scholarly journals Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces

Bernoulli ◽  
2009 ◽  
Vol 15 (4) ◽  
pp. 925-954 ◽  
Author(s):  
Stefan Geiss ◽  
Anni Toivola
Keyword(s):  
Weak Convergence ◽  
Besov Spaces ◽  
Stochastic Integrals

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.

Weak Convergence to Stochastic Integrals

2021 ◽  
pp. 724-756
Author(s):  
James Davidson
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Final Section ◽  
Stochastic Integrals ◽  
Martingale Difference ◽  
Mixing Processes ◽  
Linear Processes ◽  
Special Cases

The main object of this chapter is to prove the convergence of the covariances of stochastic processes with their increments to stochastic integrals with respect to Brownian motion. Some preliminary theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.

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