scholarly journals Hidden Geometry of Bidirectional Grid-Constrained Stochastic Processes

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Aldo Taranto ◽  
Shahjahan Khan
Keyword(s):  
Interest Rates ◽  
Stochastic Processes ◽  
Chemical Reactions ◽  
Parabolic Cylinder ◽  
Concentration Gradients ◽  
Uhlenbeck Process ◽  
Geometric Properties ◽  
Extensive Simulation ◽  
Ornstein Uhlenbeck Process

Bidirectional Grid Constrained (BGC) stochastic processes (BGCSPs) are constrained Itô diffusions with the property that the further they drift away from the origin, the more the resistance to movement in that direction they undergo. The underlying characteristics of the BGC parameter Ψ X t , t are investigated by examining its geometric properties. The most appropriate convex form for Ψ , that is, the parabolic cylinder is identified after extensive simulation of various possible forms. The formula for the resulting hidden reflective barrier(s) is determined by comparing it with the simpler Ornstein–Uhlenbeck process (OUP). Applications of BGCSP arise when a series of semipermeable barriers are present, such as regulating interest rates and chemical reactions under concentration gradients, which gives rise to two hidden reflective barriers.

scholarly journals Limiting Distribution of the Present Value of a Portfolio

1994 ◽  
Vol 24 (1) ◽  
pp. 47-60 ◽  
Author(s):  
Gary Parker
Keyword(s):  
Interest Rates ◽  
Limiting Distribution ◽  
Present Value ◽  
Uhlenbeck Process ◽  
Insurance Contracts ◽  
Ornstein Uhlenbeck Process ◽  
Force Of Interest ◽  
The One

AbstractAn approximation of the distribution of the present value of the benefits of a portfolio of temporary insurance contracts is suggested for the case where the size of the portfolio tends to infinity. The model used is the one presented in Parker (1922b) and involves random interest rates and future lifetimes. Some justifications of the approximation are given. Illustrations for limiting portfolios of temporary insurance contracts are presented for an assumed Ornstein-Uhlenbeck process for the force of interest.

scholarly journals Non-Linear prediction problems for Ornstein-Uhlenbeck process

1983 ◽  
Vol 91 ◽  
pp. 173-184 ◽  
Author(s):  
Sheu-San Lee
Keyword(s):  
Stochastic Processes ◽  
Linear Functional ◽  
Linear Prediction ◽  
Basic Process ◽  
Prediction Theory ◽  
Uhlenbeck Process ◽  
Linear Predictor ◽  
Ornstein Uhlenbeck Process ◽  
Non Linear ◽  
Prediction Problems

We shall discuss in this paper some problems in non-linear prediction theory. An Ornstein-Uhlenbeck process {U(t)} is taken to be a basic process, and we shall deal with stochastic processes X(t) that are transformed by functions f satisfying certain condition. Actually, observed processes are expressed in the form X(t) = f(U(t)). Our main problem is to obtain the best non-linear predictor X̂(t, τ) for X(t + τ), τ > 0, assuming that X(s), s ≤t, are observed. The predictor is therefore a non-linear functional of the values X(s), s ≤ t.

Fluctuations near homogeneous states of chemical reactions with diffusion

1987 ◽  
Vol 19 (2) ◽  
pp. 352-370 ◽  
Author(s):  
Peter Kotelenez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Limit Theorem ◽  
Chemical Reactions ◽  
Stochastic Partial Differential Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Homogeneous State ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process

Conditions are given under which a space-time jump Markov process describing the stochastic model of non-linear chemical reactions with diffusion converges to the homogeneous state solution of the corresponding reaction-diffusion equation. The deviation is measured by a central limit theorem. This limit is a distribution-valued Ornstein–Uhlenbeck process and can be represented as the mild solution of a certain stochastic partial differential equation.

On American Derivatives and Related Obstacle Problems

2003 ◽  
Vol 06 (06) ◽  
pp. 565-591 ◽  
Author(s):  
Jörg Kampen
Keyword(s):  
Interest Rates ◽  
Stochastic Volatility ◽  
Obstacle Problems ◽  
Stochastic Volatility Models ◽  
Uhlenbeck Process ◽  
Typical Situation ◽  
Motion Models ◽  
Volatility Models ◽  
Ornstein Uhlenbeck Process ◽  
Classical Models

We derive obstacle problems for pricing of American derivatives with multiple underlyings heuristically using only a few postulates such that classical (Brownian motion) models as well as models based on Levy processes can be considered in our frame. For the classical models we define a "signed measure" which allows to compute the exercise region near maturity and obtain a generic condition for continuity of the free boundary and prove some more general features of exercise regions for classical models. Especially, we investigate the exercise regions of the most important American derivatives with one and multiple underlyings where we include dependence of volatility and interest rates on time and the underlyings extending and recovering some classical results. Further applications include stochastic volatility models. It is shown that in classical stochastic volatility models where volatility is driven by an Ornstein-Uhlenbeck process an American compound call has a nonempty exercise region and compute the exercise region near expiration in a typical situation.

scholarly journals Beyond Brownian motion and the Ornstein-Uhlenbeck process: Stochastic diffusion models for the evolution of quantitative characters

2016 ◽  
Author(s):  
Simon Phillip Blomberg
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Comparative Methods ◽  
Diffusion Models ◽  
Mathematical Framework ◽  
Phylogenetic Comparative Methods ◽  
Uhlenbeck Process ◽  
Trait Evolution ◽  
Ornstein Uhlenbeck Process ◽  
Non Gaussian

AbstractGaussian processes such as Brownian motion and the Ornstein-Uhlenbeck process have been popular models for the evolution of quantitative traits and are widely used in phylogenetic comparative methods. However, they have drawbacks which limit their utility. Here I describe new, non-Gaussian stochastic differential equation (diffusion) models of quantitative trait evolution. I present general methods for deriving new diffusion models, and discuss possible schemes for fitting non-Gaussian evolutionary models to trait data. The theory of stochastic processes provides a mathematical framework for understanding the properties of current, new and future phylogenetic comparative methods. Attention to the mathematical details of models of trait evolution and diversification may help avoid some pitfalls when using stochastic processes to model macroevolution.

scholarly journals Joint law of an Ornstein–Uhlenbeck process and its supremum

2020 ◽  
Vol 57 (2) ◽  
pp. 541-558
Author(s):  
Christophette Blanchet-Scalliet ◽  
Diana Dorobantu ◽  
Laura Gay
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Brownian Motion ◽  
Monte Carlo Method ◽  
Density Distribution ◽  
Parabolic Cylinder ◽  
Density Distribution Function ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Parabolic Cylinder Functions

AbstractLet X be an Ornstein–Uhlenbeck process driven by a Brownian motion. We propose an expression for the joint density / distribution function of the process and its running supremum. This law is expressed as an expansion involving parabolic cylinder functions. Numerically, we obtain this law faster with our expression than with a Monte Carlo method. Numerical applications illustrate the interest of this result.

scholarly journals Stochastic differential of Ito – Levy processes

2016 ◽  
Vol 19 (2) ◽  
pp. 80-83
Author(s):  
Dam Ton Duong ◽  
Phung Ngoc Nguyen
Keyword(s):  
Information Technology ◽  
Stochastic Processes ◽  
Lévy Process ◽  
Jump Process ◽  
Levy Processes ◽  
Levy Process ◽  
Product Rule ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Geometric Levy Process

In this paper, we continue to expand some results to get the product rule for differential of stochastic processes with jump, and apply for some special processes like pure jump process, Levy-Ornstein-Uhlenbeck process, geometric Levy process, in models of finance, ecomomics, and information technology.

Fluctuations near homogeneous states of chemical reactions with diffusion

1987 ◽  
Vol 19 (02) ◽  
pp. 352-370
Author(s):  
Peter Kotelenez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Limit Theorem ◽  
Chemical Reactions ◽  
Stochastic Partial Differential Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Homogeneous State ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process

Conditions are given under which a space-time jump Markov process describing the stochastic model of non-linear chemical reactions with diffusion converges to the homogeneous state solution of the corresponding reaction-diffusion equation. The deviation is measured by a central limit theorem. This limit is a distribution-valued Ornstein–Uhlenbeck process and can be represented as the mild solution of a certain stochastic partial differential equation.

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