First Attempt Time Algorithm (FATA)

• FATA is an unbiased algorithm for simulating first-passage times in stochastic processes by avoiding time-discretization bias through candidate generation and Girsanov correction.
• It adapts to both Itô diffusions and non-Markovian systems like fractional Brownian motion using techniques such as adaptive bisection to control error and efficiency.
• The method is applicable in fields such as computational neuroscience, offering significant improvements in computational efficiency and precision compared to traditional simulation methods.

The First Attempt Time Algorithm (FATA) is an acceptance–rejection approach for unbiased simulation of the first-passage time (FPT) in stochastic processes, particularly for Itô diffusions and fractional Brownian motion, to constant or time-dependent boundaries. FATA eliminates the bias associated with time-discretization by simulating candidate first-passage events under a simpler reference process (standard Brownian motion) and correcting via a likelihood ratio derived from Girsanov's theorem. Extensions exist for non-Markovian systems such as fractional Brownian motion, employing adaptive bisection and conditional sampling to control error and computational cost. FATA is exact, flexible, and particularly well-suited when high numerical precision is required, as in computational neuroscience or fine-grained stochastic modeling (Khurana et al., 2024, Walter et al., 2019).

1. Problem Formulation and Mathematical Framework

FATA addresses the exact simulation of the first-passage time

$\tau = \inf \{ t \geq 0 : X_t = L(t) \}$

for a one-dimensional diffusion $X$ governed by the Itô SDE

$dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$

to a (possibly time-dependent) boundary $L : [0,\infty) \to \mathbb{R}$. For many systems, discretizing either the path $X$ or the boundary $L$ introduces bias in the estimate of $\tau$. FATA circumvents this by transforming the simulation problem into an acceptance–rejection process over candidate FPTs drawn from the law of a reference process (most often standard Brownian motion) to the target boundary (Khurana et al., 2024).

The algorithm can also be adapted to fractional Brownian motion (fBm) and general Gaussian processes, where the Markov property no longer holds. In these systems, adaptive refinement of the simulation mesh is used to retain accuracy and efficiency (Walter et al., 2019).

2. Core Algorithmic Structure

FATA for Itô diffusions combines candidate generation and correction:

1. Candidate generation: Sample a candidate $\tau_B$ from the known first-passage distribution of a standard Brownian motion $W_t$ to $L(t)$. For constant and linear boundaries, closed forms exist, e.g., for $X$0,

$X$1

For general $X$2, iterative schemes such as the Herrmann–Tanré approach are employed (Khurana et al., 2024).

2. Girsanov correction: Accept $X$3 as the FPT for the original drifted process with probability

$X$4

where $X$5 denotes the Wiener measure and \begin{align*} \gamma_1(s) &= -A'(L(s)) L'(s),\ \gamma_2(x) &= \frac{1}{2}[\alpha'(x)+\alpha^2(x)], \end{align*} with $X$6. $X$7 is a Brownian bridge conditioned to start at $X$8 and hit $X$9 at time $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$0 (Khurana et al., 2024).

For fBm and other Gaussian processes, FATA (also termed Adaptive Bisection Sectioning, "ABSec") uses a two-phase procedure: initialize on a coarse grid via the Davies–Harte algorithm, truncate beyond the first crossing, and recursively bisect "critical" intervals by sampling conditional midpoints. Refinement continues until the probability of missing a crossing on any interval is bounded by a user-specified $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$1 (Walter et al., 2019).

3. Extensions to Time-Dependent and Nonlinear Boundaries

FATA naturally handles time-dependent boundaries by updating the candidate generation step to sample from the FPT of a Brownian motion to a moving boundary $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$2. For boundaries that are not piecewise linear or constant, iterative inverse-Gaussian schemes are applied. The acceptance probability and related Girsanov correction terms are recomputed, with the only requirement being that $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$3 and its derivative $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$4 are available for computing $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$5 (Khurana et al., 2024).

For fBm, the approach adapts to the loss of the Markov property by replacing candidate generation and acceptance steps with adaptive mesh refinement. Bridges that could conceal a crossing are recursively bisected, and at each bisection, conditional distributions are updated and crossing probabilities are re-evaluated. Error per bridge is at most $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$6, and overall false-miss probability $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$7 can be set arbitrarily small with complexity increasing only logarithmically in $dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$8 (Walter et al., 2019).

4. Computational Complexity and Efficiency

For Itô diffusions, the expected number of FATA candidate draws needed for acceptance is

$dX_t = \alpha(X_t)\,dt + dW_t, \quad X_0 = x_0,$9

and is exponentially sensitive to the worst-case value $L : [0,\infty) \to \mathbb{R}$0. An exponential upper bound follows: $L : [0,\infty) \to \mathbb{R}$1 To reduce cost when $L : [0,\infty) \to \mathbb{R}$2 is large, techniques such as shifting $L : [0,\infty) \to \mathbb{R}$3 by their mininum or "space splitting"—dividing the boundary into $L : [0,\infty) \to \mathbb{R}$4 sublevels—can bring costs to $L : [0,\infty) \to \mathbb{R}$5 per run (Khurana et al., 2024).

For fBm, FATA attains an effective computational complexity of $L : [0,\infty) \to \mathbb{R}$6 for time resolution $L : [0,\infty) \to \mathbb{R}$7, compared to $L : [0,\infty) \to \mathbb{R}$8 for the Davies–Harte method on a uniform grid. Empirically, for $L : [0,\infty) \to \mathbb{R}$9 and $X$0, FATA is $X$1 faster and $X$2 more memory efficient. With $X$3 and $X$4, the speedup is $X$5 and memory cost $X$6 times smaller. The complexity increases only logarithmically with decreasing target global error $X$7 (Walter et al., 2019).

Algorithm	Time Complexity	Memory Complexity	Bias Control
FATA (diffusion)	Random (potentially $X$8)	Problem-dependent	No bias (exact law)
FATA (fBm/ABSec)	$X$9	$L$0	Tunable, bounded error
Davies–Harte	$L$1	$L$2	Discretization bias

5. Numerical Results and Empirical Behavior

Representative numerical experiments for FATA have been reported for diverse SDEs and boundary types:

• Sinusoidal drift SDE with linear boundary: FATA produces unbiased FPT samples. Euler–Maruyama and improved EM methods exhibit $L$3 or $L$4 bias that persists unless exceedingly fine discretization is used; FATA achieves exactness regardless of $L$5 (Khurana et al., 2024).
• SDE with exponential boundary: When explicit sampling is unavailable, iterative inverse-Gaussian schemes allow FATA to converge to the true density as tolerance $L$6, with run-time scaling moderately as $L$7 decreases (Khurana et al., 2024).
• Neuronal spiking models: For a quadratic leaky integrate-and-fire model with adaptive threshold $L$8, FATA produces exact spike-time samples, including under nonlinear SDEs for which EM can fail (blow-up). Reported performance is $L$9 per $\tau$0 spikes, with first and second moments matching theoretical predictions (Khurana et al., 2024).
• Fractional Brownian motion: For $\tau$1 and effective grid size $\tau$2, Davies–Harte is infeasible (beyond $\tau$3), while FATA requires $\tau$4 per sample and $\tau$5 RAM. Similar drastic improvements are reported for more subdiffusive $\tau$6 and even higher resolutions (Walter et al., 2019).

6. Limitations and Remedies

The principal limitation of FATA for drifted diffusions is the potentially large exponential cost with respect to the maximal Girsanov correction $\tau$7 and large candidate FPTs $\tau$8. Remedies include recentering correction terms or partitioning the boundary range to reduce per-step rejection rates (Khurana et al., 2024). For fBm and adaptive bisection, double-precision subtraction when updating variances leads to floating-point limitations; this manifests as practical bounds $\tau$9, with extended precision arithmetic offering a solution (Walter et al., 2019).

7. Comparison with Alternative Approaches

Conventional methods such as Euler–Maruyama, improved EM, and Davies–Harte are fundamentally limited by time-discretization bias and computational cost scaling linearly with target resolution or step size. In contrast, FATA (for both Markovian and non-Markovian systems) achieves either unbiased or provably bounded-error FPT estimates with substantially reduced computational resource requirements at fine resolutions. For applications demanding extreme precision or unbiased estimates—such as theoretical validation, rare event simulation, or computational neuroscience—FATA provides strong advantages, provided the potential for exponential algorithmic cost is managed for the specific problem instance (Khurana et al., 2024, Walter et al., 2019).