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Andrius Grigutis

Publications and source records attributed to Andrius Grigutis.

At least 19 recordsLinked to original sources
arXiv2026

Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function

We prove four explicit continued fraction representations for the Lerch transcendent $\Phi(z, s, M+1)$, where $\Re M>0$ and $(z,s)\in\{(-1,1),(-1,2),(1,2),(1,3)\}$. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter $M$. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For $z=1$, the corresponding representations give continued fractions for the values of the Hurwitz zeta function $\zeta(2, M+1)$ and $\zeta(3, M+1)$.

math.NT↗
arXiv2026

Generalizing Markowitz Portfolio Optimization by a Quadratic Risk Measure

We show that the key optimization results of the classical Markowitz portfolio selection theory, originally formulated for variance as the risk measure, remain available in explicit closed form under a broader class of strictly convex quadratic risk measures. The proposed framework replaces the covariance matrix with an arbitrary symmetric positive definite matrix and allows additional linear and constant terms, thereby containing various models arising in transaction cost optimization, benchmark relative optimization, covariance regularization, and factor models. Closed-form formulas are obtained for the efficient frontier, the global minimum risk portfolio, the maximum Sharpe ratio portfolio, the Capital Market Curve, the tangency portfolio, and the maximum utility portfolio. In contrast to the classical Markowitz model, the tangency portfolio does not coincide with the maximum Sharpe ratio portfolio, revealing a new geometric phenomenon. A numerical example confirms the derived formulas.

q-fin.PM↗
arXiv2026

The limit law of the maximum of discrete partial-sums distribution II

Let $X_1,\,X_2,\,\ldots,\,X_N$, $N\in\mathbb N$ be independent, discrete, integer-valued random variables. Assume that $X_j\geqslant m_j$ almost surely for each $j=1,\,2,\,\ldots,\,N$, where $m_1,\,m_2,\,\ldots,\,m_N\in\mathbb{Z}$ satisfy $m_1+\cdots+m_N<0$. Furthermore, suppose that the sequence $X_1,\,X_2,\,\ldots$ is periodic in distribution, i.e. $X_k{\buildrel d \over =} X_{k+N}$ for all $k\in\mathbb N$. We derive computable representations for the distribution functions of $\max\{X_1,\,X_1+X_2,\,\ldots\}$, $\max\{X_2,\,X_2+X_3,\,\ldots\}$, $\ldots$, $\max\{X_N,\,X_N+X_{N+1},\,\ldots\}$. The obtained formulas are based on a linear recurrence whose initial values are determined from a linear system that involves the roots of an associated characteristic equation and the distributions of $X_1,\,X_2,\,\ldots,\,X_N$. Several examples are presented, including a biseasonal-biased Rademacher random walk for which the distribution, generating functions, and all moments admit explicit closed-form expressions. In addition, we identify and correct several inaccuracies in the results reported in \cite{Grigutis2024}.

math.PR↗
arXiv2026

Several expressions of the net single premiums under the constant force of mortality

In this article, we present several formulas that make it easier to compute the net single premiums when the mortality force over the fractional ages is assumed to be constant (C). More precisely, we compute the moments of the random variables $\nu^{T_x}$, $T_x$, $T_x\nu^{T_x}$, etc., where $T_x$ denotes the future lifetime of a person who is $x\in\{0,\,1,\,\ldots\}$ years old, and $\nu$ is the annual discount multiplier. We verify the obtained formulas on the real data from the human mortality table and the Gompertz survival law. The obtained numbers are compared with the corresponding ones when the survival function over fractional ages is interpolated using the uniform distribution of deaths (UDD) and Balducci's (B) assumptions. We also formulate and prove the statement on the comparison of the moments of the mentioned random variables under assumptions (C), (UDD), and (B).

math.PR↗
arXiv2025

Note on the positivity of the real part of the log-derivative of the Riemann $\xi$-function near the critical line

In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $\xi$-function in the region $1/2+1/\sqrt{\log t}<\sigma<1$, where $t$ is sufficiently large. We provide an explicit lower bound for $\mathfrak{R}\sum_{\rho}1/(s-\rho)$, where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the log-derivative of the Riemann $\xi$-function in the provided region, assuming that there are non-trivial zeta-zeros off the critical line.

math.NT↗
arXiv2025

Picturesque convolution-like recurrences and partial sums' generation

Let ${\pmb b}=\{b_0,\,b_1,\,\ldots\}$ be the known sequence of numbers such that $b_0\neq0$. In this work, we develop methods to find another sequence ${\pmb a}=\{a_0,\,a_1,\,\ldots\}$ that is related to ${\pmb b}$ as follows: $a_n=a_0\,b_{n+m}+a_1\,b_{n+m-1}+\ldots+a_{n+m}\,b_0$, $n\in\mathbb{N}\cup\{0\}$, $m\in\mathbb{N}$. We show the connection of $\lim_{n\to\infty}a_n$ with $a_0,\,a_1,\,\ldots,\,a_{m-1}$ and provide varied examples of finding the sequence ${\pmb a}$ when ${\pmb b}$ is given. We demonstrate that the sequences ${\pmb a}$ may exhibit pretty patterns in the plane or space. Also, we show that the properly chosen sequence ${\pmb b}$ may define ${\pmb a}$ as some famous sequences, such as the partial sums of the Riemann zeta function, etc.

math.NT↗
arXiv2025

Several facts about Theodor Wittstein, Gaetano Balducci, and some expressions of the net single premiums under their mortality assumption

The mathematical essence in life insurance spins around the search of the nu\-me\-ri\-cal characteristics of the random variables $T_x$, $\nu^{T_x}$, $T_x\nu^{T_x}$, etc., where $\nu$ (deterministic) denotes the discount multiplier and $T_x$ (random) is the future lifetime of an in\-di\-vi\-dual being of $x\in\{0,\,1,\,\ldots\}$ years old. This work provides some historical facts about T. Wittstein and G. Balducci and their mortality assumption. We also develop some formulas that make it easier to compute the moments of the mentioned random variables assuming that the survival function is interpolated according to Balducci's assumption. Derived formulas are verified using some hypothetical mortality data.

math.PR↗
arXiv2024

The limit law of certain discrete multivariate distributions

Let $X_1,\,X_2,\,\ldots,\,X_N$, $N\in\mathbb{N}$ be independent but not necessarily identically distributed discrete and integer-valued random variables. Assume that $X_1\geqslant m_1$, $X_2\geqslant m_2$, $\ldots$, $X_N\geqslant m_N$ almost surely, where $m_1,\,m_2,\ldots,\,m_N$ are some integer numbers such that $m_1+m_2+\ldots+m_N<0$, and $X_k$ is identically distributed as $X_{k+N}$, for all $k\in\mathbb{N}$ in the sequence $X_1,\,X_2,\,\ldots$ In this communication, we make use of some of the known results to provide the closed-form expression of the limit multivariate distribution function $\mathbb{P}(X_1\leqslant x,\,X_1+X_2\leqslant x,\,\ldots)$, $x\in\mathbb{Z}$ via: (1) inclusion-exclusion principle based product of the roots of $G_{N}(s)=1$, where $G_{N}(s)$ is the probability generating function of $S_N=X_1+X_2+\ldots+X_N$, (2) the probability mass function of $S_N$, and (3) the expectation $\mathbb{E}S_N$.

math.PR↗
arXiv2023

On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model

In this work, we propose a simplification of the Pollaczek-Khinchine formula for the ultimate time survival (or ruin) probability calculation in exchange for a few assumptions on the random variables which generate the renewal risk model. More precisely, we show the expressibility of the distribution function $$ \mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-cθ_i) 0$, $X$ and $cθ$ are independent non-negative and integer-valued, and the support of $θ$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions.

math.PR↗
arXiv2023

Ruin probability for renewal risk models with neutral net profit condition

In ruin theory, the net profit condition intuitively means that the incurred random claims on average do not occur more often than premiums are gained. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims' inter-occurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims on average occur equally as the premiums are gained. We study the discrete-time risk model with $N\in\mathbb{N}$ periodically occurring independent distributions, the classical risk model, also known as the Cramér-Lundberg risk process, and the more general E. Sparre Andersen model.

math.PR↗
arXiv2023

Probabilistic Overview of Probabilities of Default for Low Default Portfolios by K. Pluto and D. Tasche

This article gives a probabilistic overview of the widely used method of default probability estimation proposed by K. Pluto and D. Tasche. There are listed detailed assumptions and derivation of the inequality where the probability of default is involved under the influence of systematic factor. The author anticipates adding more clarity, especially for early career analysts or scholars, regarding the assumption of borrowers' independence, conditional independence and interaction between the probability distributions such as binomial, beta, normal and others. There is also shown the relation between the probability of default and the joint distribution of $\sqrt{\varrho}X-\sqrt{1-\varrho}Y$, where $X$, including but not limiting, is the standard normal, $Y$ admits, including but not limiting, the beta-normal distribution and $X,\,Y$ are independent.

q-fin.RM↗
arXiv2022

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

In this paper, we set up the distribution function $$ φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right), $$ and the generating function of $φ(u+1)$, where $u\in\mathbb{N}_0$, $κ\in\mathbb{N}$, the random walk $\left\{\sum_{i=1}^{n}X_i, n\in\mathbb{N}\right\},$ consists of $N\in\mathbb{N}$ periodically occurring distributions, and the integer-valued and non-negative random variables $X_1,\,X_2,\,\ldots$ are independent. This research generalizes two recent works where $\{κ=1,\,N\in\mathbb{N}\}$ and $\{κ\in\mathbb{N},\,N=1\}$ were considered respectively. The provided sequence of sums $\left\{\sum_{i=1}^{n}\left(X_i-κ\right),\,n\in\mathbb{N}\right\}$ generates so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to calculate the ultimate time ruin probability $1-φ(u)$ or survival probability $φ(u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $φ(u)$ and its generating function when $\{κ=2,\,N=2\}$, $\{κ=3,\,N=2\}$, $\{κ=5,\,N=10\}$ and $X_i$ admits Poisson and some other distributions. We also conjecture the non-singularity of certain matrices.

math.PR↗
arXiv2022

Distribution of Shifted Discrete Random Walk and Vandermonde matrices

In this work we set up the generating function of the ultimate time survival probability $φ(u+1)$, where $$φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right)$$ and $u\in\mathbb{N}_0,\,κ\in\mathbb{N}$, and the random walk $\left\{\sum_{i=1}^{n}X_i,\,n\in\mathbb{N}\right\}$ consists of independent and identically distributed random variables $X_i$, which are non-negative and integer valued. We also give expressions of $φ(u)$ via the roots of certain polynomials. Based on the proven theoretical statements, we give several examples on $φ(u)$ and its generating function expressions, when random variables $X_i$ admit Bernoulli, Geometric and some other distributions.

math.PR↗
arXiv2022

On $\mathbf{2\times2}$ determinants originating from survival probabilities in homogeneous discrete time risk model

We analyze $2\times 2$ Hankel-like determinants $D_n$ that arise in the initial values problem for the ultimate time survival probability $φ(u)$ in a homogeneous discrete time risk model $W(n)=u+κn+\sum_{i=1}^nZ_i$, where $Z_i$ are positive integer valued i.i.d. random claims, the initial surplus $u \in \mathbb{N}_0$ and the income rate $κ=2$. We prove the asymptotic version of a recent conjecture on the non--vanishing and monotonicity of $D_n$ and derive explicit formulas for the initial values $φ(0)$, $φ(1)$ of a recurrence that yields survival probabilities. In cases when $Z_i$ are Bernoulli or Geometrically distributed, the conjecture on $D_n$ is shown to hold for all $n\in\mathbb{N}_0$. Additionally, a generating function $Ξ(s)$ for ultimate survival probabilities $φ(u)$ is derived.

math.PR↗
arXiv2022

Multi seasonal discrete time risk model revisited

In this work we set up the distribution function of $\mathcal{M}:=\sup_{n\geqslant1}\sum_{i=1}^{n}{(Z_i-1)}$, where the random walk $\sum_{i=1}^{n}Z_i, n\in\mathbb{N},$ is generated by $N$ periodically occurring distributions and the integer-valued and non-negative random variables $Z_1,\,Z_2,\,\ldots$ are independent. The considered random walk generates so-called multi seasonal discrete time risk model, and a known distribution of random variable $\mathcal{M}$ enables to calculate ultimate time ruin or survival probability. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\mathbb{P}(\mathcal{M}< u)$ when $N=2,\,3$ or $10$.

math.PR↗
arXiv2022

On a positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function

In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function for $1/2<\sigma<1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\Re\sum_{\rho} 1/(s-\rho)$, where the sum runs over the zeros of $\zeta(s)$ on the line $1/2+it$. We also check the positivity of $\Re \xi'/\xi(s)$ for $1/2<\sigma<1$ assuming that there occur a non-trivial zeros of $\zeta(s)$ off the critical line.

math.NT↗
arXiv2021

Bi-seasonal discrete time risk model with income rate two

This paper proceeds an approximate calculation of ultimate time survival probability for bi-seasonal discrete time risk model when premium rate equals two. The same model with income rate equal to one was investigated in 2014 by Damarackas and Šiaulys. In general, discrete time and related risk models deal with possibility for a certain version of random walk to hit a certain threshold at least once in time. In this research, the mentioned threshold is the line $u+2t$ and random walk consists from two interchangeably occurring independent but not necessarily identically distributed random variables. Most of proved theoretical statements are illustrated via numerical calculations. Also, there are raised a couple of conjectures on a certain recurrent determinants non-vanishing.

math.PR↗