A Karhunen–Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables

Deheuvels, Paul; Martynov, G. V.

doi:10.1016/j.jfa.2008.07.021

Cited by 24 publications

(18 citation statements)

References 43 publications

(50 reference statements)

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“…It also allows the modeler to choose regularity and self-similarity properties independently of each other, which offers more flexibility than the models considered in [11,10,38]. More examples of Gaussian self-similar process can be found in [8,18]. Interestingly, many of the Gaussian self-similar models share the same path regularity properties as fBm, because it can be shown that there are positive finite constants c, C for which c |t − s|…”

Section: Specific Motivations and Modeling Choicesmentioning

confidence: 99%

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

2018

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We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter H, its first Karhunen-Loève eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for H result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1 2 and 3 2 , and the estimator for H sees an affine adjustment, while remaining model-free. AMS 2010 Classification: 60G15, 91G20, 40E05.

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Section: Specific Motivations and Modeling Choicesmentioning

confidence: 99%

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

2018

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“…The spectral decomposition (1.2) is a useful tool in both theory and applications (see, e.g., [1], [16]). However explicit solutions to the eigenproblem (1.1) are notoriously hard to find and they are available only in special cases [12], [9,8], [21], [22,23], including the Brownian motion with K(s, t) = t ∧ s: λ n = 1 (n − 1 2 )π 2 and ϕ n (t) = √ 2 sin(n − 1 2 )πt (1.3) and the Brownian bridge with covariance function K(s, t) = s ∧ t − st: λ n = 1 πn 2 and ϕ n (t) = √ 2 sin πnt. (1.4) These formulas are obtained by reduction of the eigenproblem for integral operators to explicitly solvable boundary value problems for ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

On the eigenproblem for Gaussian bridges

Chigansky¹,

Kleptsyna²,

Marushkevych³

2020

Bernoulli

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Spectral decomposition of the covariance operator is one of the main building blocks in the theory and applications of Gaussian processes. Unfortunately it is notoriously hard to derive in a closed form. In this paper we consider the eigenproblem for Gaussian bridges. Given a base process, its bridge is obtained by conditioning the trajectories to start and terminate at the given points. What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.Various aspects of general Gaussian bridges are discussed in [11], [25]. Aside of being interesting mathematical objects, they are important ingredients in applications, such as statistical hypothesis testing [15], exact sampling of diffusions [3], etc.The covariance operator of the bridge with kernel (1.5) is a rank one perturbation of the covariance operator of its base process. This explains similarity between (1.3) and (1.4) and suggests that the spectra of the two processes must be closely related in general. This is indeed the case and one can find an exact expression for the Fredholm determinant of K in terms of the Fredholm determinant of K even for more general finite rank perturbations (see, e.g., [26], Ch. II, 4.6 in [13]). As mentioned above, the precise formulas for the eigenvalues and eigenfunctions of K are rarely known; however, the exact asymptotic approximations can be more tractable. This raises the following question:Can the exact asymptotics of the eigenvalues and the eigenfunctions for the bridge be deduced from that of the base process ?A rough answer to this question is given by the general perturbation theory [14], which implies that the eigenvalues of K and K agree in the leading asymptotic term, as it happens for (1.3) and (1.4) (see, e.g., the proof of Lemma 2 in [4]). More delicate spectral discrepancies are harder to exhibit and seem to be highly dependent on the perturbation structure. This is vividly demonstrated in the paper [21], where the kernels of the following form are considered, cf. (1.5):K Q (s, t) = K(s, t) + Qψ(s)ψ(t).(1.6)Here Q is a scalar real valued parameter and ψ is a function in the range of K. It turns out that for any Q greater than a certain critical value Q * , the spectrum of K Q coincides

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“…In 1982, he published a short book in French called "Probability, Chance and Certainty," destined for the educated general public (Deheuvels 1982), now in its fourth edition (Deheuvels 2008). This is a true intellectual tour de force.…”

Section: Advocate For Statisticsmentioning

confidence: 99%

“…The main aim of Deheuvels and Martynov (2008) was to enlarge the class of Gaussian processes for which an explicit Karhunen-Loeve decomposition may be computed (see also Pycke (2001)). This class is not so large.…”

Section: Brownian Quadratic Functionals With the Same Law (Deheuvels mentioning

confidence: 99%

Mathematical Statistics and Limit Theorems

Yu.

Ahsanullah²

2015

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A Karhunen–Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables

Cited by 24 publications

References 43 publications

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

On the eigenproblem for Gaussian bridges

Mathematical Statistics and Limit Theorems

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