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9 - Martingale Representation, Girsanov and Kolmogorov
from Part I - Point Processes
Tomas Björk, Stockholm School of Economics
Book:

Point Processes and Jump Diffusions

Published online:

27 May 2021

Print publication:

17 June 2021, pp 82-86
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Summary

The Girsanov theorem is proved and the martingale representation theorem is presented. We again study the relation between SDEs and PIDEs, but now in the framework of a marked point process.

5 - Martingale Representations and Girsanov Transformations
from Part I - Point Processes
Tomas Björk, Stockholm School of Economics
Book:

Point Processes and Jump Diffusions

Published online:

27 May 2021

Print publication:

17 June 2021, pp 43-55
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Summary

In this chapter we present and prove the Girsanov theorem for counting process. We also discuss (without proof) the martingale representation theorem. We then apply the Girsanov theory to Cox processes and maximum-likelihood estimation.

An Introduction with Finance Applications
Tomas Björk
Published online:

27 May 2021

Print publication:

17 June 2021
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The theory of marked point processes on the real line is of great and increasing importance in areas such as insurance mathematics, queuing theory and financial economics. However, the theory is often viewed as technically and conceptually difficult and has proved to be a block for PhD students looking to enter the area. This book gives an intuitive picture of the central concepts as well as the deeper results, while presenting the mathematical theory in a rigorous fashion and discussing applications in filtering theory and financial economics. Consequently, readers will get a deep understanding of the theory and how to use it. A number of exercises of differing levels of difficulty are included, providing opportunities to put new ideas into practice. Graduate students in mathematics, finance and economics will gain a good working knowledge of point-process theory, allowing them to progress to independent research.

A general recursive discrete-time filter
Part of
- Stochastic systems and control
- Stochastic processes
Robert J. Elliott
Journal:

Journal of Applied Probability / Volume 30 / Issue 3 / September 1993

Published online by Cambridge University Press:

14 July 2016, pp. 575-588

Print publication:

September 1993
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A reference probability is explicitly constructed under which the signal and observation processes are independent. A simple, explicit recursive form is then obtained for the conditional density of the signal given the observations. Both non-linear and linear filters are considered, as well as two different information patterns.

Transition probabilities for some ‘special' diffusions
Václav E. Beneš, Ioannis Karatzas
Journal:

Journal of Applied Probability / Volume 24 / Issue 4 / December 1987

Published online by Cambridge University Press:

14 July 2016, pp. 888-898

Print publication:

December 1987
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We study the transition probabilities of the diffusions dXt = (1 – exp(Xt))dt + dWt and dXt = – tanh Xtdt + dWt, in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).

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