Numerical solution of stochastic differential equations with jumps in finance

Bruti-Liberati, N

Numerical solution of stochastic differential equations with jumps in finance

Bruti-Liberati, N

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Publication Type:: Thesis
Issue Date:: 2007

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Field	Value	Language
dc.contributor.author	Bruti-Liberati, N
dc.date.accessioned	2010-10-19T04:52:59Z
dc.date.accessioned	2012-12-15T03:53:14Z
dc.date.available	2010-10-19T04:52:59Z
dc.date.available	2012-12-15T03:53:14Z
dc.date.issued	2007
dc.identifier.uri	http://hdl.handle.net/10453/20293
dc.description	University of Technology, Sydney. Faculty of Business.
dc.description.abstract	This thesis concerns the design and analysis of new discrete time approximations for stochastic differential equations (SDEs) driven by Wiener processes and Poisson random measures. In financial modelling, SDEs with jumps are often used to describe the dynamics of state variables such as credit ratings, stock indices, interest rates, exchange rates and electricity prices. The jump component can capture event-driven uncertainties, such as corporate defaults, operational failures or central bank announcements. The thesis proposes new, efficient, and numerically stable strong and weak approximations. Strong approximations provide efficient tools for problems such as filtering, scenario analysis and hedge simulation, while weak approximations are useful for handling problems such as derivative pricing, the evaluation of moments, and the computation of risk measures and expected utilities. The discrete time approximations proposed are divided into regular and jump-adapted schemes. Regular schemes employ time discretizations that do not include the jump times of the Poisson measure. Jump-adapted time discretizations, on the other hand, include these jump times. The first part of the thesis introduces stochastic expansions for jump diffusions and proves new, powerful lemmas providing moment estimates of multiple stochastic integrals. The second part presents strong approximations with a new strong convergence theorem for higher order general approximations. Innovative strong derivative-free and predictor-corrector schemes are derived. Furthermore, the strong convergence of higher order schemes for pure jump SDEs is established under conditions weaker than those required for jump diffusions. The final part of the thesis presents a weak convergence theorem for jump-adapted higher order general approximations. These approximations include new derivative-free, predictor-corrector, and simplified schemes. Finally, highly efficient implementations of simplified weak schemes based on random bit generators and hardware accelerators are developed and tested.	en
dc.format	Thesis (PhD)	en
dc.language.iso	en	en
dc.relation	https://opus.lib.uts.edu.au/bitstream/10453/20293/9/02Whole.pdf
dc.relation.replaces	http://hdl.handle.net/2100/1157
dc.rights	au.edu.uts.lib/ppc
dc.rights	info:eu-repo/semantics/openAccess
dc.rights	The author owns the copyright in this thesis including all reproduction and reuse rights for the work. The work may not be altered without the permission of the copyright owner. Attribution is essential when quoting or paraphrasing from this thesis.
dc.subject	Stochastic differential equations.	en
dc.subject	Jump processes.	en
dc.title	Numerical solution of stochastic differential equations with jumps in finance	en
dc.type	Thesis
utslib.copyright.status	open_access

Abstract:

This thesis concerns the design and analysis of new discrete time approximations for stochastic differential equations (SDEs) driven by Wiener processes and Poisson random measures. In financial modelling, SDEs with jumps are often used to describe the dynamics of state variables such as credit ratings, stock indices, interest rates, exchange rates and electricity prices. The jump component can capture event-driven uncertainties, such as corporate defaults, operational failures or central bank announcements. The thesis proposes new, efficient, and numerically stable strong and weak approximations. Strong approximations provide efficient tools for problems such as filtering, scenario analysis and hedge simulation, while weak approximations are useful for handling problems such as derivative pricing, the evaluation of moments, and the computation of risk measures and expected utilities. The discrete time approximations proposed are divided into regular and jump-adapted schemes. Regular schemes employ time discretizations that do not include the jump times of the Poisson measure. Jump-adapted time discretizations, on the other hand, include these jump times. The first part of the thesis introduces stochastic expansions for jump diffusions and proves new, powerful lemmas providing moment estimates of multiple stochastic integrals. The second part presents strong approximations with a new strong convergence theorem for higher order general approximations. Innovative strong derivative-free and predictor-corrector schemes are derived. Furthermore, the strong convergence of higher order schemes for pure jump SDEs is established under conditions weaker than those required for jump diffusions. The final part of the thesis presents a weak convergence theorem for jump-adapted higher order general approximations. These approximations include new derivative-free, predictor-corrector, and simplified schemes. Finally, highly efficient implementations of simplified weak schemes based on random bit generators and hardware accelerators are developed and tested.

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http://hdl.handle.net/10453/20293