Description Usage Arguments Details Value See Also Examples
rdtq
implements density tracking by quadrature (DTQ) algorithms
to compute the probability density function of a stochastic differential
equation with userspecified drift and diffusion functions.
1 2 
h 
Time step size, a positive numeric scalar. 
k 
Spatial grid spacing, a positive numeric scalar.
(Must be specified if 
bigm 
If k is specified, then bigm is a positive integer such that
 
a 
Left boundary, a numeric scalar.
(Must be specified if 
b 
Right boundary, a numeric scalar.
(Must be specified if 
init 
A numeric scalar indicating either a fixed initial condition of
the form X(0)= 
fT 
The final time, a positive numeric scalar. The computation
assumes an initial time of t=0 and then computes the PDF at time
t= 
drift 
When the user chooses the
We expect the drift function to be a C++ function, implemented using Rcpp, of
type When the user chooses the 
diffusion 
When the user chooses the When the user chooses the 
thresh 
This is an optional numeric scalar parameter that is only used
for the 
method 
A string that indicates which DTQ algorithm to use. There are two choices:

Consider the stochastic differential equation (SDE)
dX(t) = f(X(t)) dt + g(X(t)) dW(t)
where W(t) is standard Brownian motion, f is the drift
function, and g is the diffusion function. Let p(x,t)
denote the timedependent probability density function (PDF) of X(t);
then rdtq
computes p(x,T) for a fixed time T.
Note that the PDF is computed on a spatial grid that can be specified in one of two ways:
Specify a real, positive value k and a positive integer
M = bigm
. In this case, the PDF will be computed on the grid
x_j = j k where j = M, M+1, ..., M1, M. In total,
there will be 2M+1 grid points.
Specify a real, positive integer M and a computational domain [a,b]. In this case, there will be exactly M equispaced grid points. The grid spacing will be k = (ba)/(M1).
The output consists of a list with two elements:
xvec
a numeric vector that contains the spatial grid
pdf
a numeric vector that contains the PDF evaluated at the grid points
H. S. Bhat and R. W. M. A. Madushani, "Density Tracking by Quadrature for Stochastic Differential Equations," arXiv:1610.09572 [stat.CO], http://bit.ly/2fbNsp5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47  # Example 1:
# Define the drift function f(x) = x and diffusion function g(x) = 1
# using C++ code:
require(Rcpp)
sourceCpp(code = '#include <Rcpp.h>
using namespace Rcpp;
double drift(double& x)
{
return(x);
}
double diff(double& x)
{
return(1.0);
}
typedef double (*funcPtr)(double& x);
// [[Rcpp::export]]
XPtr<funcPtr> driftXPtr()
{
return(XPtr<funcPtr>(new funcPtr(&drift)));
}
// [[Rcpp::export]]
XPtr<funcPtr> diffXPtr()
{
return(XPtr<funcPtr>(new funcPtr(&diff)));
}')
# Solve for the PDF (at final time fT=1) of the SDE with drift f,
# diffusion g, and deterministic initial condition X(0) = 0.
# First we solve using the grid specified by k and bigm.
# Then we solve using the grid specified by a, b, and bigm.
# We then check that we get the same PDF either way.
k = 0.01
M = 250
test1 = rdtq(h=0.1,k,bigm=M,init=0,fT=1,
drift=driftXPtr(),diffusion=diffXPtr(),method="cpp")
test2 = rdtq(h=0.1,a=2.5,b=2.5,bigm=501,init=0,fT=1,
drift=driftXPtr(),diffusion=diffXPtr(),method="cpp")
print(k*sum(abs(test1$pdftest2$pdf)))
# Example 2:
# We again use the drift function f(x) = x and diffusion function g(x) = 1.
# This time, we use the method="sparse" version of DTQ.
# This requires us to define the drift and diffusion functions in R:
mydrift = function(x) { x }
mydiff = function(x) { rep(1,length(x)) }
test = rdtq(h=0.1,k=0.01,bigm=250,init=0,fT=1,
drift=mydrift,diffusion=mydiff,method="sparse")
plot(test$xvec,test$pdf,type='l')

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