Thesis

Numerical approximation and parametric statistical inference of stochastic differential equations, with applications to finance

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Awarding institution
  • University of Strathclyde
Date of award
  • 2014
Thesis identifier
  • T13851
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Abstract
  • Stochastic differential equations (SDEs) have become an indispensable tool for modelling the dynamics of key state variables in mathematical finance such as instantaneous short rates of interest, share prices, and volatility processes. The appropriate application of SDEs requires reliable methods of generating sample paths from the equations, e.g. for use in Monte Carlo simulations, and robust parameter estimation methods to calibrate the SDEs to observed market data. Proposed stochastic models for financial variables are becoming increasingly complex in an effort to produce more realistic models, but only on rare occasions are the analytic expressions for the processes' transition densities available. Consequently, it is rare to be able to simulate sample data from the exact process, or conduct full likelihood-based inference. This difficulty motivates the need for approximation methods that are capable of simulating approximate sample paths with desirable convergence properties such that approximation errors can be controlled; and flexible parameter estimation methods that are not materially hampered by a paucity of analytic results associated with intractible SDEs. In this thesis we introduce a numerical approximation scheme for a class of SDEs that are widely applicable to finance. We prove the strong convergence of the numerical scheme and provide a lower bound on the convergence rate associated with the scheme. We also explore the subject of parameter estimation in the context of SDEs, and present three new parameter estimation techniques. By an application of approxiate Bayesian computation (ABC) we develop two sampling algorithms that are capable of producing high quality approximations to the posterior distribution of model parameters, without any need to evaluate model likelihoods.
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DOI
Date Created
  • 2014
Former identifier
  • 1039508

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