28,452 research outputs found
A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations
The implementation of the convolution method for the numerical solution of
backward stochastic differential equations (BSDEs) introduced in [19] uses a
uniform space grid. Locally, this approach produces a truncation error, a space
discretization error, and an additional extrapolation error. Even if the
extrapolation error is convergent in time, the resulting absolute error may be
high at the boundaries of the uniform space grid. In order to solve this
problem, we propose a tree-like grid for the space discretization which
suppresses the extrapolation error leading to a globally convergent numerical
solution for the (F)BSDE. On this alternative grid the conditional expectations
involved in the BSDE time discretization are computed using Fourier analysis
and the fast Fourier transform (FFT) algorithm as in the initial
implementation. The method is then extended to higher-order time
discretizations of FBSDEs. Numerical results demonstrating convergence are also
presented.Comment: 28 pages, 8 figures; Previously titled 'Global convergence and
stability of a convolution method for numerical solution of BSDEs'
(1410.8595v1
Exact Simulation of Wishart Multidimensional Stochastic Volatility Model
In this article, we propose an exact simulation method of the Wishart
multidimensional stochastic volatility (WMSV) model, which was recently
introduced by Da Fonseca et al. \cite{DGT08}. Our method is based onanalysis of
the conditional characteristic function of the log-price given volatility
level. In particular, we found an explicit expression for the conditional
characteristic function for the Heston model. We perform numerical experiments
to demonstrate the performance and accuracy of our method. As a result of
numerical experiments, it is shown that our new method is much faster and
reliable than Euler discretization method.Comment: 27 page
Discretization of Fractional Differential Equations by a Piecewise Constant Approximation
There has recently been considerable interest in using a nonstandard
piecewise approximation to formulate fractional order differential equations as
difference equations that describe the same dynamical behaviour and are more
amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the
fundamental papers, the difference equations formulated through this process do
not capture the dynamics of the fractional order equations. We show that the
correct application of this nonstandard piecewise approximation leads to a one
parameter family of fractional order differential equations that converges to
the original equation as the parameter tends to zero. A closed formed solution
exists for each member of this family and leads to the formulation of a
difference equation that is of increasing order as time steps are taken. Whilst
this does not lead to a simplified dynamical analysis it does lead to a
numerical method for solving the fractional order differential equation. The
method is shown to be equivalent to a quadrature based method, despite the fact
that it has not been derived from a quadrature. The method can be implemented
with non-uniform time steps. An example is provided showing that the difference
equation can correctly capture the dynamics of the underlying fractional
differential equation
Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies
In this paper, we give a procedure of how to discretize the recursion
operators by considering unified bilinear forms of integrable hierarchies. As
two illustrative examples, the unified bilinear forms of the AKNS hierarchy and
the KdV hierarchy are presented from their recursion operators. Via the
compatibility between soliton equations and their auto-B\"acklund
transformations, the bilinear integrable hierarchies are discretized and the
discrete recursion operators are obtained. The discrete recursion operators
converge to the original continuous forms after a standard limit.Comment: 11Page
New definitions of exponential, hyperbolic and trigonometric functions on time scales
We propose two new definitions of the exponential function on time scales.
The first definition is based on the Cayley transformation while the second one
is a natural extension of exact discretizations. Our eponential functions map
the imaginary axis into the unit circle. Therefore, it is possible to define
hyperbolic and trigonometric functions on time scales in a standard way. The
resulting functions preserve most of the qualitative properties of the
corresponding continuous functions. In particular, Pythagorean trigonometric
identities hold exactly on any time scale. Dynamic equations satisfied by
Cayley-motivated functions have a natural similarity to the corresponding
diferential equations. The exact discretization is less convenient as far as
dynamic equations and differentiation is concerned.Comment: 27 page
A partial Fourier transform method for a class of hypoelliptic Kolmogorov equations
We consider hypoelliptic Kolmogorov equations in spatial dimensions,
with , where the differential operator in the first spatial
variables featuring in the equation is second-order elliptic, and with respect
to the st spatial variable the equation contains a pure transport term
only and is therefore first-order hyperbolic. If the two differential
operators, in the first and in the st co-ordinate directions, do not
commute, we benefit from hypoelliptic regularization in time, and the solution
for is smooth even for a Dirac initial datum prescribed at . We
study specifically the case where the coefficients depend only on the first
variables. In that case, a Fourier transform in the last variable and standard
central finite difference approximation in the other variables can be applied
for the numerical solution. We prove second-order convergence in the spatial
mesh size for the model hypoelliptic equation subject to
the initial condition , with and , proposed by Kolmogorov, and for an
extension with . We also demonstrate exponential convergence of an
approximation of the inverse Fourier transform based on the trapezium rule.
Lastly, we apply the method to a PDE arising in mathematical finance, which
models the distribution of the hedging error under a mis-specified derivative
pricing model
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