency for any $\frac{1}{2}<H<1$ on the one hand, and on the other, we also discuss the asy...
Article
Full-text available
Stationary processes have been extensively studied in the literature. Their applications include modelling and forecasting numerous real life phenomena such as natural disasters, sales and market movements. When stationary processes are considered, modelling is traditionally based on fitting an autoregressive moving average (ARMA) process. However,...
Article
Full-text available
We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black-Scholes type pricing models where...
Article
Full-text available
In this article, we show that general weakly stationary time series can be modeled applying Gaussian subordinated processes. We show that, for any given weakly stationary time series $(z_t)_{z\in\mathbb{N}}$ with given equal one-dimensional marginal distribution, one can always construct a function $f$ and a Gaussian process $(X_t)_{t\in\mathbb{N}}...
Article
Full-text available
In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and H\"older continuous random processes $X$ and $Y$. We discuss a notion of sufficient variability for the process $X$ which ensures that the paths of the composite process $t \maps...
Article
We calculate the regular conditional future law of the fractional Brownian motion with index $H\in(0,1)$ conditioned on its past. We show that the conditional law is continuous with respect to the conditioning path. We investigate the path properties of the conditional process and the asymptotic behavior of the conditional covariance.
Chapter
We review the use of fractional Brownian motion in financial modeling with an emphasis on arbitrage and hedging in the (mixed) fractional Black–Scholes model for stock prices. Moreover, short-rate models and volatility models based on fractional Ornstein–Uhlenbeck processes are briefly discussed.
Article
Full-text available
We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. In particular, we prove an It\^o formula that is, as far as we know, the most general Malliavin...
Article
In this paper, we study linear backward stochastic differential equations driven by a class of centered Gaussian non-martingales, including fractional Brownian motion with Hurst parameter $H\in (0,1)\setminus \{\frac12\}$. We show that, for every choice of deterministic coefficient functions, there is a square integrable terminal condition such tha...
Article
Full-text available
We show that every multi-parameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square-root of the covariance. We analyze the equivalence of multi-parameter Gaussian...
Article
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Quadratic variations of Gaussian processes play important role in both stochastic analysis and in applications such as estimation of model parameters, and for this reason the topic has been extensively studied in the literature. In this article we study the problem for general Gaussian processes and we provide sufficient and necessary conditions fo...
Article
Full-text available
We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, i.e., when the Hurst parameter $H$ of the fractional part satisfies $H\in(3/4,1)$ the central limit theorem holds. In the non-semimartingale case, i.e., when $H\in(1/2,3/4]$, the...
Article
It is well-known that Langevin equation $d U_t = -\theta U_td t + d W_t$ defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian motion $W_t$ with some other process $G$ with stationary increments. In this article we prove that all conti...
Article
Full-text available
This paper deals with stochastic integrals of form $\int_0^T f(X_u)d Y_u$ in a case where the function $f$ has discontinuities, and hence the process $f(X)$ is usually of unbounded $p$-variation for every $p\geq 1$. Consequently, integration theory introduced by Young or rough path theory introduced by Lyons cannot be applied directly. In this pape...
Article
Full-text available
In this note, we provide a general approach to obtain very satisfactory upper bounds for small deviations $ \Pro(\Vert y \Vert \le \epsilon)$ in different norms, namely the supremum and $\beta$- H\"older norms. The large class of processes $y$ under consideration take the form $y_t= X_t + \int_0^t a_s \ud s$, where $X$ and $a$ stand for any two sto...
Article
Full-text available
We study integral representations of random variables with respect to general H\"older continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distr...
Article
The continuity of Gaussian processes is extensively studied topic and it culminates in the Talagrand's notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the H\"older continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form tha...
Article
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We show that if a random variable is a final value of an adapted Holder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.
Article
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In this note, we consider European options of type h(X T 1 ,X T 2 ,...,X T n ) depending on several underlying assets. We give a multidimensional version of the result of D. T. Breeden and R. H. Litzenberger [“Prices of state-contingent claims implicit in option prices”, J. Bus. 51, No. 4, 621–651 (1978), http://www.jstor.org/stable/2352653] on the...
Article
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In this article, we consider European options of type $h(X^1_T, X^2_T,\ldots, X^n_T)$ depending on several underlying assets. We study how such options can be valued in terms of simple vanilla options in non-specified market models. We consider different approaches related to static hedging and derive several pricing formulas for a wide class of pa...
Article
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It was shown in \cite{m-s-v} that any random variable can be represented as improper pathwise integral with respect to fractional Brownian motion. In this paper we extend this result to cover a wide class of Gaussian processes. In particular, we consider a wide class of processes that are H\"{o}lder continuous of order $\alpha>{1}{2}$ and show that...
Article
Full-text available
We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by F\"ollmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Z\"ahle. As an applicatio...
Article
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Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -\theta X_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ \theta >0$ with driving noise $ Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}; \ a_t= H e^{{t}{H}}$ where $B$ is a fractional Brownian motion with Hurst parameter $H \in...
Article
Full-text available
In this article, we study the rate of convergence of prices when a model is approximated by some simplified model. We also provide a method how explicit error formula for more general options can be obtained if such formula is available for digital option prices. We illustrate our results by considering convergence of binomial prices to Black-Schol...
Article
There exist several methods how more general options can be priced with call prices. In this article, we extend these results to cover a wider class of options and market models. In particular, we introduce a new pricing formula which can be used to price more general options if prices for call options and digital options are known for every strike...
Article
Full-text available
In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$ is considered. In Statistics & Decisions, 27, 129-143, for any convex function $f$, the almost sure convergence of uniform disc...

Projects

Projects (3)
Project
Parameter estimation, limit central theorem and fractional processes via Malliavin calculus