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(1 - 3 of 3)
- Title
- SIMULATING THE HESTON MODEL VIA THE QE METHOD WITH A SPECIFIED ERROR TOLERANCE
- Creator
- Zhao, Xiaoyang
- Date
- 2017, 2017-05
- Description
-
The Quadratic Exponential (QE) model is a market standard simulation method for the Heston stochastic volatility model. We identify certain...
Show moreThe Quadratic Exponential (QE) model is a market standard simulation method for the Heston stochastic volatility model. We identify certain numerical problems with the standard discretization and modify the original method to correct these problems. We implement our modified QE scheme for the Heston model in the Guaranteed Automatic Integration Library (GAIL)|a suite of algorithms that includes Monte Carlo and quasi-Monte Carlo methods for multidimensional integration and computation of means. GAIL computes answers to satisfy user-defined error tolerances. We also implement variance reduction techniques for our modified QE scheme in GAIL. The numerical results show that our modified scheme is fast and accurate, and satisfies the user-defined error tolerances.
M.S. in Applied Mathematics, May 2017
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- Title
- ADAPTIVE QUASI-MONTE CARLO CUBATURE
- Creator
- Jimenez Rugama, Lluis Antoni
- Date
- 2016, 2016-12
- Description
-
In some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the...
Show moreIn some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the solution with numerical methods that estimate the value of the integral. However, for high dimensional integrals many techniques suffer from the curse of dimensionality. This can be solved if we use quasi-Monte Carlo methods which do not suffer from this phenomenon. Section 2.2 describes digital sequences and rank-1 lattice node sequences, two of the most common points used in quasi-Monte Carlo. If one uses quasi-Monte Carlo, there is still another problem to address: how many points are needed to estimate the integral within a particular absolute error tolerance. In this dissertation, we propose two automatic cubatures based on digital sequences and rank-1 lattice node sequences that estimate high dimensional problems. These new algorithms are constructed in Chapter 3 and the user-specified absolute error tolerance is guaranteed to be satisfied for a specific set of integrands. In Chapter 4 we define a new estimator that satisfies a generalized tolerance function and includes a relative error tolerance option. An important property of quasi-Monte Carlo methods is that they are effective when the function has low effective dimension. In [1], Sobol’ defined the global sensitivity indices, which measure what part of the variance is explained by each dimension. We can use these indices to measure the effective dimensionality of a function. In Chapter 5 we extend our digital sequences cubature to estimate first order and total effect Sobol’ indices.
Ph.D. in Applied Mathematics, December 2016
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- Title
- Fast Automatic Bayesian Cubature Using Matching Kernels and Designs
- Creator
- Rathinavel, Jagadeeswaran
- Date
- 2019
- Description
-
Automatic cubatures approximate multidimensional integrals to user-specified error tolerances. In many real-world integration problems, the...
Show moreAutomatic cubatures approximate multidimensional integrals to user-specified error tolerances. In many real-world integration problems, the analytical solution is either unavailable or difficult to compute. To overcome this, one can use numerical algorithms that approximately estimate the value of the integral. For high dimensional integrals, quasi-Monte Carlo (QMC) methods are very popular. QMC methods are equal-weight quadrature rules where the quadrature points are chosen deterministically, unlike Monte Carlo (MC) methods where the points are chosen randomly.The families of integration lattice nodes and digital nets are the most popular quadrature points used. These methods consider the integrand to be a deterministic function. An alternative approach, called Bayesian cubature, postulates the integrand to be an instance of a Gaussian stochastic process. For high dimensional problems, it is difficult to adaptively change the sampling pattern. But one can automatically determine the sample size, $n$, given a fixed and reasonable sampling pattern. We take this approach using a Bayesian perspective. We assume a Gaussian process parameterized by a constant mean and a covariance function defined by a scale parameter and a function specifying how the integrand values at two different points in the domain are related. These parameters are estimated from integrand values or are given non-informative priors. This leads to a credible interval for the integral. The sample size, $n$, is chosen to make the credible interval for the Bayesian posterior error no greater than the desired error tolerance. However, the process just outlined typically requires vector-matrix operations with a computational cost of $O(n^3)$. Our innovation is to pair low discrepancy nodes with matching kernels, which lowers the computational cost to $O(n \log n)$. We begin the thesis by introducing the Bayesian approach to calculate the posterior cubature error and define our automatic Bayesian cubature. Although much of this material is known, it is used to develop the necessary foundations. Some of the major contributions of this thesis include the following: 1) The fast Bayesian transform is introduced. This generalizes the techniques that speedup Bayesian cubature when the kernel matches low discrepancy nodes. 2) The fast Bayesian transform approach is demonstrated using two methods: a) rank-1 lattice sequences and shift-invariant kernels, and b) Sobol' sequences and Walsh kernels. These two methods are implemented as fast automatic Bayesian cubature algorithms in the Guaranteed Automatic Integration Library (GAIL). 3) We develop additional numerical implementation techniques: a) rewriting the covariance kernel to avoid cancellation error, b) gradient descent for hyperparameter search, and c) non-integer kernel order selection.The thesis concludes by applying our fast automatic Bayesian cubature algorithms to three sample integration problems. We show that our algorithms are faster than the basic Bayesian cubature and that they provide answers within the error tolerance in most cases. The Bayesian cubatures that we develop are guaranteed for integrands belonging to a cone of functions that reside in the middle of the sample space. The concept of a cone of functions is also explained briefly.
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