Semi-analytical methods for solving Ito stochastic models on the notion of Karhunen–Loéve Brownian motion transform

2021 ◽  
Author(s):  
Opeyemi Ogundile ◽  
Sunday Edeki
Keyword(s):  
Brownian Motion ◽  
Stochastic Models ◽  
Analytical Methods

scholarly journals MULTIFRACTAL FLUCTUATIONS IN FINANCE

2000 ◽  
Vol 03 (03) ◽  
pp. 361-364 ◽  
Author(s):  
FRANCOIS SCHMITT ◽  
DANIEL SCHERTZER ◽  
SHAUN LOVEJOY
Keyword(s):  
Brownian Motion ◽  
Exchange Rate ◽  
Foreign Exchange ◽  
Stochastic Models ◽  
Evolution Equations ◽  
Foreign Exchange Rate ◽  
Scaling Exponent ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Multiplicative Processes

We consider the structure functions S(q)(τ), i.e. the moments of order q of the increments X(t + τ)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S(q)(τ)∝τζ(q)). We demonstrate that the nonlinearity of the observed scaling exponent ζ(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity of ζ(q) corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponent qD > 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.

scholarly journals Beyond multifractional Brownian motion: new stochastic models for geophysical modelling

2013 ◽  
Vol 20 (5) ◽  
pp. 643-655 ◽  
Author(s):  
J. Lévy Véhel
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Local Regularity ◽  
Multifractional Brownian Motion ◽  
Digital Elevation ◽  
Alternative Type ◽  
Temperature Records ◽  
Local Intensity ◽  
Alternative Processes

Abstract. Multifractional Brownian motion (mBm) has proved to be a useful tool in various areas of geophysical modelling. Although a versatile model, mBm is of course not always an adequate one. We present in this work several other stochastic processes which could potentially be useful in geophysics. The first alternative type is that of self-regulating processes: these are models where the local regularity is a function of the amplitude, in contrast to mBm where it is tuned exogenously. We demonstrate the relevance of such models for digital elevation maps and for temperature records. We also briefly describe two other types of alternative processes, which are the counterparts of mBm and of self-regulating processes when the intensity of local jumps is considered in lieu of local regularity: multistable processes allow one to prescribe the local intensity of jumps in space/time, while this intensity is governed by the amplitude for self-stabilizing processes.

Stochastic Processes for the Risk Management

2015 ◽  
pp. 188-200
Author(s):  
Gamze Özel
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Financial Markets ◽  
Interest Rates ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Stock Prices ◽  
Financial Risk ◽  
Future Events ◽  
Software Codes

The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.

scholarly journals On the Transient Behavior of Ehrenfest and Engset Processes

2012 ◽  
Vol 44 (02) ◽  
pp. 562-582 ◽  
Author(s):  
Mathieu Feuillet ◽  
Philippe Robert
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Heat Exchange ◽  
Communication Networks ◽  
Stochastic Models ◽  
Transient Behavior ◽  
Hitting Times ◽  
Kinetic Theory Of Gases ◽  
Two Bodies ◽  
Number Of Particles

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

Stochastic Processes for the Risk Management

2018 ◽  
pp. 473-488
Author(s):  
Gamze Özel
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Financial Markets ◽  
Interest Rates ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Stock Prices ◽  
Financial Risk ◽  
Future Events ◽  
Software Codes

The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.

Stochastic Processes for the Risk Management

2018 ◽  
pp. 174-189
Author(s):  
Gamze Özel
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Financial Markets ◽  
Interest Rates ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Stock Prices ◽  
Financial Risk ◽  
Future Events ◽  
Software Codes

The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.

Sign in / Sign up

Export Citation Format

Share Document