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Lauri Viitasaari

Publications and source records attributed to Lauri Viitasaari.

84 records · Page 5Linked to original sources
arXiv2015

Sufficient and Necessary Conditions for Limit Theorems for Quadratic Variations of Gaussian Sequences

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 for different types of convergence which include convergence in probability, almost sure convergence, $L^p$-convergence as well as convergence in law. Furthermore, we study general Gaussian vectors from which different interesting cases including first or second order quadratic variations can be studied by appropriate choice of the underlying vector. Finally, we provide a practical and simple approach to attack the problem which simplifies the existing methodology considerably.

math.PR↗
arXiv2014

Pathwise integrals and Ito-Tanaka Formula for Gaussian processes

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 Föllmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of Ito-Tanaka formula for pricing and hedging of financial derivatives.

math.PR↗
arXiv2014

Pathwise stochastic integrals and Itô formula for multidimensional Gaussian processes

In this article we study existence of pathwise stochastic integrals with respect to a general class of $n$-dimensional Gaussian processes and a wide class of adapted integrands. More precisely, we study integrands which are functions that are of locally bounded variation with respect to all variables. Moreover, multidimensional Itô formula is derived.

math.PR↗
arXiv2014

Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind

Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -θX_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ θ>0$ with Gaussian driving noise $ Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}$, where $ a_t= H e^{\frac{t}{H}}$ and $B$ is a fractional Brownian motion with Hurst parameter $H \in (0,1)$. In this article, we consider the case $H>\frac{1}{2}$. Then using the ergodicity of $\text{fOU}_{2}$ process, we construct consistent estimators of drift parameter $θ$ based on discrete observations in two possible cases: $(i)$ the Hurst parameter $H$ is known and $(ii)$ the Hurst parameter $H$ is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range $H \in (\frac{1}{2},1)$.

math.PR↗
arXiv2014

Adapted integral representations of random variables

We study integral representations of random variables with respect to general Hölder 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 distribution. If in addition the random variable is a final value of an adapted Hölder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.

math.PR↗
arXiv2014

Necessary and Sufficient Conditions for Hölder Continuity of Gaussian Processes

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 Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov-Čentsov condition.

math.PR↗
arXiv2014

Multidimensional Breeden-Litzenberger representation for state price densities and static hedging

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 payoff functions $h:\R_+^n\rightarrow \R$. We also give new relations between prices of different options both in one dimensional and multidimensional case.

math.PR↗
arXiv2014

Note on multidimensional Breeden-Litzenberger representation for state price densities

In this note, we consider European options of type $h(X^1_T, X^2_T,\ldots, X^n_T)$ depending on several underlying assets. We give a multidimensional version of the result of Breeden and Litzenberger \cite{Breeden} on the relation between derivatives of the call price and the risk-neutral density of the underlying asset. The pricing measure is assumed to be absolutely continuous with respect to the Lebesgue measure on the state space.

math.PR↗
arXiv2013

Rate of convergence for discrete approximation of option prices

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-Scholes prices. We also consider smooth convergence in which the approximation does not oscillate for general class of payoff functions.

math.PR↗
arXiv2012

Option prices with call prices

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 price. Moreover, we derive similar results for barrier type options. As a consequence, we obtain a static hedging for general options in the general class of models. Our result can be utilised in several significant applications. As a simple example, we derive an upper bound for the value of a general American option with convex payoff and characterise conditions under which the value of this option equals to the value of the corresponding European option.

q-fin.PR↗
arXiv2012

Rate of convergence for discretization of integrals with respect to Fractional Brownian motion

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 discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as possible to $H - 1/2$.

math.PR↗