fptdApprox (version 2.5)

diffproc: Diffusion Processes

Description

diffproc creates an object of class “diffproc” from the given set of values.

as.diffproc attempts to turn its argument into an object of class “diffproc”.

is.diffproc tests if its argument is an object of class “diffproc”.

print shows an object of class “diffproc”.

Usage

diffproc(text)
as.diffproc(text)
is.diffproc(obj)
# S3 method for diffproc
print(x, ...)

Value

If possible, this function returns an object of class “diffproc” that defines a family of diffusion processes. It is a four-component list:

mean

character of length 1 with the mathematical expression of the infinitesimal mean of the process.

var

character of length 1 with the mathematical expression of the infinitesimal variance of the process.

tpdf

character of length 1 with the mathematical expression of the transition probability density function of the process.

tpdF

character of length 1 with the mathematical expression of the transition probability distribution function of the process.

is.diffproc returns TRUE or FALSE depending on whether its argument is an object of class “diffproc” or not.

Arguments

text

a character vector or list of length four to be coerced, containing the infinitesimal mean, infinitesimal variance, transition probability density function and transition probability distribution function of a diffusion process.

obj

an R object to be tested.

x

an object of class “diffproc”.

...

additional arguments potentially passed (currently none is considered).

Author

Patricia Román-Román, Juan J. Serrano-Pérez and Francisco Torres-Ruiz.

Details

The main goal of the diffproc function is to validate the mathematical expressions in the given character string vector or list. If no errors happen, the function converts the input into an object of class “diffproc”. Otherwise, the function reports the errors.

The mathematical expressions of the infinitesimal mean and variance should be expressions in \(x\) and \(t\), \(A_1(x,t)\) and \(A_2(x,t)\), respectively.

The mathematical expressions of the transition probability density and distribution functions should be expressions in \(x\), \(t\), \(y\) and \(s\), \(f(x,t|y,s)\) and \(F(x,t|y,s)\), respectively.

In addition, all mathematical expressions may depend on generic parameters and functions of \(t\), or \(s\) and \(t\), referred by name. To name the expression of a function \(h(u)\) we can use `h(u)` = value if we want to show its dependence on \(u\), or h otherwise.

The function checks if the mathematical expressions show syntax errors and if R can compute the symbolic derivative with respect to \(x\) for the infinitesimal variance and transition probability density function.

References

P. Román-Román, J.J. Serrano-Pérez, F. Torres-Ruiz. (2012) An R package for an efficient approximation of first-passage-time densities for diffusion processes based on the FPTL function. Applied Mathematics and Computation, 218, 8408--8428.

P. Román-Román, J.J. Serrano-Pérez, F. Torres-Ruiz. (2014) More general problems on first-passage times for diffusion processes: A new version of the fptdApprox R package. Applied Mathematics and Computation, 244, 432--446.

See Also

D to know more about the simple functions and expressions that support symbolic derivative in R, and in particular about the functions provided in R to evaluate probability density functions.

FPTL to locate the first-passage-time of a diffusion process through a continuous boundary.

Examples

## Creating a diffproc object for the lognormal diffusion process
x <- c("m*x", "sigma^2*x^2",
       "dnorm((log(x)-(log(y)+(m-sigma^2/2)*(t-s)))/(sigma*sqrt(t-s)),0,1)/
       (sigma*sqrt(t-s)*x)", "plnorm(x,log(y)+(m-sigma^2/2)*(t-s),
       sigma*sqrt(t-s))") 

Lognormal <- diffproc(x)
Lognormal

## Creating a diffproc object for the Ornstein Uhlenbeck diffusion process
x <- c("alpha*x + beta", "sigma^2", "dnorm((x-(y*exp(alpha*(t-s))-beta*
       (1-exp(alpha*(t-s)))/alpha))/(sigma*sqrt((exp(2*alpha*(t-s)) - 1)/
       (2*alpha))),0,1)/(sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha)))",
       "pnorm(x, y*exp(alpha*(t-s)) - beta*(1 - exp(alpha*(t-s)))/alpha,
       sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha)))")
			 
OU <- diffproc(x)
OU

## Creating a diffproc object for the lognormal diffusion process with exogenous factors
x <- c("`h(t)`*x", "sigma^2*x^2", "dnorm((log(x)-(log(y)+`H(s,t)`-(sigma^2/2)*
       (t - s)))/(sigma*sqrt(t-s)),0,1)/(sigma*sqrt(t-s)*x)", "plnorm(x,log(y)+
       `H(s,t)`-(sigma^2/2)*(t-s),sigma*sqrt(t-s))")
    
LognormalFEx <- diffproc(x)
LognormalFEx

## Testing diffproc objects
is.diffproc(Lognormal)
is.diffproc(OU)
is.diffproc(LognormalFEx)