Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Stochastic differential equations and diffusion processes

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Applied Probability and Queues.

Stochastic models in biology.

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Ito processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Ito processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.

/pdf/on-the-moments-of-the-modulus-of-continuity-of-ito-processes-41h6xxu80y.pdf

On the Moments of the Modulus of Continuity of Itô Processes

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

Rate of convergence to equilibrium of marked Hawkes processes

Kendall distributions and level sets in bivariate exchangeable survival models

Let X be a Markov process characterized as the solution of a martingale problem with generator A, and let Y be a related observation process. The conditional distribution t of X(t) given observations of Y up to time t satisfies certain martingale properties, and it is shown that any probability-measure-valued process with the appropriate martingale properties can be interpreted as the conditional distribution of X for some observation process. In particular, if Y (t) = (X(t)) for some measurable mapping , the conditional distribution of X(t) given observations of Y up to time t is characterized as the solution of a filtered martingale problem. Uniqueness for the original martingale problem implies uniqueness for the filtered martingale problem which in turn implies the Markov property for the conditional distribution considered as a probability-measure-valued process. Other applications include a Markov mapping theorem and uniqueness for filtering equations. MSC 2000 subject classifications: 60J25, 93E11, 60G35, 60J35, 60G44

The filtered martingale problem

We consider in brownian particles which interact with each other and eventually die. Each particle is represented by the process ( x i"~, ~in~) (i = 1, , ..., n) where the is brownian motion and ~in) is a process in D((0, T); ~0, 1}) which defines the state of the particle: death or life. We prove propagation of chaos and a fluctuation theorem for the empirical distribution using a martingale method.

http://archive.numdam.org/article/AIHPB_1988__24_3_319_0.pdf

Giovanna Nappo

Papers

On the Moments of the Modulus of Continuity of Itô Processes

Rate of convergence to equilibrium of marked Hawkes processes

Kendall distributions and level sets in bivariate exchangeable survival models

The filtered martingale problem

Limit laws for a coagulation model of interacting random particles