Weak Solutions for Semi-Martingales

1981 ◽  
Vol 33 (5) ◽  
pp. 1165-1181 ◽  
Author(s):  
J. Pellaumail
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Weak Solution ◽  
Stochastic Differential Equation ◽  
Marginal Distribution ◽  
Uniform Norm ◽  
Section 8 ◽  
Initial Space ◽  
Sample Paths ◽  
The Given

The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.For such an equation, the existence of a “weak solution” is stated; actually, the notion of weak solution here considered is more precise than this one introduced by Strook and Varadhan (cf. [30], [31], [23]).Namely, this weak solution is a probability, so-called “rule,” defined on (DH × Ω), DH being the classical Skorohod space of all the cadlag sample paths and Ω is the initial space which Z is defined on: the marginal distribution of R on Ω is the given probability P on Ω. This concept of rule is defined in Section 3.

Continuous-time threshold AR(1) processes

1996 ◽  
Vol 28 (03) ◽  
pp. 728-746 ◽  
Author(s):  
O. Stramer ◽  
P. J. Brockwell ◽  
R. L. Tweedie
Keyword(s):  
Differential Equation ◽  
Weak Solution ◽  
Stochastic Differential Equation ◽  
Strong Solution ◽  
Piecewise Linear ◽  
Transition Probability ◽  
Unique Strong Solution ◽  
Direct Definition ◽  
Definition Of ◽  
Boundary Width

A threshold AR(1) process with boundary width 2δ > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process with δ = 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution with δ = 0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ > 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.

scholarly journals The numerical solution of stochastic differential equations

1977 ◽  
Vol 20 (1) ◽  
pp. 8-12 ◽  
Author(s):  
P. E. Kloeden ◽  
R. A. Pearson
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Iterative Scheme ◽  
Correction Term ◽  
Second Order ◽  
Runge Kutta ◽  
The Given

AbstractA method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.

Continuous-time threshold AR(1) processes

1996 ◽  
Vol 28 (3) ◽  
pp. 728-746 ◽  
Author(s):  
O. Stramer ◽  
P. J. Brockwell ◽  
R. L. Tweedie
Keyword(s):  
Differential Equation ◽  
Weak Solution ◽  
Stochastic Differential Equation ◽  
Strong Solution ◽  
Piecewise Linear ◽  
Transition Probability ◽  
Unique Strong Solution ◽  
Direct Definition ◽  
Definition Of ◽  
Boundary Width

A threshold AR(1) process with boundary width 2δ > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process with δ = 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution with δ = 0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ > 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.

On the Existence and Uniqueness of a Solution to a Stochastic Differential Equation in a Banach Space

2004 ◽  
Vol 11 (3) ◽  
pp. 515-526
Author(s):  
B. Mamporia
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Covariance Operator ◽  
Sufficient Condition ◽  
Existence Of A Solution ◽  
Arbitrary Banach Space ◽  
Uniqueness Of A Solution

Abstarct A sufficient condition is given for the existence of a solution to a stochastic differential equation in an arbitrary Banach space. The method is based on the concept of covariance operator and a special construction of the Itô stochastic integral in an arbitrary Banach space.

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