Applied Stochastic Processes and Control for Jump diffusion.pdf

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Applied Stochastic Processes and Control for
Jump-Diffusions: Modeling, Analysis and
Computation
Floyd B. Hanson
University of Illinois
Chicago, Illinois, USA
Copyright c 2007 by the Society for Industrial and Applied Mathematics.

January 7, 2007
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To five generations of women in my life,
Margaret Geiger, Violet Bliss, Ethel Hutchins, Lisa Hanson
and Chiara Hanson Whitehurst
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Contents
Preface xvii
1 Stochastic Jump and Diffusion Processes 1
Poisson and Wiener Processes Basics . . . . . . . . . . . . . . . 1
Wiener Process Basic Properties . . . . . . . . . . . . . . . . . . 3
More Wiener Process Moments . . . . . . . . . . . . . . . . . . 6
Wiener Process Non-Differentiability . . . . . . . . . . . . . . . 9
Wiener Process Expectations Conditioned on Past . . . . . . . . 10
Poisson Process Basic Properties . . . . . . . . . . . . . . . . . . 11
Poisson Process Moments . . . . . . . . . . . . . . . . . . . . . . 16
Poisson Poisson Zero-One Jump Law . . . . . . . . . . . . . . . 18
Temporal, Non-Stationary Poisson Process . . . . . . . . . . . . 21
Poisson Process Expectations Conditioned on Past . . . . . . . 24
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Stochastic Integration for Diffusions 31
Ordinary or Riemann Integration . . . . . . . . . . . . . . . . . 32
Stochastic Integration in W (t): The Foundations . . . . . . . . 35
Stratonovich and other Stochastic Integration Rules . . . . . . . 56
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Stocha