[2][3][4] Modeling the probability of financial ruin as a first passage time was an early application in the field of insurance.
[5] An interest in the mathematical properties of first-hitting-times and statistical models and methods for analysis of survival data appeared steadily between the middle and end of the 20th century.
In this example, an entity (often described as a gambler or an insurance company) has an amount of money which varies randomly with time, possibly with some drift.
The model considers the event that the amount of money reaches 0, representing bankruptcy.
First-hitting-time models can be applied to expected lifetimes, of patients or mechanical devices.
When the process reaches an adverse threshold state for the first time, the patient dies, or the device breaks down.
[11][12] One of the simplest and omnipresent stochastic systems is that of the Brownian particle in one dimension.
Given that Brownian motion is used often as a tool to understand more complex phenomena, it is important to understand the probability of a first passage time of the Brownian particle of reaching some position distant from its start location.
The probability density function (PDF) for a particle in one dimension is found by solving the one-dimensional diffusion equation.
(This equation states that the position probability density diffuses outward over time.
After a long time the cream has diffused throughout the entire drink evenly.)
The diffusion equation states that the rate of change over time in the probability of finding the particle at
It can be shown that the one-dimensional PDF is This states that the probability of finding the particle at
More specifically the Full Width at Half Maximum (FWHM) – technically, this is actually the Full Duration at Half Maximum as the independent variable is time – scales like Using the PDF one is able to derive the average of a given function,
The First Passage Time Density (FPTD) is the probability that a particle has first reached a point
is an abbreviation for cliff used in many texts as an analogy to an absorption point).
If one uses the first-order Taylor approximation, the definition of the FPTD follows): By using the diffusion equation and integrating, the explicit FPTD is The first-passage time for a Brownian particle therefore follows a Lévy distribution.
This equation states that the probability for a Brownian particle achieving a first passage at some long time (defined in the paragraph above) becomes increasingly small, but is always finite.
The first moment of the FPTD diverges (as it is a so-called heavy-tailed distribution), therefore one cannot calculate the average FPT, so instead, one can calculate the typical time, the time when the FPTD is at a maximum (
The state of the stochastic process may represent, for example, the strength of a physical system, the health of an individual, or the financial condition of a business firm.
The system, individual or firm fails or experiences some other critical endpoint when the process reaches a threshold state for the first time.
The critical event may be an adverse event (such as equipment failure, congested heart failure, or lung cancer) or a positive event (such as recovery from illness, discharge from hospital stay, child birth, or return to work after traumatic injury).
In some applications, the threshold is a set of multiple states so one considers competing first hitting times for reaching the first threshold in the set, as is the case when considering competing causes of failure in equipment or death for a patient.
Practical applications of theoretical models for first hitting times often involve regression structures.
[13] The threshold state, parameters of the process, and even time scale may depend on corresponding covariates.
Threshold regression as applied to time-to-event data has emerged since the start of this century and has grown rapidly, as described in a 2006 survey article [13] and its references.
Connections between threshold regression models derived from first hitting times and the ubiquitous Cox proportional hazards regression model [14] was investigated in.
[15] Applications of threshold regression range over many fields, including the physical and natural sciences, engineering, social sciences, economics and business, agriculture, health and medicine.
[16][17][18][19][20] In many real world applications, a first-hitting-time (FHT) model has three underlying components: (1) a parent stochastic process
It is very important to distinguish whether the sample path of the parent process is latent (i.e., unobservable) or observable, and such distinction is a characteristic of the FHT model.