Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin's calculus

1996 ◽  
Vol 106 (1) ◽  
pp. 105-135 ◽  
Author(s):  
Peter Imkeller
Keyword(s):  
Random Variable ◽  
Continuous Random Variable ◽  
Absolutely Continuous ◽  
Malliavin’S Calculus

scholarly journals On Cumulative Entropies in Terms of Moments of Order Statistics

2021 ◽  
Author(s):  
Narayanaswamy Balakrishnan ◽  
Francesco Buono ◽  
Maria Longobardi
Keyword(s):  
Order Statistics ◽  
Random Variable ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Continuous Random Variable ◽  
Absolutely Continuous ◽  
Moments Of Order Statistics

AbstractIn this paper, relations between some kinds of cumulative entropies and moments of order statistics are established. By using some characterizations and the symmetry of a non-negative and absolutely continuous random variable X, lower and upper bounds for entropies are obtained and illustrative examples are given. By the relations with the moments of order statistics, a method is shown to compute an estimate of cumulative entropies and an application to testing whether data are exponentially distributed is outlined.

OPTIMIZATION OF MARKOV SYSTEMS OF QUEUEING WITH WAITING TIME IN THE MATLAB SYSTEM

2017 ◽  
pp. 39-47
Author(s):  
Viktor Afonin ◽  
Vladimir Valer'evich Nikulin
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Random Variable ◽  
Model Systems ◽  
Queuing Systems ◽  
Input Flow ◽  
Continuous Random Variable ◽  
Input Stream ◽  
Time Intervals ◽  
Markov Systems

The article focuses on attempt to optimize two well-known Markov systems of queueing: a multichannel queueing system with finite storage, and a multichannel queueing system with limited queue time. In the Markov queuing systems, the intensity of the input stream of requests (requirements, calls, customers, demands) is subject to the Poisson law of the probability distribution of the number of applications in the stream; the intensity of service, as well as the intensity of leaving the application queue is subject to exponential distribution. In a Poisson flow, the time intervals between requirements are subject to the exponential law of a continuous random variable. In the context of Markov queueing systems, there have been obtained significant results, which are expressed in the form of analytical dependencies. These dependencies are used for setting up and numerical solution of the problem stated. The probability of failure in service is taken as a task function; it should be minimized and depends on the intensity of input flow of requests, on the intensity of service, and on the intensity of requests leaving the queue. This, in turn, allows to calculate the maximum relative throughput of a given queuing system. The mentioned algorithm was realized in MATLAB system. The results obtained in the form of descriptive algorithms can be used for testing queueing model systems during peak (unchanged) loads.

Distribution of values of classic singular Cantor function of random argument

2018 ◽  
Vol 26 (4) ◽  
pp. 193-200 ◽  
Author(s):  
Mykola Pratsiovytyi ◽  
Iryna Lysenko ◽  
Oksana Voitovska
Keyword(s):  
Random Variable ◽  
Absolutely Continuous ◽  
Cantor Function ◽  
Distribution Of Values

Abstract Let X be a random variable with independent ternary digits and let {y=F(x)} be a classic singular Cantor function. For the distribution of the random variable {Y=F(X)} , the Lebesgue structure (i.e., the content of discrete, absolutely continuous and singular components), the structure of its point and the continuous spectra are exhaustively studied.

scholarly journals Distributional Replication

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1063
Author(s):  
Brendan K. Beare
Keyword(s):  
Hedge Fund ◽  
Minimum Cost ◽  
Random Variable ◽  
Optimal Approximation ◽  
Regularity Conditions ◽  
Return Distribution ◽  
Continuous Random Variable ◽  
Hedge Fund Replication ◽  
Market Index ◽  
Fund Return

A function which transforms a continuous random variable such that it has a specified distribution is called a replicating function. We suppose that functions may be assigned a price, and study an optimization problem in which the cheapest approximation to a replicating function is sought. Under suitable regularity conditions, including a bound on the entropy of the set of candidate approximations, we show that the optimal approximation comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, the optimal approximation corresponds to the cheapest portfolio of market index options which delivers the hedge fund return distribution.

Two-player zero-sum stochastic differential games with random horizon

2019 ◽  
Vol 51 (4) ◽  
pp. 1209-1235
Author(s):  
M. Ferreira ◽  
D. Pinheiro ◽  
S. Pinheiro
Keyword(s):  
Viscosity Solution ◽  
Nonlinear Partial Differential Equation ◽  
Random Variable ◽  
Planning Horizon ◽  
Continuous Random Variable ◽  
State Variable ◽  
Random Horizon ◽  
Hamilton Jacobi Bellman ◽  
Zero Sum ◽  
Random Planning Horizon

AbstractWe consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

Geometric bounds on certain sublinear functionals of geometric Brownian motion

2003 ◽  
Vol 40 (4) ◽  
pp. 893-905 ◽  
Author(s):  
Per Hörfelt
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Borel Measure ◽  
Random Variable ◽  
Geometric Brownian Motion ◽  
Upper And Lower Bounds ◽  
Tail Probabilities ◽  
Absolutely Continuous ◽  
Basket Options ◽  
Sublinear Functionals

Suppose that {Xs, 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝm → [0,∞) is a (weighted) lq(ℝm)-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the Lp(μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(Xs), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.

A Note on Nonparametric Estimation of the CTE

2009 ◽  
Vol 39 (2) ◽  
pp. 717-734 ◽  
Author(s):  
Bangwon Ko ◽  
Ralph P. Russo ◽  
Nariankadu D. Shyamalkumar
Keyword(s):  
Order Term ◽  
Random Variable ◽  
Small Samples ◽  
Closed Form Expression ◽  
Form Expression ◽  
Continuous Random Variable ◽  
Underlying Distribution ◽  
First Order ◽  
Bootstrap Approach ◽  
Conditional Tail Expectation

AbstractThe α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > qα} where qα represents its α-level quantile. It is well known that the empirical CTE (the average of the n (1 – α) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests an alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

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