Geometric bounds on certain sublinear functionals of geometric Brownian motion

2003 ◽  
Vol 40 (04) ◽  
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 {X s , 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ m → [0,∞) is a (weighted) l q (ℝ m )-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L p (μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(X s ), 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.

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.

Kinetic model of wealth distribution by trading stocks with geometric brownian motion

2021 ◽  
pp. 2250001
Author(s):  
Ryosuke Yano ◽  
Hisayasu Kuroda
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Kinetic Equation ◽  
Kinetic Model ◽  
Wealth Distribution ◽  
Geometric Brownian Motion ◽  
Total Asset ◽  
Two Agents ◽  
Specific Number ◽  
Specific Amount

In this paper, we consider the wealth distribution obtained by trading (buying–selling) stocks whose prices follow the geometric Brownian motion (GBM), when both number of the ticker symbol of the stock and maximum number of the traded stock are limited to unity. The binary exchange of the cash and stock between two agents is expressed with the Boltzmann-type kinetic equation. The distribution function of the number of the agents with the specific number of the stock or specific amount of the cash can be demonstrated, theoretically, when the price of the stock is constant. The distribution function of the number of the agents with the specific amount of the total asset can be approximated by [Formula: see text]-distribution, when the price of the stock follows the GBM. Finally, the rule in the binary-exchange-game approximates the distribution function of the number of the agents with the specific amount of the total asset to the Feller–Pareto-like distribution at the high wealth tail.

scholarly journals An exponential functional of random walks

2003 ◽  
Vol 40 (2) ◽  
pp. 413-426 ◽  
Author(s):  
Tamás Szabados ◽  
Balázs Székely
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Random Variable ◽  
Financial Mathematics ◽  
Asian Options ◽  
Discrete Approximations ◽  
Absolutely Continuous ◽  
Exponential Functional

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

Real-valued Random Processes

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Brownian Motion ◽  
Random Variables ◽  
Random Variable ◽  
Random Processes ◽  
Geometric Brownian Motion ◽  
Independent Random Variables ◽  
Benford’S Law ◽  
Random Maps ◽  
Random Samples ◽  
Benford's Law

Benford's law arises naturally in a variety of stochastic settings, including products of independent random variables, mixtures of random samples from different distributions, and iterations of random maps. This chapter provides the concepts and tools to analyze significant digits and significands for these basic random processes. Benford's law also arises in many other important fields of stochastics, such as geometric Brownian motion, random matrices, and Bayesian models, and the chapter may serve as a preparation for specialized literature on these advanced topics. By Theorem 4.2 a random variable X is Benford if and only if log ¦X¦ is uniformly distributed modulo one.

scholarly journals An exponential functional of random walks

2003 ◽  
Vol 40 (02) ◽  
pp. 413-426 ◽  
Author(s):  
Tamás Szabados ◽  
Balázs Székely
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Random Variable ◽  
Financial Mathematics ◽  
Asian Options ◽  
Discrete Approximations ◽  
Absolutely Continuous ◽  
Exponential Functional

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (04) ◽  
pp. 929-932
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

Asymptotic behavior of tail and local probabilities for sums of subexponential random variables

2004 ◽  
Vol 41 (01) ◽  
pp. 108-116 ◽  
Author(s):  
Kai W. Ng ◽  
Qihe Tang
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Random Variables ◽  
Random Variable ◽  
Subexponential Distribution ◽  
Tail Probabilities

Let {X k , k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X (τ), S τ and S (τ), where X 0 = 0 by convention and h > 0 is arbitrarily fixed.

Asymptotic behavior of tail and local probabilities for sums of subexponential random variables

2004 ◽  
Vol 41 (1) ◽  
pp. 108-116 ◽  
Author(s):  
Kai W. Ng ◽  
Qihe Tang
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Random Variables ◽  
Random Variable ◽  
Subexponential Distribution ◽  
Tail Probabilities

Let {Xk, k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {Xk, k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X(τ), Sτ and S(τ), where X0 = 0 by convention and h > 0 is arbitrarily fixed.

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

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