Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case

1996 ◽  
pp. 197-285 ◽  
Author(s):  
Thomas G. Kurtz ◽  
Philip E. Protter
Keyword(s):  
Weak Convergence ◽  
Differential Equations ◽  
Dimensional Case ◽  
Stochastic Integrals ◽  
Infinite Dimensional

scholarly journals BSDE and generalized Dirichlet forms: The infinite dimensional case

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Rong-Chan Zhu
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Parabolic System ◽  
Dirichlet Forms ◽  
Dimensional Case ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

AbstractWe consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space

On the regularity of stochastic difference equations in hyperfinite-dimensional vector spaces and applications to -valued stochastic differential equations

1994 ◽  
Vol 124 (6) ◽  
pp. 1089-1117 ◽  
Author(s):  
Jiang-Lun Wu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Difference Equations ◽  
Nonstandard Analysis ◽  
Existence Theorems ◽  
Vector Spaces ◽  
Stochastic Integrals ◽  
Dimensional Vector ◽  
Stochastic Difference Equations ◽  
Infinite Dimensional

Nonstandard analysis is used, in this paper, to give a construction of a Wiener -process Wt, t ∈ [0, ∞). From this, a hyperfinite representation of stochastic integrals for operatorvalued processes with respect to Wt is derived, and existence theorems in the spirit of Keisler are proved for (infinite-dimensional) stochastic differential equations of Itô's type one and a certain kind of Itô's type two, via regularity of hyperfinite stochastic difference equations.

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Moderate deviation principle for multivalued stochastic differential equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050015 ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Moderate Deviation ◽  
Reflected Brownian Motion ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

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