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.

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.

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.

scholarly journals Sterile Neutrinos as Dark Matter: Alternative Production Mechanisms in the Early Universe

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 264
Author(s):  
Daniel Boyanovsky
Keyword(s):  
Dark Matter ◽  
Distribution Function ◽  
Kinetic Equation ◽  
Standard Model ◽  
Early Universe ◽  
Quantum Kinetic Equation ◽  
Sterile Neutrinos ◽  
The Standard Model ◽  
Production Mechanisms ◽  
Freeze Out

We study various production mechanisms of sterile neutrinos in the early universe beyond and within the standard model. We obtain the quantum kinetic equations for production and the distribution function of sterile-like neutrinos at freeze-out, from which we obtain free streaming lengths, equations of state and coarse grained phase space densities. In a simple extension beyond the standard model, in which neutrinos are Yukawa coupled to a Higgs-like scalar, we derive and solve the quantum kinetic equation for sterile production and analyze the freeze-out conditions and clustering properties of this dark matter constituent. We argue that in the mass basis, standard model processes that produce active neutrinos also yield sterile-like neutrinos, leading to various possible production channels. Hence, the final distribution function of sterile-like neutrinos is a result of the various kinematically allowed production processes in the early universe. As an explicit example, we consider production of light sterile neutrinos from pion decay after the QCD phase transition, obtaining the quantum kinetic equation and the distribution function at freeze-out. A sterile-like neutrino with a mass in the keV range produced by this process is a suitable warm dark matter candidate with a free-streaming length of the order of few kpc consistent with cores in dwarf galaxies.

scholarly journals Geometric Brownian motion with affine drift and its time-integral

2021 ◽  
Vol 395 ◽  
pp. 125874
Author(s):  
Runhuan Feng ◽  
Pingping Jiang ◽  
Hans Volkmer
Keyword(s):  
Brownian Motion ◽  
Geometric Brownian Motion ◽  
Time Integral
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