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First-Hitting-Time Model

A model that represents an event time as the first instant a latent stochastic process reaches or crosses a specified boundary, translating path dynamics into a distribution of survival and failure times.

Version
v1 · 2026-09-28 · History
Domain-specific #
9471
Domain group
Formal Sciences
Origin domain
Experimental Design & Statistics
Subdomain
Survival Analysis → Experimental Design & Statistics
Aliases
First passage time model, Threshold regression model, First-hitting-time survival model, FHT model

Core Idea

A first-hitting-time model converts trajectory dynamics into duration data. The event occurs at the first time a stochastic state reaches a defined boundary, so survival means that the entire path has remained outside the target set so far.

This mechanistic appeal carries assumptions: the latent state, dynamics, starting point, and boundary can be difficult to observe separately. Similar event-time distributions may conceal different path explanations.

Structural Signature

Sig role-phrases:

  • Latent stochastic process — Represents evolving health, reserve, stress, or position. It is state trajectory. Counterfactual: The process need not be directly observed.
  • Initial condition — Sets distance and orientation relative to failure. It is start state. Counterfactual: Different starting reserves change duration distributions.
  • Drift and variability — Govern systematic movement and random fluctuation. It is dynamics. Counterfactual: A threshold alone does not determine hitting time.
  • Boundary or target set — Defines the event-producing state. It is event rule. Counterfactual: Moving and fixed boundaries yield different models.
  • First-entry operator — Chooses the earliest qualifying crossing. It is stopping rule. Counterfactual: Later returns do not redefine the event time.
  • Observation and censoring — Connect latent paths to recorded durations. It is data interface. Counterfactual: Non-events can be right-censored rather than infinite.

What It Is Not

  • Every survival model is not a first-hitting-time model.
  • A later recrossing does not change the first passage.
  • The threshold need not be constant, but it must be specified.
  • A fitted duration distribution alone does not identify the latent mechanism.
  • Closest near-miss. A hazard model specifies instantaneous event risk conditional on survival; a first-hitting-time model derives that risk from stochastic path dynamics and a boundary.

Scope of Application

  • Reliability. Models failure as degradation reaching a limit.
  • Finance and economics. Represents default, ruin, or exit thresholds.
  • Biostatistics. Provides conceptual latent-health survival models.
  • Ecology and physics. Studies absorption and first-passage events.

Clarity

State the stochastic process, state space, initial distribution, drift and diffusion or transition law, target set, crossing convention, censoring, covariates, and identifiability assumptions. Distinguish latent paths from observed event times.

Manages Complexity

Path dependence makes a one-time event encode an entire unobserved trajectory. Boundary shape, discontinuous jumps, multiple absorbing sets, and measurement error can make analytic formulas unavailable and causal interpretation fragile.

Abstract Reasoning

  1. Define the latent state and why its first crossing corresponds to the event.
  2. Specify initial condition, stochastic dynamics, covariates, and boundary geometry.
  3. Derive or approximate the hitting-time distribution and implied hazard.
  4. Connect paths to measurement, truncation, and censoring.
  5. Test alternative dynamics, boundary choices, and ordinary survival formulations.

Knowledge Transfer

Stopping-time mathematics transfers across domains, but the latent state and boundary interpretation do not. A ruin process, degradation path, and ecological threshold can share equations while requiring different evidence and identifiability arguments.

Examples

Canonical

A firm's reserve follows a noisy process with operating drift, and failure time is modeled as the first moment reserve reaches zero; surviving firms at study end are right-censored.

Mapped back: process → reserve; boundary → zero; event → ruin; observation → right censoring.

Applied / In Practice

A customer contract ending on a fixed calendar date is a duration outcome but not a first-hitting time unless some modeled trajectory determines that date by crossing a threshold.

Mapped back: event → scheduled expiry; latent process → absent; boundary crossing → absent; verdict → not FHT.

Structural Tensions

T1 — Mechanistic Interpretation versus Latent Nonidentifiability. Different process-and-boundary combinations can produce similar duration distributions.

Diagnostic: Which path assumptions are empirically distinguishable?

T2 — Tractable Boundary versus Realistic Event Rule. Simple constant thresholds aid inference while degradation or policy limits may move over time.

Diagnostic: Is the boundary physics or convenience?

Structural–Framed Character

First-Hitting-Time Model is structural as first boundary entry by a stochastic path and framed by survival analysis. The stopping rule links process dynamics to duration.

Structural Core vs. Domain Accent

The general pattern is event generation through threshold crossing. Stochastic modeling adds path laws, stopping times, absorption, censoring, and hitting distributions.

This entry presupposes Stochastic Process.

  • Approved unparented root. No reviewed parent entails this stopping-time survival construction.

  • Related — first-passage process, hazard model, and threshold model. They share mathematical pieces but differ in what generates the event time.

Relationships to Other Abstractions

Local relationship map for First-Hitting-Time ModelParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.First-Hitting-TimeModelDOMAINPrime abstraction: Stochastic Process — presupposesStochasticProcessPRIME

Current abstraction First-Hitting-Time Model Domain-specific

Parents (1) — more general patterns this builds on

  • First-Hitting-Time Model presupposes Stochastic Process Prime

    First-Hitting-Time Model presupposes Stochastic Process because event time is defined as the first boundary crossing of a latent stochastic process.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

First-Hitting-Time Model sits in a crowded region of the domain-specific corpus (28th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Decision & System Modeling Frameworks (30 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Hazard model. Tell: Specifies conditional event rate without necessarily positing a threshold path.
  • First return time. Tell: Measures return after a state has already been visited.
  • Threshold regression. Tell: Can refer to covariate regimes without a stochastic first passage.
  • Competing-risks model. Tell: Uses multiple event causes and may or may not derive them from boundaries.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/First-hitting-time_model (revision 1339054830).
  • Preserved source candidate: http://www.numdam.org/articles/10.24033/asens.476/
  • Preserved source candidate: http://projecteuclid.org/euclid.aoms/1177706881

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.