Approximation of First-Passage-Time Densities for Diffusion Processes

Description

Efficient approximation of first-passage-time (f.p.t.) densities for diffusion processes based on the First-Passage-Time Location (FPTL) function.

For a complete list of functions, use library(help=“fptdApprox”).

Details

The fptdApprox package allows to approximate efficiently the f.p.t. density for a diffusion process through a continuous time-dependent boundary in the cases of conditioned and unconditioned f.p.t. problems.

For an unconditioned f.p.t. problem, a step by step study can be performed. First, the diffusion process under consideration must be defined by using the function diffproc. Then, the FPTL function will be used to calculate the FPTL function for the specified process and boundary. The information provided by the function is then extracted by the method summary.fptl and used to find the range of variation of the f.p.t. variable. Finally, such information is used by the function Approx.cfpt.density to obtain the approximation of the f.p.t. density.

In the general case (conditioned and unconditioned f.p.t. problems) the function Approx.fpt.density allows to obtain directly the approximate f.p.t. density.

Author(s)

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

Maintainer: Juan J. Serrano-Pérez, jjserra@ugr.es

References

Buonocore, A., Nobile, A.G. and Ricciardi, L.M. (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Probab., 19, 784–800.

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

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.

Approximating First-Passage-Time Densities for Conditioned Problems

Description

Approx.cfpt.density computes values of the approximate first-passage-time (f.p.t.) density, for a conditioned problem, from an object of class “summary.fptl” that contains the information provided by First-Passage-Time Location (FPTL) function.

Usage

Approx.cfpt.density(sfptl, variableStep = TRUE, from.t0 = FALSE, 
                   to.T = FALSE, skip = TRUE, n = 250, p = 0.2, 
                   alpha = 1, tol = 1e-03, it.max = 50000L)                      

Arguments

Details

For a diffusion process \{X(t), t_0 \leq t \leq T \}, the f.p.t. variable, conditioned to X(t_0) = x_0, through a continuous boundary S(t) is defined as

T_{S(t), x_0} = \left\{ \begin{array}{lll} Inf \ \{ t \geq t_0 \ : \ X(t) > S(t) \mid X(t_0)=x_0 \} & & if \ x_0 < S(t_0) \\[7pt] Inf \ \{ t \geq t_0 \ : \ X(t) < S(t) \mid X(t_0)=x_0 \} & & if \ x_0 > S(t_0) \end{array} \right. .

Its density function is the solution to a Volterra integral equation of the second kind. The kernel of this equation depends on the infinitesimal moments of the process, the transition probability density function and the boundary.

Nevertheless, and apart from some particular processes and boundaries, closed-form solutions for the integral equation are not available. For this reason, in the cases without explicit solutions, numerical procedures are required. That is the situation considered here and the numerical procedure implemented by the Approx.fpt.density function is the one proposed by Buonocore et al. (1987), based on the composite trapezoid method.

The Approx.cfpt.density function computes efficiently the approximate f.p.t. density by using the information provided by the FPTL function contained in the sfptl object. See the function summary.fptl for details.

By default the function does not compute the approximate f.p.t. density from the time instant t_0, but from a more suitable time instant t_1^* provided by the FPTL function. It also uses a variable integration step.

The function makes an internal call to Integration.Steps function in order to determine the subintervals and integration steps to be used in the application of the numerical algorithm according to the variableStep, from.t0, to.T, n, p and alpha arguments.

In addition, if skip = TRUE, the function checks the approximate density value for each t_{max,i}^+, and, if it is almost 0, the application of the numerical algorithm in the subinterval [t_{max,i}^+, t_{i+1}^*] is avoided, and then continued from instant t_{i+1}^* considering a zero value of the approximate density.

Similarly, if to.T = FALSE, the function checks the cumulative value of the integral for each t_{max,i}^+ provided by the FPTL function and, if it is greater than or equal to 1 - tol, the numerical algorithm is stopped. In any case, the algorithm is stopped in the final t_{max,i}^+, and if the cumulative value of the integral is less than 1 - tol the function issues a warning.

Value

The Approx.cfpt.density function computes and returns an object of class “fpt.density”. It is a three-component list:

It also includes six additional attributes:

x is the vector result of the concatenation of the sequences of equally spaced values in the suitable subintervals determined by the Integration.Steps function.

Author(s)

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

References

Buonocore, A., Nobile, A.G. and Ricciardi, L.M. (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Probab., 19, 784–800.

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

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

summary.fptl to locate the f.p.t. variable and create objects of class “summary.fptl”.

is.fpt.density to test for objects of class “fpt.density”.

print.fpt.density to show objects of class “fpt.density”.

report.fpt.density to generate a report.

plot.fpt.density for graphical display.

FPTL to evaluate the FPTL function and create objects of class “fptl”.

Examples

## Continuing the summary.fptl(.) example:

## Making an efficient approximation of the f.p.t. density 
## (optimal variable integration steps and small computational cost)
yyy <- Approx.cfpt.density(yy)
yyy

zzz <- Approx.cfpt.density(zz)
zzz

## Making a less efficient approximation of the f.p.t. density 
## (optimal fixed integration step but high computational cost related to 
##  the efficient approximation)
## Not run: 
yyy1 <- Approx.cfpt.density(yy, variableStep = FALSE, from.t0 = TRUE, to.T = 
                         TRUE, skip = FALSE)
yyy1
## End(Not run)

Approximating First-Passage-Time Densities

Description

Approx.fpt.density computes values of the approximate first-passage-time (f.p.t.) density, for conditioned and unconditioned problems.

For the unconditioned case, values of the approximate densities for f.p.t. problems conditioned to suitable values of the initial distribution are also calculated.

Usage

Approx.fpt.density(dp, t0, T, id, S, env = NULL, variableStep = TRUE, 
                   from.t0 = FALSE, to.T = FALSE, r = 4000, zeroSlope = 0.01, 
                   p0.tol = 8, k = 3, m = 100, n = 250, p = 0.2, alpha = 1, 
                   skip = TRUE, tol = 0.001, it.max)               

Arguments

Details

For an unconditioned f.p.t. problem, this function allows to compute directly the approximate f.p.t. density from an object of class “diffproc”. In contrast to the approximation of a conditioned f.p.t. density by using the Approx.cfpt.density function, in this case it is not necessary for users to make previous calls to the functions FPTL and summary.FPTL.

For a diffusion process \{X(t), t_0 \leq t \leq T \} with non-degenerate initial distribution the unconditioned f.p.t. variable, through a continuous boundary S(t), is defined as

T_{S(t), X(t_0)} = \left\{ \begin{array}{lll} Inf \ \{ t \geq t_0 \ : \ X(t) > S(t) \} & & if \ X(t_0) < S(t_0) \\[7pt] Inf \ \{ t \geq t_0 \ : \ X(t) < S(t) \} & & if \ X(t_0) > S(t_0) \end{array} \right. ,

Its density function can be obtained by means of the numerical integration, in the range of variation of X(t_0), of the corresponding f.p.t. densities conditioned to values of X(t_0), weighted by the initial density function.

For each conditioned problem related to an unconditioned f.p.t. problem, the Approx.fpt.density function makes internal calls to FPTL and summary.fptl functions, in order to localize each conditioned f.p.t. variable, according to the zeroSlope, p0.tol and k arguments. Then, the function makes a internall call to Integration.Steps function, in order to determine the suitable subintervals and integration steps to be used to approximate the unconditioned f.p.t. density and conditioned f.p.t. densities according to the variableStep, from.t0, to.T, n, p and alpha arguments.

From this information, a suitable sequence of time instants in [t_0, T] for the approximation of the f.p.t. densities is available. For each time instant t in such sequence, the Approx.fpt.density function computes the value of the f.p.t. density conditioned to m values of the initial distribution (equally spaced in its range of variation), and then it computes the value of the unconditioned f.p.t density.

For the approximation of each conditioned f.p.t density, the numerical procedure proposed by Buonocore et al. (1987), based on the composite trapezoid method, has been implemented. This method has also been used in the numerical integration for obtaining the unconditioned f.p.t. density.

The mathematical expression of the boundary S should be a function of t and may include the argument t0, the name x0, to refer to an initial value of the process, and the parameters specified in the env argument. The function checks if the mathematical expression shows syntax errors and if R can compute its symbolic derivative with respect to t.

The env argument is a list of tagged values in name = value form with name other than x, t, y and s. To name the expression of a function h(u) as a character string we can use `h(u)` = value if we want to show its dependence on u, or h = value otherwise.

The env argument is copied into a temporary environment for evaluating the mathematical expressions in objects dp and S. R looks for the objects not found into this temporary environment in the parent.frame() environment.

By default the function does not compute the approximate f.p.t. density from the time instant t_0, but from a more suitable time instant provided by the First-Passage-Time Location (FPTL) function. It also uses a variable integration step.

If skip = TRUE, for each subinterval in which it could be possible to avoid calculating all the conditioned f.p.t. densities, the function checks the value of the approximate unconditioned f.p.t. density at the lower end of such subinterval. If it is almost 0, the approximate unconditioned f.p.t. density calculation is avoided in the subinterval, and it continues from the upper end of the subinterval considering a zero value of the approximate unconditioned f.p.t. density.

Similarly, if to.T = FALSE, the function checks the cumulative value of the integral for each upper end of the subintervals for which it is not possible to avoid the application of the numerical algorithm for each conditioned problem. If it is greater than or equal to 1 - tol, the approximation procedure is stopped. In any case, the procedure is stopped at the upper end of the last subinterval, and if the cumulative value of the integral is less than 1 - tol the function issues a warning.

Value

The Approx.fpt.density function computes and returns an object of class “fpt.density”. This object is a three-component list:

It also includes six additional attributes:

x is the vector result of the concatenation of the sequences of equally spaced values in the suitable subintervals determined by the Integration.Steps function.

Author(s)

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

References

Buonocore, A., Nobile, A.G. and Ricciardi, L.M. (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Probab., 19, 784–800.

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

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

diffproc about creation of class “diffproc” objects.

is.fpt.density to test for objects of class “fpt.density”.

print.fpt.density to show objects of class “fpt.density”.

report.fpt.density to generate a report.

plot.fpt.density for graphical display.

FPTL to evaluate the FPTL function and create objects of class “fptl”.

summary.fptl to locate the f.p.t. variable and create objects of class “summary.fptl”.

Examples

## Continuing the diffproc(.) example:

## Making an efficient approximation of the f.p.t. density
## in the case of a conditioned f.p.t. problem. (optimal 
## variable integration steps and small computational cost)
yyy.cp <- Approx.fpt.density(dp = Lognormal, t0 = 0, T = 18, id = 1, S = 
                        "4.5 + 4*t^2 + 7*t*sqrt(t)*sin(6*sqrt(t))", 
                        env = list(m = 0.48, sigma = 0.07))
yyy.cp

## Not run: 
## Making a less efficient approximation of the f.p.t. density 
## (optimal fixed integration step but high computational cost related to 
##  the efficient approximation)
yyy1.cp <- Approx.fpt.density(dp = Lognormal, t0 = 0, T = 18, id = 1, S = 
                        "4.5 + 4*t^2 + 7*t*sqrt(t)*sin(6*sqrt(t))", 
                        env = list(m = 0.48, sigma = 0.07), 
                        variableStep = FALSE, from.t0 = TRUE, to.T = 
                        TRUE, skip = FALSE)
yyy1.cp

## Making an efficient approximation of the f.p.t. density 
## in the case of an unconditioned f.p.t. problem.
yyy.ucp <- Approx.fpt.density(dp = Lognormal, t0 = 0, T = 18, id = 
                list("dlnorm(x,-0.005,0.1)", "Lambda(-0.005,0.1)", 
                "\\Lambda(-0.005,0.1)", "Lognormal(-0.005,0.1)"),
                S = "4.5 + 4*t^2 + 7*t*sqrt(t)*sin(6*sqrt(t))", 
                env = list(m = 0.48, sigma = 0.07), m=25)                 
yyy.ucp
## End(Not run)

First-Passage-Time Location Function

Description

FPTL computes values of the First-Passage-Time Location (FPTL) function of a diffusion process for a continuous boundary.

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

print.fptl shows an object of class “fptl”.

Usage

FPTL(dp, t0, T, x0, S, env = NULL, n = 4000)

is.fptl(obj)

## S3 method for class 'fptl'
print(x, ...)

Arguments

Details

The FPTL function for the problem of the first-passage-time of a diffusion process \{X(t), t_0 \leq t \leq T \}, conditioned to X(t_0) = x_0, through a continuous boundary S(t) is defined as

FPTL(t) = \left\{ \begin{array}{lll} P[ X(t)>S(t) \, | \, X(t_0)=x_0] = 1 - F(S(t),t \, | \, x_0,t_0) & & if \ x_0 < S(t_0) \\[7pt] P[ X(t)<S(t) \, | \, X(t_0)=x_0] = F(S(t),t \, | \, x_0,t_0) & & if \ x_0 > S(t_0) \end{array} \right. ,

where F(x,t|y,s) is the transition probability distribution function of the process.

Initially, the FPTL function is evaluated at a sequence of n equally spaced values from t0 to T. Then the FPTL function makes an internal call to the growth.intervals function in order to study the growth of the evaluation vector. Finally, the FPTL function is evaluated at a more adequate sequence of values from t0 to T according to the abovementioned study.

The mathematical expression of the boundary S should be a function of t and may include arguments t0, x0 and the parameters specified in the env argument. The FPTL function checks if the mathematical expression shows syntax errors and if R can compute its symbolic derivative with respect to t.

The env argument is a list of tagged values in name = value form with name other than x, t, y and s. To name the expression of a function h(u) as a character string we can use `h(u)` = value if we want to show its dependence on u, or h = value otherwise.

The env argument is copied into a temporary environment for evaluating the mathematical expressions in objects dp and S. R looks for the objects not found into this temporary environment in the parent.frame() environment.

Value

The function FPTL computes and returns an object of class “fptl”. It is a two-component list:

It also includes three additional attributes:

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

Since n is usually large, the print.fptl function does not display an object of class “fptl” as a list, but in its ‘basic’ structure instead. However, each component can be displayed separately in the usual way.

Author(s)

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

References

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

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

diffproc about creation of class “diffproc” objects.

summary.fptl for summaries and plot.fptl for graphical display.

Examples

## Continuing the diffproc(.) examples:

## Specifying a boundary
b <- "4.5 + 4*t^2 + 7*t*sqrt(t)*sin(6*sqrt(t))"

## Computing FPTL functions and creating objects of class fptl
y <- FPTL(dp = Lognormal, t0 = 0, T = 18, x0 = 1, S = b, env = list(m = 0.48,
          sigma = 0.07))
y

z <- FPTL(dp = LognormalFEx, t0 = 1, T = 10, x0 = 1, S = 15, env = list(sigma=0.1, 
          `h(t)` = "t/4", `H(s,t)` = "(t^2-s^2)/8"))
z
          
## Testing fptl objects
is.fptl(y)
is.fptl(z)

Subintervals and Integration Steps To Approximate First-Passage-Time Densities

Description

According to the First-Passage-Time Location (FPTL) function and the arguments in the function call, this function calculates suitable subintervals and integration steps in order to approximate the first-passage-time (f.p.t.) density.

Usage

Integration.Steps(sfptl, variableStep = TRUE, from.t0 = FALSE,
                  to.T = FALSE, n = 250, p = 0.2, alpha = 1)

Arguments

Details

Based on the information provided by the FPTL function contained in the sfptl object, this function computes and returns suitable subintervals and integration steps in order to approximate the density function of the f.p.t. variable according to the other arguments in the function call.

When the sfptl object is of length greater than 1, it comes from an unconditioned f.p.t. problem. Each component is associated with the same f.p.t. problem conditioned on different values of the initial distribution (equally spaced in the range of the distribution). Let x_{0,j}, j=1, \ldots, N, these values. For each initial value x_{0,j} let t_{i,j}^*, t_{max,i,j}^- and t_{max,i,j}^+, i=1,\ldots \ m_j, the interesting time instants provided by the FPTL function and stored in the instants component of the j-th list in the sfptl object. Then, the time instants \{t_{i,j}, i=1, 2, \ldots, 2m_j \}, where

t_{i,j} = \left\{ \begin{array}{ll} t_{(i+1)/2, \, j}^* & for \ i \ odd \\[7pt] t_{max, \, i/2, \thinspace j}^+ & for \ i \ even \end{array} \right. ,

provide a suitable partition of interval [t_0, T] to approximate the f.p.t density for the fixed value x_{0,j} of the initial distribution.

If the sfptl object is of length 1, it comes from a conditioned f.p.t. problem. In this case we denote the interesting time instants provided by the FPTL function and stored in the sfptl object by t_{i,1}^*, t_{max,i,1}^- and t_{max,i,1}^+.

In what follows, \lceil x \rceil is the integer part of x.

For each list in the sfptl object the function computes

h_{i,j} = \displaystyle{\frac{t_{max,i,j}^{+} - \, t_{i,j}^*}{n_{i,j}}} , i=1, \ldots, m_j,

where

n_{i,j} = \lceil n \, k_{i,j} \rceil

and

k_{i,j} = \displaystyle{\frac{t_{max,i,j}^+ - t_{i,j}^{*}}{t_{max,i,j}^{-} - \, t_{i,j}^*}} \ .

If variableStep = TRUE, for 0 < p and \alpha \leq 1, also computes

• h_{i,j}^* = \displaystyle{\frac{t_{i+1,j}^* - t_{max,i,j}^+}{n_{i,j}^*}} , i=1, \ldots, m_j-1, where

  n_{i,j}^* = \left\{ \begin{array}{cc} \lceil n \, p \, k_{i,j}^* \rceil & if \ k_{i,j}^* \leq 1 \\[7pt] \lceil n \, p \, {k_{i,j}^{*}}^{\alpha} \rceil & if \ k_{i,j}^* > 1 \end{array} \right.

  with

  k_{i,j}^* = \displaystyle{\frac{t_{i+1,j}^* - \, t_{max,i,j}^+}{t_{max,i,j}^{-} - \, t_{i,j}^*}} \ .

  If h_{i,j}^* < h_{i,j}, we then set t_{max,i,j}^+ equal to t_{i+1,j}^* and h_{i,j} is recalculated.

• h_{0,j}^* = \displaystyle{\frac{t_{1,j}^* - t_0}{n_{0,j}^*}} , where

  n_{0,j}^* = \left\{ \begin{array}{cc} \lceil n \, p \, k_{0,j}^* \rceil & if \ k_{0,j}^* \leq 1 \\[7pt] \lceil n \, p \, {k_{0,j}^{*}}^{\alpha} \rceil & if \ k_{0,j}^* > 1 \end{array} \right.

  with

  k_{0,j}^* = \displaystyle{\frac{t_{1,j}^* - \, t_0}{t_{max,1,j}^{-} - \, t_{1,j}^*}} \ ,

  when the sfptl object is of length 1 and from.t0 = TRUE, or the sfptl object is of length greater than 1.

  If h_{0,j}^* < h_{1,j}, we then set t_{1,j}^* equal to t_0 and h_{1,j} is recalculated.

• h_{m_j,j}^* = \displaystyle{\frac{T - \, t_{max,m_j,j}^+}{n_{m_j,j}^*}} , where

  n_{m_j,j}^* = \left\{ \begin{array}{cc} \lceil n \, p \, k_{m_j,j}^* \rceil & if \ k_{m_j,j}^* \leq 1 \\[7pt] \lceil n \, p \, {k_{m_j,j}^{*}}^{\alpha} \rceil & if \ k_{m_j,j}^* > 1 \end{array} \right.

  with

  k_{m_j,j}^* = \displaystyle{\frac{T - \, t_{max,m_j,j}^+}{t_{max,m_j,j}^{-} - \, t_{m_j,j}^*}} \ ,

  when the sfptl object is of length 1 and to.T = TRUE, or the sfptl object is of length greater than 1.

  If h_{m_j,j}^* < h_{m_j,j}, we then set t_{max,m_j,j}^+ equal to T and h_{m_j,j} is recalculated.
  

p \geq 0.1 and 0.75 \leq \alpha \leq 1 are recommended; otherwise, some integration steps can be excessively large.

If the sfptl object is of length 1 (conditioned f.p.t. problem), the suitable subintervals and integration steps that the function provides are:

• If variableStep = TRUE,

  • h_{i,1} in subintervals [t_{i,1}^*, t_{max,i,1}^+], i=1, \ldots, m_1.

  • h_{i,1}^* in subintervals [t_{max,i,1}^+, t_{i+1,1}^*], i=1, \ldots, m_1-1. In these subintervals is possible to avoid applying the numerical algorithm to approximate the f.p.t. density provided that the value of the approximate density at the time instant t_{max,i,1}^+ is almost 0.

  • h_{0,1}^* in subinterval [t_0, t_{1,1}^*], if from.t0 = TRUE.

  • h_{m_1,1}^* in subinterval [t_{max,m_1,1}^{+}, T], if to.T = TRUE.
    

• If variableStep = FALSE the function computes

  h = min \left\{ h_{i,1} \ , \ i=1, \ldots, m_1 \right\} \thinspace .

  Then

  • If from.t0 = FALSE and to.T = FALSE, h is readjusted to exactly split the interval [t_{1,1}^*, t_{max,m_1,1}^+].

  • If from.t0 = TRUE and to.T = FALSE, h is readjusted to exactly split the interval [t_0, t_{max,m_1,1}^+].

  • If from.t0 = FALSE and to.T = TRUE, h is readjusted to exactly split the interval [t_{1,1}^*, T].

  • If from.t0 = TRUE and to.T = TRUE, h is readjusted to exactly split the interval [t_0, T].

  h is a suitable fixed integration step in subintervals [t_{i,1}^*, t_{max,i,1}^+], i=1, \ldots, m_1, and [t_{max,i,1}^+, t_{i+1,1}^*], i=1, \ldots, m_1-1; in subintervals [t_0, t_{1,1}^*] if from.t0 = TRUE, and in [t_{max,m_1,1}^{+}, T] if to.T = TRUE. The endpoints of such subintervals are readjusted according to this integration step.
  

If the sfptl object is a list of length greater than 1 (unconditioned f.p.t problem), a common partition of the interval [t_0, T] is calculated from the suitable partitions of this interval for each fixed value of the initial distribution.

Let, in unified form, H_{r,j}, r=1, \ldots, 2m_j+1, the suitable integration steps (calculated for each j in similar manner to the case of the sfptl object is of length 1) in subintervals I_{r,j} = [t_{r-1,j}, t_{r,j}], with t_{0,j}=t_0 and t_{2m_j+1,j} = T, j=1, \ldots, N. Then, the ordered values of all time instants in the suitable partitions, t_{(1)}, \ldots, t_{(M)}, provide a common suitable partition of the interval [t_0, T] in subintervals [t_{(i-1)}, t_{(i)}], i=1, \ldots, M \negthinspace + \negthinspace 1, where t_{(0)} = t_0 and t_{(M \negthinspace + \negthinspace 1)} = T.

For this partition, the function computes

• H_{i} = min \, \{ H_{r,j} : j=1, \ldots, N, \ and \ [t_{(i-1)}, t_{(i)}] \subseteq I_{r,j} \} , \ i=2, \ldots, M.

• H_{1} = min \, \{ H_{r,j} : j=1, \ldots, N, \ and \ [t_0, t_{(1)}] \subseteq I_{r,j} \}, if from.t0 = TRUE.

• H_{M \negthinspace + \negthinspace 1} = min \, \{ H_{r,j} : j=1, \ldots, N, \ and \ [t_{(M)}, T] \subseteq I_{r,j} \}, if to.T = TRUE.
  

Thus,

• If variableStep = TRUE, the suitable subintervals and integrations steps that the function provides are

  • H_i in subintervals [t_{(i-1)}, t_{(i)}], i=2, \ldots, M.

  • H_1 in subinterval [t_0, t_{(1)}], if from.t0 = TRUE.

  • H_{M \negthinspace + \negthinspace 1} in subinterval [t_{(M)}, T], if to.T = TRUE.

  Each integration step is readjusted to exactly split the corresponding subinterval.
  

• If variableStep = FALSE, a suitable fixed integration step for any subinterval [t_{(i-1)}, t_{(i)}] is

  h = min \left\{ H_{i} : i=1, \ldots, M \negthinspace + \negthinspace 1 \right\}.

  In this case it is not possible to avoid applying the approximation algorithm in [t_{(i-1)}, t_{(i)}] \ \forall \ i=1, \ldots, M \negthinspace + \negthinspace 1 .
  

  Then

  • If from.t0 = FALSE and to.T = FALSE, h is readjusted to exactly split the interval [t_{(1)}, t_{(M)}].

  • If from.t0 = TRUE and to.T = FALSE, h is readjusted to exactly split the interval [t_0, t_{(M)}].

  • If from.t0 = FALSE and to.T = TRUE, h is readjusted to exactly split the interval [t_{(1)}, T].

  • If from.t0 = TRUE and to.T = TRUE, h is readjusted to exactly split the interval [t_0, T].

  h is a suitable fixed integration step in subintervals [t_{(i-1)}, t_{(i)}], i=2, \ldots, M, in subintervals [t_0, t_{(1)}] if from.t0 = TRUE, and in [t_{(M)}, T] if to.T = TRUE. The endpoints of such subintervals are readjusted according to this integration step.
  

Value

A two-component list:

Author(s)

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

References

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

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

Approx.fpt.density to approximate f.p.t. densities from objects of class “summary.fptl” and create objects of class “fpt.density”.

summary.fptl to locate the f.p.t. variable and create objects of class “summary.fptl” from objects of class “fptl”.

FPTL to evaluate the FPTL function and create objects of class “fptl”.

Examples

## Continuing the summary.fptl(.) example:

Integration.Steps(yy)
Integration.Steps(yy, from.t0 = TRUE)
Integration.Steps(yy, to.T = TRUE, n = 100, p = 0.25)

Integration.Steps(zz)

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 class 'diffproc'
print(x, ...)

Arguments

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.

Value

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

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

Note

The transition probability density functions specified by the dnorm function should be expressed in terms of the standard normal distribution, since only its derivative is considered in R.

Author(s)

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

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)

Studying the Growth of a Vector

Description

For the vector of values resulting from the evaluation of a function, this function determines the positions of the values between which the vector grows.

Usage

growth.intervals(x, y, zeroSlope = 0.01)

Arguments

Details

The growth.intervals function ignores the pairs of values between which the vector y starts and stops growing if the slope is less than zeroSlope.

Value

This function returns NULL if the vector y is not growing. Otherwise, the function computes a matrix of indexes with two columns. The first column contains the positions of the values from which the vector y starts growing and the second column those at which the vector y stops growing.

Author(s)

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

Examples

u <- seq(0, 5, length = 200)
v <- sin(u)
w <- growth.intervals(u, v)
w

plot(u, v, type = "l", las = 1)
abline(v = u[as.vector(w)])

## Continuing the FPTL(.) examples:

growth.intervals(y$x, y$y)
growth.intervals(y$x, y$y, zeroSlope = 0.001)

Testing for objects of class “fpt.density”

Description

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

Usage

is.fpt.density(obj)

Arguments

Value

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

An object of class “fpt.density” is a three-component list:

It also includes six additional attributes. For more details, see the values of Approx.cfpt.density and Approx.fpt.density functions.

Author(s)

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

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.

Examples

## Testing fpt.density objects

## Continuing the Approx.cfpt.density example:
is.fpt.density(yyy)

## Continuing the Approx.fpt.density example:
is.fpt.density(yyy.cp)
## Not run: 
is.fpt.density(yyy.ucp)
## End(Not run)

Plotting Method for fpt.density Objects

Description

This function creates a plot of the approximate density function for the conditioned or unconditioned f.p.t. problem at hand. Optionally, it displays the information provided by the First-Passage-Time Location (FPTL) function for a conditioned problem. Moreover, for an unconditioned problem creates an additional plot with all the approximate f.p.t. densities conditioned to each value x_0 selected from the initial ditribution.

Usage

## S3 method for class 'fpt.density'
plot(x, from.t0, to.T, dp.legend = TRUE, dp.legend.cex = 1,  
     ylab = TRUE, growth.points = FALSE, instants = FALSE, ...)

Arguments

Details

If from.t0 or to.T arguments are missing the function considers the corresponding arguments used in the call to Approx.cfpt.density or Approx.fpt.density functions, which in turn generated the x object.

If the approximate density functions were calculated from the lower end of the interval considered and from.t0 = FALSE, the approximate density functions should be plotted from the lower endlimit of the first subinterval in the attribute Steps of the x object.

If the approximate density functions were calculated to the upper end of the interval considered and to.T = FALSE, the approximate density functions should be plotted to the upper endlimit of last subinterval in the attribute Steps of the x object.

If dp.legend = TRUE, a legend is placed in the top inside of each plot frame.

Additional graphical arguments as cex, lwd and ps can be specified.

Author(s)

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

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

FPTL to know more about the FPTL function and objects of class “fptl”.

summary.fptl to extract the information contained in objects of class “fptl” and create objects of class “summary.fptl”.

Approx.cfpt.density to approximate the f.p.t. density from objects of class “summary.fptl” and create objects of class “fpt.density”.

Approx.fpt.density to approximate the f.p.t. density from objects of class “dp” and create objects of class “fpt.density”.

Examples

## Continuing the Approx.cfpt.density(.) example:
plot(yyy)
plot(yyy, from.t0 = TRUE)
plot(yyy, growth.points = TRUE)
plot(yyy, growth.points = TRUE, instants = TRUE)
plot(yyy, dp.legend = FALSE, growth.points = TRUE, instants = TRUE)
plot(yyy, cex = 1.25, cex.main = 1.15)
plot(yyy, cex = 1.25, cex.main = 1.15, dp.legend.cex = 0.8, growth.points = TRUE, instants = TRUE)

## Continuing the Approx.fpt.density example:
plot(yyy.cp)
## Not run: 
plot(yyy.ucp)
## End(Not run)

Plotting Method for fptl Objects

Description

This function creates a plot of the First-Passage-Time Location (FPTL) function for a first-passage-time problem, displaying the information of interest contained in an object of class “fptl” and a corresponding object of class “summary.fptl”.

Usage

## S3 method for class 'fptl'
plot(x, sfptl, from.t0 = TRUE, to.T = TRUE, dp.legend = TRUE,  
     dp.legend.cex = 1, ylab = TRUE, growth.points = TRUE, 
     instants = TRUE, ...)

Arguments

Details

If the sfptl object is missing, the function makes an internal call to the summary.fptl function in order to identify the points of interest provided by the FPTL function.

If the FPTL function shows at least a local maximum and from.t0 = FALSE, the FPTL function should be plotted from the first time instant from which the function starts growing.

If the FPTL function shows at least a local maximum and to.T = FALSE, the FPTL function should be plotted to the t_{max,m}^{+} value related to the last local maximum t_{max,m} if the function does not decrease subsequently, or to the local minimum following the last local maximum t_{max,m} if the function decreases subsequently.

If dp.legend = TRUE, a legend is placed in the top inside of the plot frame.

Additional graphical arguments as cex, lwd and ps can be specified.

Author(s)

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

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.

See Also

FPTL to know more about the FPTL function and objects of class “fptl”.

summary.fptl for extract the information of interest in an object of class “fptl”.

Examples

## Continuing the FPTL(.) example:

plot(y)
plot(y, cex.main = 1.4, growth.points = FALSE)
plot(y, cex.main = 1.4, growth.points = FALSE, instants = FALSE)
plot(y, cex.main = 1.4, dp.legend = FALSE, growth.points = FALSE, instants = FALSE)
plot(y, cex = 1.25, cex.main = 1.25)
plot(y, cex = 1.25, cex.main = 1.25, dp.legend.cex = 0.8)

plot(z)
plot(z, from.t0 = FALSE)
plot(z, to.T = FALSE)
plot(z, from.t0 = FALSE, to.T = FALSE)

Printing First-Passage-Time Densities

Description

print.fpt.density shows an object of class “fpt.density”.

Usage

## S3 method for class 'fpt.density'
print(x, ...)

Arguments

Value

Since the length of components of an object of class “fpt.density” is usually large, the print.fpt.density function does not display such object as a list, but in its ‘basic’ structure instead. However, each component can be displayed separately in the usual way.

Author(s)

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

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.

Examples

## Continuing the Approx.cfpt.density example:
yyy
print(yyy, digits=10)

## Continuing the Approx.fpt.density example:
yyy.cp
## Not run: 
yyy.ucp
## End(Not run)

Writing a Report of an Object

Description

report is a function used to generate a report of the results of various statistical methods. The function invokes particular ‘methods’ which depend on the ‘class’ of the first argument.

Usage

report(obj, ...)

Arguments

Details

The functions report.summary.fptl and report.fpt.density are examples of particular methods which report objects of class “summary.fptl” and “fpt.density”.

Value

The form of the value returned by report depends on the class of its argument.

Author(s)

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

See Also

report.summary.fptl

report.fpt.density

Writing a Report of a fpt.density Object

Description

A report is generated with the information contained in an object of class “fpt.density”.

Usage

## S3 method for class 'fpt.density'
report(obj, report.sfptl = FALSE, tex = FALSE, digits = 8, ...)

Arguments

Details

The resulting report can be recicled by copy-and-pasting into a document (if tex = FALSE) or directly included into a LaTeX file (if tex = TRUE).

Value

Send the report to the command windows.

Author(s)

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

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

Approx.cfpt.density to approximate the f.p.t. density from objects of class “summary.fptl” and to create objects of class “fpt.density”.

Approx.fpt.density to approximate the f.p.t. density from objects of class “dp” and to create objects of class “fpt.density”.

is.fpt.density to test whether an object is an object of class “fpt.density”.

Examples

## Continuing the Approx.cfpt.density(.) example:
report(yyy, digits = 4)
report(yyy, report.sfptl = TRUE, digits = 4)
report(yyy, tex = TRUE, digits = 4)
report(yyy, report.sfptl = TRUE, tex = TRUE, digits = 4)

## Continuing the Approx.fpt.density(.) example:
report(yyy.cp)
## Not run: 
report(yyy.ucp)
## End(Not run)

Writing a Report of a summary.fptl Object

Description

A report is generated with the information contained in an object of class “summary.fptl”.

Usage

## S3 method for class 'summary.fptl'
report(obj, tex = FALSE, digits = 8, heading = TRUE, ...)

Arguments

Details

The resulting report can be recicled by copy-and-pasting into a document (if tex = FALSE) or directly included into a LaTeX file (if tex = TRUE).

Value

Send the report to the command windows.

Author(s)

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

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

summary.fptl to create objects of class “summary.fptl” from objects of class “fptl”.

Approx.fpt.density to create objects of class “fpt.density” with the attribute summary.fptl of class “summary.fptl”

is.summary.fptl to test whether an object is of class “summary.fptl”.

Examples

## Continuing the summary.fptl(.) example:

report(yy, digits = 4)
report(yy, tex = TRUE, digits = 4)

report(zz)
report(zz, tex = TRUE)

Locating a Conditioned First-Passage-Time Variable

Description

summary.fptl summary method for class “fptl”.

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

print.summary.fptl shows an object of class “summary.fptl”.

Usage

## S3 method for class 'fptl'
summary(object, zeroSlope = 0.01, p0.tol = 8, k = 3, ...)

is.summary.fptl(obj)

## S3 method for class 'summary.fptl'
print(x, ...)

Arguments

Details

The summary.fptl function extracts the information contained in object about the location of the variation range of a conditioned first-passage-time (f.p.t.) variable.

It makes an internal call to growth.intervals function in order to determine the time instants t_i, \ i=1, \ldots, m, from which the First-Passage-Time Location (FPTL) function starts growing, and its local maximums t_{max,i}. For this, zeroSlope argument is considered.

If there is no growth subinterval, the execution of the function summary.fptl is stopped and an error is reported. Otherwise, for each of the subintervals I_{i} = [t_{i},t_{i+1}] the function determines:

• The first time instant t_i^* \in [t_i, t_{max,i}] at which the function is bigger than or equal to

  p_i^* = p_i + 10^{-p0.tol}(p_{max,i} - p_i) \ ,

  where p_i = FPTL(t_i) and p_{max,i} = FPTL(t_{max,i}) \ .

10^{-p0.tol} is the ratio of the global increase of the function in the growth subinterval [t_i, t_{max,i}] that should be reached to consider that it begins to increase significantly.

• The first time instant t_{max,i}^{-} \in [t_i, t_{max,i}] at which the FPTL function is bigger than or equal to

  p_{max,i}^{-} = p_{max,i} \big( 1 - 0.05(p_{max,i} - p_i) \big) \ .

• The last time instant t_{max,i}^{+} \in \big[ t_{max,i}, \thinspace T_i \big] at which the FPTL function is bigger than or equal to

  p_{max,i}^{+} = max \left\{ 1 - (1 - p_{max,i}^2)^{(1+q)/2}, FPTL(T_i) \right\},

  where

  T_i = min \big\{ t_{max,i} + k \thinspace (t_{max,i}-t_i^*)(1 - p_{max,i}), \thinspace t_{i+1} \big\}

  and

  q = \displaystyle{\frac{p_{max,i} - p_i}{p_{max,i}}} \ .

print.summary.fptl displays an object of class “summary.fptl” for immediate understanding of the information it contains.

Value

The summary.fptl function computes and returns an object of class “summary.fptl” and length 1.

An object of class “summary.fptl” is a list of length 1 for a conditioned f.p.t problem, or of the same length as the number of values selected from the non-degenerate initial distribution for an unconditioned f.p.t problem. Each component of the list is again a named list with two components:

It also includes four additional attributes:

For an unconditioned f.p.t problem, the object includes the additional attribute id specifying the non-degenerate initial distribution.

The attribute “summary.fptl” of the value (of class “fpt.density”) of the Approx.fpt.density function is an object of class summary.fptl of length 1 for a conditioned problem, and of length greather than 1 for an unconditioned problem. It is created from one or successive internal calls to the summary.fptl function.

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

Author(s)

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

References

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

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

Approx.cfpt.density to approximate densities of f.p.t. variables conditioned to a fixed initial value from objects of class “summary.fptl” and create objects of class “fpt.density”.

Approx.fpt.density to approximate densities of conditioned or unconditioned f.p.t. variables and create objects of class “fpt.density” from objects of class “dp”.

FPTL to evaluate the FPTL function and create objects of class “fptl”.

report.summary.fptl to generate a report.

growth.intervals to study the growth of the vector of values resulting from the evaluation of a function.

Examples

## Continuing the FPTL(.) example:

## Summarizing an object of class fptl
yy <- summary(y)
yy
print(yy, digits=10)
yy1 <- summary(y, zeroSlope = 0.001)
yy1
yy2 <- summary(y, zeroSlope = 0.001, p0.tol = 10)
yy2

zz <- summary(z)
zz

## Testing summary.fptl objects
is.summary.fptl(yy)
is.summary.fptl(zz)