First-hitting-time model

First-Hitting-Time Model

The first-hitting-time model is a concept used in stochastic processes and probability theory to describe the time it takes for a stochastic process to reach a certain state for the first time. This model is particularly useful in various fields such as finance, biology, and medicine, where it can be used to model the time until an event of interest occurs.

Definition

In mathematical terms, the first-hitting-time is defined for a stochastic process \( \{X(t), t \geq 0\} \) with state space \( S \). The first-hitting-time \( T_A \) to a set \( A \subseteq S \) is defined as:

\[ T_A = \inf \{ t \geq 0 : X(t) \in A \} \]

where \( \inf \) denotes the infimum, or greatest lower bound, and \( X(t) \) is the state of the process at time \( t \).

Applications

Medicine

In medicine, the first-hitting-time model can be used to predict the time until a patient reaches a critical health state, such as the onset of a disease or the occurrence of a medical event like a heart attack. This can help in planning interventions and treatments.

Finance

In finance, the first-hitting-time model is used to determine the time until a financial asset reaches a certain price level, which can be crucial for option pricing and risk management.

Biology

In biology, this model can be applied to understand the time until a population reaches a certain size or until a species becomes extinct.

Mathematical Properties

The distribution of the first-hitting-time can often be difficult to determine analytically, but it can be characterized using various techniques such as:

• Laplace transforms
• Markov processes
• Brownian motion

For example, if \( X(t) \) is a Brownian motion with drift, the first-hitting-time to a level \( a \) can be found using the reflection principle and is known to have an inverse Gaussian distribution.

Related Concepts

• First-passage-time: Similar to the first-hitting-time, but specifically refers to the time it takes for a process to reach a certain state for the first time, starting from a given initial state.
• Absorption time: In the context of absorbing Markov chains, the time until the process is absorbed in an absorbing state.

See Also

• Stochastic process
• Markov chain
• Brownian motion
• Probability theory

References

• Karlin, S., & Taylor, H. M. (1975). A First Course in Stochastic Processes. Academic Press.
• Ross, S. M. (1996). Stochastic Processes. Wiley.