Manipulating Stochastic Differential Equations and Stochastic Integrals

2015 ◽  
pp. 93-110
Author(s):  
Carl Chiarella ◽  
Xue-Zhong He ◽  
Christina Sklibosios Nikitopoulos
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Integrals

On effects of discretization on estimators of drift parameters for diffusion processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimators ◽  
Statistical Properties ◽  
Stochastic Integrals

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

scholarly journals Stochastic Differential Equations in a Differentiable Manifold

1950 ◽  
Vol 1 ◽  
pp. 35-47 ◽  
Author(s):  
Kiyosi Itô
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Differentiable Manifold ◽  
Stochastic Integrals

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

scholarly journals Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

2009 ◽  
Vol 46 (4) ◽  
pp. 1116-1129 ◽  
Author(s):  
David Applebaum ◽  
Michailina Siakalli
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Noise ◽  
Stochastic Integrals ◽  
Martingale Inequality ◽  
Moment Estimates ◽  
Exponential Martingale ◽  
Exponentially Stable ◽  
Levy Noise

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

scholarly journals Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion

2019 ◽  
Vol 17 (1) ◽  
pp. 1515-1525
Author(s):  
Yazid Alhojilan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Transport Theory ◽  
Numerical Solutions ◽  
Optimal Transport ◽  
Stochastic Integrals ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

Quantum stochastic integration and quantum stochastic differential equations

1994 ◽  
Vol 116 (3) ◽  
pp. 535-553 ◽  
Author(s):  
Chris Barnett ◽  
Stanisław Goldstein ◽  
Ivan Wilde
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Stochastic Integrals ◽  
Explicit Solutions ◽  
Space Theory ◽  
Stochastic Integration ◽  
Uniqueness Of The Solution

AbstractQuantum stochastic integrals are constructed using the non-commutative Lp-space theory of Haagerup. The existence and uniqueness of the solution to quantum stochastic differential equations driven by quasi-Wiener noises, or noises satisfying generalized standing hypotheses, is established as is the Markov behaviour of the solution. Various examples of the theory are discussed, and quantum Ornstein-Uhlenbeck processes are obtained as explicit solutions.

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