A counterexample related to Ap-weights in martingale theory

The aim of the present paper is to introduce some techniques, based on the change of variable formula for processes of finite variation, for establishing (integro) differential equations for evaluating the distribution of jump processes for a fixed period of time. This is of interest in insurance mathematics for evaluating the distribution of the total amount of claims occurred over some period of time, and attention will be given to such issues. Firstly we will study some techniques when the process has independent increments, and then a more refined martingale technique is discussed. The building blocks are delivered by the theory of marked point processes and associated martingale theory. A simple numerical example is given.

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Martingale Theory and the Stochastic Integral for Point Processes

Theory of Stochastic Differential Equations with Jumps and Applications ◽

10.1007/0-387-25175-8_1 ◽

2005 ◽

pp. 3-38

Keyword(s):

Point Processes ◽

Stochastic Integral ◽

Martingale Theory

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Martingale Theory - Potential Theory

Potential Theory ◽

10.1007/978-3-642-11084-9_5 ◽

2010 ◽

pp. 203-206

Author(s):

J.L. Doob

Keyword(s):

Potential Theory ◽

Martingale Theory

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Martingales

10.1093/oso/9780192844507.003.0016 ◽

2021 ◽

pp. 313-343

Author(s):

James Davidson

Keyword(s):

Martingale Theory ◽

Basic Properties ◽

Martingale Inequalities ◽

Martingale Convergence ◽

Doob Decomposition

This chapter summarizes the essentials of sequential conditioning and martingale theory. After a review with examples of the basic properties of martingales and semi‐martingales, including the Doob decomposition, the upcrossing inequality and martingale convergence are studied and also the role of the conditional variances in establishing convergence. The important martingale inequalities of Kolmogorov, Doob, Burkholder, and Azuma are proved.

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