scholarly journals Uniform approximation of the Cox-Ingersoll-Ross process

2015 ◽  
Vol 47 (4) ◽  
pp. 1132-1156 ◽  
Author(s):  
Grigori N. Milstein ◽  
John Schoenmakers
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wiener Process ◽  
Point Of View ◽  
First Passage ◽  
Cir Process ◽  
Quality Of Approximation ◽  
Time Grid ◽  
Passage Times

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

scholarly journals Uniform approximation of the Cox-Ingersoll-Ross process

2015 ◽  
Vol 47 (04) ◽  
pp. 1132-1156 ◽  
Author(s):  
Grigori N. Milstein ◽  
John Schoenmakers
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wiener Process ◽  
Point Of View ◽  
First Passage ◽  
Cir Process ◽  
Quality Of Approximation ◽  
Time Grid ◽  
Passage Times

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

Uniform approximation of the Cox–Ingersoll–Ross process via exact simulation at random times

2016 ◽  
Vol 48 (4) ◽  
pp. 1095-1116 ◽  
Author(s):  
Grigori N. Milstein ◽  
John Schoenmakers
Keyword(s):  
Uniform Approximation ◽  
Point Of View ◽  
Exact Simulation ◽  
Cir Process ◽  
Quality Of Approximation ◽  
Random Times ◽  
Time Grid

AbstractIn this paper we uniformly approximate the trajectories of the Cox–Ingersoll–Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view, the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact, simulation of the CIR dynamics at some deterministic time grid.

Optimal reserves growth under a Wiener process

1975 ◽  
Vol 12 (03) ◽  
pp. 457-465
Author(s):  
W. F. Foster
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Wiener Process ◽  
Order Differential Equation ◽  
Optimality Criterion ◽  
First Passage ◽  
First Order ◽  
Bounded Below ◽  
Passage Times

This paper considers a body whose funds accumulate according to a Wiener Process that has parameters which can be controlled at any stage. The process is bounded above by a level at which dividends (or savings) are set aside, and it is bounded below by a level at which a ‘rescue’ policy is invoked to avoid insolvency. Taking long-term dividend maximisation as the optimality criterion, first passage times are used to derive a general first order differential equation for the optimal control of the system at any reserves level, and this equation is solved fully for a certain class of problems. Examples are given of insurance and investment applications.

Optimal reserves growth under a Wiener process

1975 ◽  
Vol 12 (3) ◽  
pp. 457-465
Author(s):  
W. F. Foster
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Wiener Process ◽  
Order Differential Equation ◽  
Optimality Criterion ◽  
First Passage ◽  
First Order ◽  
Bounded Below ◽  
Passage Times

This paper considers a body whose funds accumulate according to a Wiener Process that has parameters which can be controlled at any stage. The process is bounded above by a level at which dividends (or savings) are set aside, and it is bounded below by a level at which a ‘rescue’ policy is invoked to avoid insolvency. Taking long-term dividend maximisation as the optimality criterion, first passage times are used to derive a general first order differential equation for the optimal control of the system at any reserves level, and this equation is solved fully for a certain class of problems. Examples are given of insurance and investment applications.

scholarly journals The ruin problem for a Wiener process with state-dependent jumps

2020 ◽  
Vol 16 (1) ◽  
pp. 13-23
Author(s):  
M. Lefebvre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Jump Diffusion ◽  
Moment Generating Function ◽  
Continuous Part ◽  
State Dependent ◽  
The Moment ◽  
First Time

AbstractLet X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

Travelling Wave Solutions for the Unsteady Flow of a Third Grade Fluid Induced Due to Impulsive Motion of Flat Porous Plate Embedded in a Porous Medium

2014 ◽  
Vol 30 (5) ◽  
pp. 527-535 ◽  
Author(s):  
T. Aziz ◽  
F. M. Mahomed ◽  
A. Shahzad ◽  
R. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Porous Plate ◽  
Travelling Wave ◽  
Point Of View ◽  
Third Grade ◽  
Third Grade Fluid ◽  
Physical Point ◽  
Grade Fluid

AbstractThis work describes the time-dependent flow of an incompressible third grade fluid filling the porous half space over an infinite porous plate. The flow is induced due to the motion of the porous plate in its own plane with an arbitrary velocityV(t). Translational type symmetries are employed to perform the travelling wave reduction into an ordinary differential equation of the governing nonlinear partial differential equation which arises from the laws of mass and momentum. The reduced ordinary differential equation is solved exactly, for a particular case, as well as by using the homotopy analysis method (HAM). The better solution from the physical point of view is argued to be the HAM solution. The essentials features of the various emerging parameters of the flow problem are presented and discussed.

The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

1998 ◽  
Vol 35 (3) ◽  
pp. 671-682 ◽  
Author(s):  
Anders Martin-Löf
Keyword(s):  
Differential Equation ◽  
Wiener Process ◽  
Critical Parameter ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Number ◽  
Airy Functions ◽  
Final Size ◽  
First Passage

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion

1969 ◽  
Vol 6 (01) ◽  
pp. 218-223
Author(s):  
M.T. Wasan
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
Kolmogorov Equation ◽  
First Passage ◽  
State Variable ◽  
Strong Markov ◽  
Passage Times

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

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