FPTL | R Documentation |

`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”.

```
FPTL(dp, t0, T, x0, S, env = NULL, n = 4000)
is.fptl(obj)
## S3 method for class 'fptl'
print(x, ...)
```

`dp` |
an object of class “diffproc” defining a family of diffusion processes. |

`obj` |
an |

`x` |
an object of class “fptl”, a result of a call to this function. |

`t0, T` |
lower and upper limits of the considered time interval. Must be finite. |

`x0` |
fixed initial value of process in the time instant specified in the |

`S` |
numerical value of a constant boundary or character string with the mathematical expression of a time-dependent boundary. |

`env` |
a named list of objects of numeric or character type specifying the values of names which occur in
the mathematical expressions in objects |

`n` |
number of points at which the FPTL function is evaluated. |

`...` |
additional arguments potentially passed (currently none is considered). |

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.

The function `FPTL`

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

`x` |
a sequence of n values from |

`y` |
the corresponding values of the FPTL function for the |

It also includes three additional attributes:

`Call` | the unevaluated function call, substituting each name in this call for its value when |

the latter has length 1. | |

`dp` | the object used as `dp` argument in the function call. |

`vars` | NULL or a list containing the values of names in the function call for those names |

with values of length greater than 1. | |

`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.

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

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.

`diffproc`

about creation of class “diffproc” objects.

`summary.fptl`

for summaries and `plot.fptl`

for graphical display.

```
## 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)
```

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