Model hitting time for Brownian motion, random walks. Choose parameters, then compute mean and risk. Keep a history and export clean reports anytime easily.
| Model | Inputs | Key output |
|---|---|---|
| Brownian | x₀=0, a=1, μ=0.5, σ=1 | P≈1, E[τ]≈2 |
| Brownian | x₀=0, a=1, μ=-0.2, σ=1 | P≈exp(-0.4)≈0.6703, E[τ]=∞ |
| Random walk | N=10, i=4, p=0.5 | P=0.4, E[T]=24 |
| Random walk | N=10, i=4, p=0.55 | P≈0.516, E[T]≈22.2 |
In threshold-driven systems, the clock to first reach a level governs decisions. If a process starts at x₀=0 and targets a=1 with μ=0.5, the mean time is 2.0 units. With μ=0.25, the mean doubles to 4.0. This scaling makes drift estimation central when planning timelines.
For μ>0 in the continuous model, E[τ]=(a−x₀)/μ. Holding (a−x₀)=1, moving μ from 0.2 to 0.8 reduces E[τ] from 5.0 to 1.25. The change is linear in 1/μ, so small drift errors near zero produce large timing errors.
When μ>0, Var[τ]=(a−x₀)σ²/μ³. With μ=0.5 and σ=1, Var[τ]=8. If σ increases to 1.5, Var[τ] becomes 18, increasing spread by 125%. Risk-aware planning should track both mean and variance, not just the average.
When μ<0 and σ>0, reaching a higher level is not guaranteed. For x₀=0, a=1, μ=−0.2, σ=1, the hit chance is exp(−0.4)≈0.6703. If μ=−0.5 with the same σ, the chance drops to exp(−1)≈0.3679.
In the random-walk option with N=10 and i=4, a fair walk p=0.5 gives P(hit 10 first)=0.4 and E[T]=24 steps. With a modest bias p=0.55, the chance rises to about 0.516 while the expected absorption time often decreases slightly, reflecting more directed motion.
For μ>0, the plotted density is an inverse-Gaussian shape centered near m=(a−x₀)/μ, with a long right tail when σ is large or μ is small. For the random walk, the curve shows how starting position i shifts success probability from 0 to 1, supporting sensitivity checks across initial states.
It is the first time a process reaches a specified level or set. In this calculator, it is either the first time X(t) reaches a barrier a, or the first time a random walk hits 0 or N.
With nonpositive drift and positive noise, the process may never reach a higher barrier, so unconditional waiting can be arbitrarily long. The tool still reports the probability of ever hitting the level, which can be below 1 when μ<0.
When μ>0, the curve approximates the hitting-time distribution using an inverse-Gaussian density with mean m and shape λ. It highlights early arrivals, the most likely time window, and the long-tail risk of late arrivals.
Let N be the top threshold and i be your current state. Larger N increases expected absorption time. If i is near 0, reaching N first is unlikely unless p is significantly above 0.5.
The expectation and probability formulas are closed-form for both models. The plotted curve is a numerical trace computed from the analytic density (continuous case) or from repeated probability evaluation across i (discrete case).
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|---|---|---|---|
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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.