Kernel estimation for Lévy driven stochastic convolutions

We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X ⁢ ( t ) = ∫ 0 t a ⁢ ( t - s ) ⁢ d Z ⁢ ( s ) X(t)=\int_{0}^{t}a(t-s)\,dZ(s) , where 𝑍 is a Lévy martingale and the kernel a ( . ) a(\,{.}\,) a deterministic function square integrable on R + \mathbb{R}^{+} . Given 𝑁 i.i.d. continuous time observations ( X i ⁢ ( t ) ) t ∈ [ 0 , T ] (X_{i}(t))_{t\in[0,T]} , i = 1 , … , N i=1,\dots,N , distributed like ( X ⁢ ( t ) ) t ∈ [ 0 , T ] (X(t))_{t\in[0,T]} , we propose two types of nonparametric projection estimators of a 2 a^{2} under different sets of assumptions. We bound the L 2 \mathbb{L}^{2} -risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.

Limiting Accuracy of Height Measurement for a Precision Radar Altimeter in a Low Altitude Flying Vehicle above the Sea Surface

This study presents mathematical analysis and numerical modeling for the estimation of measurement errors of height estimation over the sea surface for a precision radar altimeter installed in a low altitude flying vehicle. Reflective properties of the electromagnetic signals from the sea surface are determined by the local backscattering patterns of the sea surface illuminated. The height estimation of the flying vehicle from the received echo signals at the output of its tracking system is the sum of three factors: the first factor is the height to the average sea level the second is the bias of the estimation of the height, which is time-varying and depends on the slope of large-scale roughness; the third is the terms related to the surface topography. For the calculation of the estimation errors of the height measurement of a low altitude precision radar altimeter, a reasonable approximation of the large roughness of the sea surface by a deterministic function is necessary. In this study, we performed the derivation of the estimation function and the analysis of the limiting accuracy of the height measurement using the calculation of the estimation errors in spectral domain method describing the large-scale sea surface roughness. The results obtained for the limiting accuracy of a flying vehicle at low altitude above the sea surface, allows to obtain reasonable system parameters minimizing height errors of the flight altitude.

OPTIMAL REGULATION OF LOCAL ENERGY SYSTEM WITH RENEWABLE ENERGY SOURCES

The paper is devoted to solving the balancing problem in local power systems with renewable energy sources. For a power system optimization problem, whose operation depends on random weather factors, a convex parameter optimization or optimal control problem was solved using controlled generation, for each individual realization of a random process as a deterministic function, and then statistical processing of results over a set of random realizations was performed and distribution density functions of the desired target function were constructed, followed by estimation of expected values and their confidence intervals. The process describing current deviations of generated power from mean value is modelled as discrete stray model and has properties of Ornstein-Uhlenbeck process, which allowed varying the duration of unit interval, in particular to select data bases of operating objects with inherent temporal discreteness of their monitoring systems. Random components are investigated and modelled, while the average values are considered to be deterministic and are provided within a predictable schedule using also traditional energy sources (centralised power grid). A mathematical model of the combined operation of renewable energy sources in a system with variable load, electric storage device and auxiliary regulating generator is implemented as a scheme of sequential generation and consumption models and random processes describing the current state of the power system. The operation of the electricity accumulators is dependent on the processes mentioned, but in the full balance, it appears together with generation or load losses, which are cumulative sums of unbalanced power and may have a different distribution from the normal one. However, these processes are internal, relating to the redistribution of energy within a generation system whose capacity is generally described satisfactorily, given the relevant criteria, by a normal law. Under this condition, it is possible to estimate the probability of different circumstances - over- or under-generation, that is, to give a numerical estimate of the reliability of energy supply.

On Two-Stage Guessing

Stationary memoryless sources produce two correlated random sequences Xn and Yn. A guesser seeks to recover Xn in two stages, by first guessing Yn and then Xn. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in n) of any positive ρ-th moment of the total number of guesses when Yn is obtained by applying a deterministic function f component-wise to Xn. We prove that, depending on f, the least exponential growth rate in the two-stage setup is lower than when guessing Xn directly. We further propose a simple Huffman code-based construction of a function f that is a viable candidate for the minimization of the least exponential growth rate in the two-stage guessing setup. (2) We characterize the least achievable exponential growth rate of the ρ-th moment of the total number of guesses required to recover Xn when Stage 1 need not end with a correct guess of Yn and without assumptions on the stationary memoryless sources producing Xn and Yn.

Determining equity-linked policy premium for family Takaful: An application of Black-Scholes option pricing with escrowed dynamic model

The premium is a deterministic function to compensate for losses due to random events and it is an essential part of an insurance company operation. Numerous issues are faced by the Takaful operator in the practice of insurance, one of them is “validity of life insurance” which is still under discussion among Islamic scholars. Their conversation leads to the issue of approach which are utilised by the Takaful operator to create a sales illustration product. Based on their current discussion, there are still some hidden elements and some missing points related to the concept of loss and surplus sharing utilised by Takaful operator. Therefore, this paper focuses on the practice of Family Takaful for producing the sale illustration product which Shariah compliant. The study develops a new model of the premium for an equity-linked policy (Unit-linked product) by considering the properties of Takaful contracts namely Tabarru and Mudarabah. It aims to ensure that a new model developed can comprehensively demonstrate Shariah compliance. The model adapted and derived from the current Takaful Business Model. We add several assumptions to implicate the approach in a real problem associated with the ratio of profit-sharing (Mudarabah) and loss-sharing (Tabarru). Secondary data are used to test and produce a sales illustration product by utilising a new integrated model of the premium developed. Based on the empirical results, a new model of premium is used to create a sales illustration product that comprehensively complies with Shariah and also more profitable and beneficial when compared with a standard approach used by Takaful operator.

A COVID-19 Model for Local Authorities of the United Kingdom

We propose a new framework to model the COVID-19 epidemic of the United Kingdom at the level of local authorities. The model fits within a general framework for semi-mechanistic Bayesian models of the epidemic, with some important innovations: we model the proportion of infections that result in reported deaths and cases as random variables. This is in contrast to standard frameworks that model the latent infection as a deterministic function of time varying reproduction number, Rt. The model is tailored and designed to be updated daily based on publicly available data. We envisage the model to be useful for now-casting and short-term projections of the epidemic as well as estimating historical trends. The model fits are available on a public website, https://imperialcollegelondon.github.io/covid19local. The model is currently being used by the Scottish government in their decisions on interventions within Scotland [1, issue 24 to now].

Implicit max-stable extremal integrals

Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function f ≥ 0. From an application point of view, one is rather interested in extreme loss events that occur relative to f than in the corresponding extreme values itself. In this context, so-called f -implicit α-Fréchet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in Goldbach (2016) by developing a stochastic integral of a deterministic function g ≥ 0 with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see Stoev and Taqqu Extremes 8, 237–266 (2005)) and, at the same time, reveals striking parallels.

Probability-theoretic Approach to Modeling a Respirator on Chemically Bound Oxygen

At present, the main prospects for improving the insulating means of respiratory protection are related to the chemical method of oxygen reservation. The arguments in favor of this choice are the high density of oxygen packaging and its self-regulating supply, depending on the physical activity of the person. Usually, the working process in devices on chemically bound oxygen is modeled using mathematical physics methods that solve the so-called sorption dynamics problem. As a result, under given boundary and initial conditions, the concentration of CO2 molecules in the regenerative cartridge turns out to be a deterministic function of time and coordinates. However, the coordinate of the elementary act of sorption is essentially a random variable. The law of its distribution evolves as the absorbing resource of the regenerative cartridge is consumed. Taking into account the above, a probability-theoretic approach to modeling the working process of an insulating breathing apparatus based on chemically bound oxygen was developed. The approach is based on the description by probability theory methods of the random coordinate of the elementary act of chemosorption of a CO2 molecule by potassium peroxide granules and the random lifetime of this molecule in the regenerative cartridge of the respiratory apparatus. The evolution of the initial and central statistical moments of these values is established. The symmetry of their probability density with respect to the permutation of dimensionless arguments is shown, which are the time and distance from the entrance to the regenerative cartridge to the considered layer of chemisorbent. The presence of symmetry increases the speed of numerical experiments by one or two orders of magnitude. Gaussian asymptotics of the process at long times and corrections to it by inverse degrees of dispersion due to asymmetries and excesses of different orders are revealed. This further increases the speed of numerical experiments in computer simulation of the working process of an insulating respirator on chemically bound oxygen.