scholarly journals Model of Continuous Random Cascade Processes in Financial Markets

2020 ◽  
Vol 8 ◽  
Author(s):  
Jun-ichi Maskawa ◽  
Koji Kuroda
Keyword(s):  
Differential Equation ◽  
Time Series ◽  
Stochastic Differential Equation ◽  
Financial Markets ◽  
Stock Price ◽  
Planck Equation ◽  
Cascade Model ◽  
The Other ◽  
Model Parameters ◽  
Cascade Processes

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

A stochastic differential equation model of diurnal cortisol patterns

2001 ◽  
Vol 280 (3) ◽  
pp. E450-E461 ◽  
Author(s):  
Emery N. Brown ◽  
Patricia M. Meehan ◽  
Arthur P. Dempster
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Plasma Cortisol ◽  
Equation Model ◽  
Plasma Cortisol Level ◽  
Model Parameters ◽  
Kinetic Properties ◽  
Simulation Studies ◽  
Physiological Factors ◽  
Diurnal Patterns

Circadian modulation of episodic bursts is recognized as the normal physiological pattern of diurnal variation in plasma cortisol levels. The primary physiological factors underlying these diurnal patterns are the ultradian timing of secretory events, circadian modulation of the amplitude of secretory events, infusion of the hormone from the adrenal gland into the plasma, and clearance of the hormone from the plasma by the liver. Each measured plasma cortisol level has an error arising from the cortisol immunoassay. We demonstrate that all of these three physiological principles can be succinctly summarized in a single stochastic differential equation plus measurement error model and show that physiologically consistent ranges of the model parameters can be determined from published reports. We summarize the model parameters in terms of the multivariate Gaussian probability density and establish the plausibility of the model with a series of simulation studies. Our framework makes possible a sensitivity analysis in which all model parameters are allowed to vary simultaneously. The model offers an approach for simultaneously representing cortisol's ultradian, circadian, and kinetic properties. Our modeling paradigm provides a framework for simulation studies and data analysis that should be readily adaptable to the analysis of other endocrine hormone systems.

scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

scholarly journals Modelling based in the stochastic dynamics for the time evolution of the COVID-19

2020 ◽  
Author(s):  
Leonardo dos Santos Lima
Keyword(s):  
Differential Equation ◽  
Evolution Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Stochastic Dynamics ◽  
Planck Equation ◽  
Time Dynamics ◽  
Health Agencies ◽  
System Of Particles ◽  
Effective Description

Abstract The stochastic differential equation (SDE) corresponding to nonlinear Fokker-Planck equation where the nonlinearity appearing in this evolution equation can be interpreted as providing an effective description of a system of particles interacting is obtained. Additionally, we propose a stochastic model for time dynamics of the COVID-19 based in the set of data supported by the Brazilian health agencies.

scholarly journals Modelling based in the stochastic dynamics for the time evolution of the COVID-19

2020 ◽  
Author(s):  
L. S. Lima
Keyword(s):  
Differential Equation ◽  
Evolution Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Stochastic Dynamics ◽  
Planck Equation ◽  
Time Dynamics ◽  
Health Agencies ◽  
System Of Particles ◽  
Effective Description

Abstract The stochastic differential equation (SDE) corresponding to nonlinear Fokker-Planck equation where the nonlinearity appearing in this evolution equation can be interpreted as providing an effective description of a system of particles interacting is obtained. Additionally, we propose a stochastic model for time dynamics of the COVID-19 based in the set of data supported by the Brazilian health agencies.

scholarly journals A Time-Series Approach to Non-Self-Financing Hedging in a Discrete-Time Incomplete Market

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
N. Josephy ◽  
L. Kimball ◽  
V. Steblovskaya
Keyword(s):  
New York ◽  
Time Series ◽  
Stock Price ◽  
Stock Exchange ◽  
Market Price ◽  
Incomplete Market ◽  
Model Parameters ◽  
York Stock Exchange ◽  
Risk Criterion ◽  
Time Series Approach

We present an algorithm producing a dynamic non-self-financing hedging strategy in an incomplete market corresponding to investor-relevant risk criterion. The optimization is a two-stage process that first determines market calibrated model parameters that correspond to the market price of the option being hedged. In the second stage, an optimal set of model parameters is chosen from the market calibrated set. This choice is based on stock price simulations using a time-series model for stock price jump evolution. Results are presented for options traded on the New York Stock Exchange.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

2015 ◽  
Vol 22 (04) ◽  
pp. 1550022
Author(s):  
Matteo Gregoratti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
The Other ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Proper Formulation

We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.

scholarly journals Fractional Fokker-Planck-Kolmogorov equations associated with SDES on a bounded domain

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Sabir Umarov
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Stochastic Differential Equation ◽  
Bounded Domain ◽  
Planck Equation ◽  
Initial Boundary ◽  
Order Differential Operator ◽  
Fokker Planck Equation ◽  
Initial Boundary Value ◽  
Fokker Planck

AbstractThis paper is devoted to the fractional generalization of the Fokker-Planck equation associated with a nonlinear stochastic differential equation on a bounded domain. The driving process of the stochastic differential equation is a Lévy process subordinated to the inverse of Lévy’s mixed stable subordinators. The Fokker-Planck equation is given through the general Waldenfels operator, while the boundary condition is given through the general Wentcel’s boundary condition. As a fractional operator a distributed order differential operator with a Borel mixing measure is considered. In the paper fractional generalizations of the Fokker-Planck equation are derived and the existence of a unique solution of the corresponding initial-boundary value problems is proved.

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