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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: closed (31 March 2021) | Viewed by 11315

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Prof. Dr. Jonas Šiaulys

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Guest Editor

Institute of Mathematics, Vilnius University, Naugarduko 24, LT-032225 Vilnius, Lithuania
Interests: stochastic processes; ruin theory; actuarial mathematics; heavy tails
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Remigijus Leipus

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Guest Editor

Institute of Applied Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
Interests: financial mathematics; econometrics; time series; heavy tails

Special Issue Information

Dear Colleagues,

Probability theory and mathematical statistics are very important mathematical tools for studying various processes in nature and society where the behaviour of underlying observed phenomena is driven by randomness. This approach, where deterministic description is inefficient and should be replaced by stochastic models, is often used in finance, insurance, physics, chemistry, medicine, and many other fields.

In particular, time series models are widely used in economics and finance, where the main goal of stochastic modelling is the forecasting of future values. Besides that, financial data possess some specific “stylized facts,” such as long memory, heavy tails, conditional heteroscedasticity, etc. On the other hand, the insurance business raises specific questions related to extreme value theory, heavy-tailed distributions, premium calculation principles, estimation of ruin probabilities, etc.

The purpose of this Special Issue is to gather a collection of articles reflecting the latest developments in different fields of applied probability such as long memory processes, stochastic processes in finance and insurance, conditional heteroscedasticity, extreme value theory, insurance risk measures, heavy-tailed distributions, and others.

Prof. Dr. Jonas Šiaulys
Prof. Dr. Remigijus Leipus
Guest Editors

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Keywords

Stochastic processes for insurance and finance
Long memory
Financial time series
Change point problem
Risk measures
Extreme value theory
Heavy-tailed distributions

Published Papers (6 papers)

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Research

16 pages, 1006 KiB

Open AccessArticle

Pathwise Convergent Approximation for the Fractional SDEs

by Kęstutis Kubilius and Aidas Medžiūnas

Mathematics 2022, 10(4), 669; https://doi.org/10.3390/math10040669 - 21 Feb 2022

Cited by 1 | Viewed by 1503

Abstract

Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional [...] Read more.

1 / 2 < γ < 1

. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate

h^{2 γ}

, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field. Full article

(This article belongs to the Special Issue Applied Probability)

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14 pages, 1311 KiB

Open AccessEditor’s ChoiceArticle

Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

by Kęstutis Kubilius and Aidas Medžiūnas

Mathematics 2021, 9(1), 18; https://doi.org/10.3390/math9010018 - 23 Dec 2020

Cited by 8 | Viewed by 2190

Abstract

We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories [...] Read more.

β > 1

are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index

H > 1 / 2

for positive solutions of FSDEs. Full article

(This article belongs to the Special Issue Applied Probability)

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21 pages, 654 KiB

Open AccessArticle

A Fast-Pivoting Algorithm for Whittle’s Restless Bandit Index

by José Niño-Mora

Mathematics 2020, 8(12), 2226; https://doi.org/10.3390/math8122226 - 15 Dec 2020

Cited by 5 | Viewed by 2261

Abstract

The Whittle index for restless bandits (two-action semi-Markov decision processes) provides an intuitively appealing optimal policy for controlling a single generic project that can be active (engaged) or passive (rested) at each decision epoch, and which can change state while passive. It further provides a practical heuristic priority-index policy for the computationally intractable multi-armed restless bandit problem, which has been widely applied over the last three decades in multifarious settings, yet mostly restricted to project models with a one-dimensional state. This is due in part to the difficulty of establishing indexability (existence of the index) and of computing the index for projects with large state spaces. This paper draws on the author’s prior results on sufficient indexability conditions and an adaptive-greedy algorithmic scheme for restless bandits to obtain a new fast-pivoting algorithm that computes the n Whittle index values of an n-state restless bandit by performing, after an initialization stage, n steps that entail

(2 / 3) n^{3} + O (n^{2})

arithmetic operations. This algorithm also draws on the parametric simplex method, and is based on elucidating the pattern of parametric simplex tableaux, which allows to exploit special structure to substantially simplify and reduce the complexity of simplex pivoting steps. A numerical study demonstrates substantial runtime speed-ups versus alternative algorithms. Full article

(This article belongs to the Special Issue Applied Probability)

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12 pages, 1131 KiB

Open AccessArticle

The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application

by Saisai Ding, Xiaoqin Li, Xiang Dong and Wenzhi Yang

Mathematics 2020, 8(12), 2113; https://doi.org/10.3390/math8122113 - 26 Nov 2020

Cited by 5 | Viewed by 1329

Abstract

In this paper, we investigate the CUSUM-type estimator of mean change-point models based on m-asymptotically almost negatively associated (m-AANA) sequences. The family of m-AANA sequences contains AANA, NA, m-NA, and independent sequences as special cases. Under some weak [...] Read more.

O_{P} (n^{1 / p - 1})

O_{P} (n^{1 / p - 1} {log}^{1 / p} n)

and

O_{P} (n^{α - 1})

, where

0 \leq α < 1

and

1 < p \leq 2

. Our rates are better than the ones obtained by Kokoszka and Leipus (Stat. Probab. Lett., 1998, 40, 385–393). In order to illustrate our results, we do perform simulations based on m-AANA sequences. As important applications, we use the CUSUM-type estimator to do the change-point analysis based on three real data such as Quebec temperature, Nile flow, and stock returns for Tesla. Some potential applications to change-point models in finance and economics are also discussed in this paper. Full article

(This article belongs to the Special Issue Applied Probability)

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35 pages, 940 KiB

Open AccessArticle

Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

by Olena Ragulina and Jonas Šiaulys

Mathematics 2020, 8(11), 1885; https://doi.org/10.3390/math8111885 - 30 Oct 2020

Viewed by 1562

Abstract

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples. Full article

(This article belongs to the Special Issue Applied Probability)

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18 pages, 855 KiB

Open AccessArticle

Martingale Approach to Derive Lundberg-Type Inequalities

by Tautvydas Kuras, Jonas Sprindys and Jonas Šiaulys

Mathematics 2020, 8(10), 1742; https://doi.org/10.3390/math8101742 - 11 Oct 2020

Viewed by 1852

Abstract

In this paper, we find the upper bound for the tail probability

P ({sup}_{n ⩾ 0} \sum_{I = 1}^{n} ξ_{I} > x)

with random summands

ξ_{1}, ξ_{2}, \dots

having light-tailed distributions. We find conditions under [...] Read more.

In this paper, we find the upper bound for the tail probability

P ({sup}_{n ⩾ 0} \sum_{I = 1}^{n} ξ_{I} > x)

with random summands

ξ_{1}, ξ_{2}, \dots

having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity

ϱ_{1} exp {- ϱ_{2} x}

with some positive constants

ϱ_{1}

and

ϱ_{2}

. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model. Full article

(This article belongs to the Special Issue Applied Probability)

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