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Lévy processes conditioned to stay in a half-space with applications to directional extremes

Jevgenijs Ivanovs

Jakob D. Thøstesen

https://doi.org/10.15559/22-VMSTA217

Pub. online: 25 Nov 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 59–75

Abstract

This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution

Luigi Amedeo Bianchi

Stefano Bonaccorsi

Luciano Tubaro

https://doi.org/10.15559/22-VMSTA216

Pub. online: 21 Nov 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 37–57

Abstract

We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/22-VMSTA94MI

Pub. online: 9 Nov 2022 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 4 (2022), pp. 503–506

Keywords index

https://doi.org/10.15559/22-VMSTA94KI

Pub. online: 9 Nov 2022 Type: Keywords Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 4 (2022), pp. 501–502

Author index

https://doi.org/10.15559/22-VMSTA94AI

Pub. online: 9 Nov 2022 Type: Author Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 4 (2022), pp. 499–500

Minimax identity with robust utility functional for a nonconcave utility

Olena Bahchedjioglou

Georgiy Shevchenko

https://doi.org/10.15559/22-VMSTA215

Pub. online: 28 Oct 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 19–35

Abstract

The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.

A limit theorem for persistence diagrams of random filtered complexes built over marked point processes

Tomoyuki Shirai Kiyotaka Suzaki

https://doi.org/10.15559/22-VMSTA214

Pub. online: 13 Sep 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 1–18

Abstract

Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $\check{\text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.

Asymptotic properties of the parabolic equation driven by stochastic measure

Boris Manikin

https://doi.org/10.15559/22-VMSTA213

Pub. online: 6 Sep 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 4 (2022), pp. 483–498

Abstract

A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as $t\to \infty $ is studied.

On the mean and variance of the estimated tangency portfolio weights for small samples

Gustav Alfelt

Stepan Mazur

https://doi.org/10.15559/22-VMSTA212

Pub. online: 2 Sep 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 4 (2022), pp. 453–482

Abstract

In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/21-VMSTA84MI

Pub. online: 3 Aug 2022 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 4 (2021), pp. 529–532

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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