271 research outputs found
Scrambled geometric net integration over general product spaces
Quasi-Monte Carlo (QMC) sampling has been developed for integration over
where it has superior accuracy to Monte Carlo (MC) for integrands of
bounded variation. Scrambled net quadrature gives allows replication based
error estimation for QMC with at least the same accuracy and for smooth enough
integrands even better accuracy than plain QMC. Integration over triangles,
spheres, disks and Cartesian products of such spaces is more difficult for QMC
because the induced integrand on a unit cube may fail to have the desired
regularity. In this paper, we present a construction of point sets for
numerical integration over Cartesian products of spaces of dimension ,
with triangles () being of special interest. The point sets are
transformations of randomized -nets using recursive geometric
partitions. The resulting integral estimates are unbiased and their variance is
for any integrand in of the product space. Under smoothness
assumptions on the integrand, our randomized QMC algorithm has variance
, for integration over -fold Cartesian
products of -dimensional domains, compared to for ordinary Monte
Carlo.Comment: 29 pages; 5 figure
Asymptotic Normality of Extensible Grid Sampling
Recently, He and Owen (2016) proposed the use of Hilbert's space filling
curve (HSFC) in numerical integration as a way of reducing the dimension from
to . This paper studies the asymptotic normality of the HSFC-based
estimate when using scrambled van der Corput sequence as input. We show that
the estimate has an asymptotic normal distribution for functions in
, excluding the trivial case of constant functions. The
asymptotic normality also holds for discontinuous functions under mild
conditions. It was previously known only that scrambled -net
quadratures enjoy the asymptotic normality for smooth enough functions, whose
mixed partial gradients satisfy a H\"older condition. As a by-product, we find
lower bounds for the variance of the HSFC-based estimate. Particularly, for
nontrivial functions in , the low bound is of order ,
which matches the rate of the upper bound established in He and Owen (2016)
On Integration Methods Based on Scrambled Nets of Arbitrary Size
We consider the problem of evaluating for a function . In situations where
can be approximated by an estimate of the form
, with a point set in
, it is now well known that the Monte Carlo
convergence rate can be improved by taking for the first
points, , of a scrambled
-sequence in base . In this paper we derive a bound for the
variance of scrambled net quadrature rules which is of order
without any restriction on . As a corollary, this bound allows us to provide
simple conditions to get, for any pattern of , an integration error of size
for functions that depend on the quadrature size . Notably,
we establish that sequential quasi-Monte Carlo (M. Gerber and N. Chopin, 2015,
\emph{J. R. Statist. Soc. B, to appear.}) reaches the
convergence rate for any values of . In a numerical study, we show that for
scrambled net quadrature rules we can relax the constraint on without any
loss of efficiency when the integrand is a discontinuous function
while, for sequential quasi-Monte Carlo, taking may only
provide moderate gains.Comment: 27 pages, 2 figures (final version, to appear in The Journal of
Complexity
High Performance Financial Simulation Using Randomized Quasi-Monte Carlo Methods
GPU computing has become popular in computational finance and many financial
institutions are moving their CPU based applications to the GPU platform. Since
most Monte Carlo algorithms are embarrassingly parallel, they benefit greatly
from parallel implementations, and consequently Monte Carlo has become a focal
point in GPU computing. GPU speed-up examples reported in the literature often
involve Monte Carlo algorithms, and there are software tools commercially
available that help migrate Monte Carlo financial pricing models to GPU.
We present a survey of Monte Carlo and randomized quasi-Monte Carlo methods,
and discuss existing (quasi) Monte Carlo sequences in GPU libraries. We discuss
specific features of GPU architecture relevant for developing efficient (quasi)
Monte Carlo methods. We introduce a recent randomized quasi-Monte Carlo method,
and compare it with some of the existing implementations on GPU, when they are
used in pricing caplets in the LIBOR market model and mortgage backed
securities
Construction of interlaced scrambled polynomial lattice rules of arbitrary high order
Higher order scrambled digital nets are randomized quasi-Monte Carlo rules
which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011),
1372--1398] and shown to achieve the optimal rate of convergence of the root
mean square error for numerical integration of smooth functions defined on the
-dimensional unit cube. The key ingredient there is a digit interlacing
function applied to the components of a randomly scrambled digital net whose
number of components is , where the integer is the so-called
interlacing factor. In this paper, we replace the randomly scrambled digital
nets by randomly scrambled polynomial lattice point sets, which allows us to
obtain a better dependence on the dimension while still achieving the optimal
rate of convergence. Our results apply to Owen's full scrambling scheme as well
as the simplifications studied by Hickernell, Matou\v{s}ek and Owen. We
consider weighted function spaces with general weights, whose elements have
square integrable partial mixed derivatives of order up to , and
derive an upper bound on the variance of the estimator for higher order
scrambled polynomial lattice rules. Employing our obtained bound as a quality
criterion, we prove that the component-by-component construction can be used to
obtain explicit constructions of good polynomial lattice point sets. By first
constructing classical polynomial lattice point sets in base and dimension
, to which we then apply the interlacing scheme of order , we obtain a
construction cost of the algorithm of order operations
using memory in case of product weights, where is the
number of points in the polynomial lattice point set
Monte Carlo Methods and Path-Generation techniques for Pricing Multi-asset Path-dependent Options
We consider the problem of pricing path-dependent options on a basket of
underlying assets using simulations. As an example we develop our studies using
Asian options. Asian options are derivative contracts in which the underlying
variable is the average price of given assets sampled over a period of time.
Due to this structure, Asian options display a lower volatility and are
therefore cheaper than their standard European counterparts. This paper is a
survey of some recent enhancements to improve efficiency when pricing Asian
options by Monte Carlo simulation in the Black-Scholes model. We analyze the
dynamics with constant and time-dependent volatilities of the underlying asset
returns. We present a comparison between the precision of the standard Monte
Carlo method (MC) and the stratified Latin Hypercube Sampling (LHS). In
particular, we discuss the use of low-discrepancy sequences, also known as
Quasi-Monte Carlo method (QMC), and a randomized version of these sequences,
known as Randomized Quasi Monte Carlo (RQMC). The latter has proven to be a
useful variance reduction technique for both problems of up to 20 dimensions
and for very high dimensions. Moreover, we present and test a new path
generation approach based on a Kronecker product approximation (KPA) in the
case of time-dependent volatilities. KPA proves to be a fast generation
technique and reduces the computational cost of the simulation procedure.Comment: 34 pages, 4 figure, 15 table
Application of Sequential Quasi-Monte Carlo to Autonomous Positioning
Sequential Monte Carlo algorithms (also known as particle filters) are
popular methods to approximate filtering (and related) distributions of
state-space models. However, they converge at the slow rate, which
may be an issue in real-time data-intensive scenarios. We give a brief outline
of SQMC (Sequential Quasi-Monte Carlo), a variant of SMC based on
low-discrepancy point sets proposed by Gerber and Chopin (2015), which
converges at a faster rate, and we illustrate the greater performance of SQMC
on autonomous positioning problems.Comment: 5 pages, 4 figure
On the dependence structure and quality of scrambled nets
In this paper we develop a framework to study the dependence structure of
scrambled -nets. It relies on values denoted by ,
which are related to how many distinct pairs of points from lie in the
same elementary interval in base . These values quantify the
equidistribution properties of in a more informative way than the
parameter . They also play a key role in determining if a scrambled set
is negative lower orthant dependent (NLOD). Indeed this property
holds if and only if for all , which in turn implies that a scrambled digital net in
base is NLOD if and only if . Through numerical examples we
demonstrate that these values are a powerful tool to
compare the quality of different -nets, and to enhance our
understanding of how scrambling can improve the quality of deterministic point
sets.Comment: 28 page
Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling
In this paper we introduce a reproducing kernel Hilbert space defined on
as the tensor product of a reproducing kernel defined on the
unit sphere in and a reproducing kernel
defined on . We extend Stolarsky's invariance principle to this
case and prove upper and lower bounds for numerical integration in the
corresponding reproducing kernel Hilbert space.
The idea of separating the direction from the distance from the origin can
also be applied to the construction of quadrature methods. An extension of the
area-preserving Lambert transform is used to generate points on
via lifting Sobol' points in to the sphere. The
-th component of each Sobol' point, suitably transformed, provides the
distance information so that the resulting point set is normally distributed in
.
Numerical tests provide evidence of the usefulness of constructing
Quasi-Monte Carlo type methods for integration in such spaces. We also test
this method on examples from financial applications (option pricing problems)
and compare the results with traditional methods for numerical integration in
.Comment: 37 pages, 6 table
Local antithetic sampling with scrambled nets
We consider the problem of computing an approximation to the integral
. Monte Carlo (MC) sampling typically attains a root
mean squared error (RMSE) of from independent random function
evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully
equispaced evaluation points can attain the rate for
any and randomized QMC (RQMC) can attain the RMSE
, both under mild conditions on . Classical
variance reduction methods for MC can be adapted to QMC. Published results
combining QMC with importance sampling and with control variates have found
worthwhile improvements, but no change in the error rate. This paper extends
the classical variance reduction method of antithetic sampling and combines it
with RQMC. One such method is shown to bring a modest improvement in the RMSE
rate, attaining for any , for
smooth enough .Comment: Published in at http://dx.doi.org/10.1214/07-AOS548 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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