Calculator Inputs
Pick a mode, enter your model inputs, then calculate. Results appear above this form after submission.
Example Data Table
These examples show the kinds of outputs this calculator can produce.
| Mode | Inputs | Sample output |
|---|---|---|
| Generator evaluation | x = 1.2, μ = 0.15, σ = 0.40, u(x) = 1 + 0.5x + 0.2x² | L[u](x) ≈ 0.179, so the earlier-time estimate rises. |
| Finite difference | u(x-Δx)=1.2, u(x)=1.0, u(x+Δx)=0.9, Δx=0.1, Δt=0.02 | Discrete generator ≈ 0.375 and the center updates to ≈ 1.0075. |
| Hitting probability | a = 0, b = 1, x = 0.35, μ = 0, σ = 0.40 | Upper-bound hitting probability = 0.35 under neutral drift. |
Formula Used
1) Backward Kolmogorov equation
For a one-dimensional diffusion with drift μ(x,t) and diffusion σ(x,t), the backward equation is:
∂u/∂t + μ(x,t)·uₓ + 0.5·σ²(x,t)·uₓₓ = 0
The spatial part L[u] = μuₓ + 0.5σ²uₓₓ is the infinitesimal generator. The calculator evaluates this operator directly.
2) Polynomial test function mode
This page uses the polynomial
u(x)=a0+a1x+a2x²+a3x³+a4x⁴.
Then
uₓ=a1+2a2x+3a3x²+4a4x³
and
uₓₓ=2a2+6a3x+12a4x².
The generator value is
L[u]=μuₓ+0.5σ²uₓₓ.
3) Finite-difference approximation
With three neighboring values, the calculator estimates:
uₓ ≈ (u(x+Δx)-u(x-Δx)) / (2Δx)
uₓₓ ≈ (u(x+Δx)-2u(x)+u(x-Δx)) / Δx²
A backward step from time t to t-Δt is estimated by
u(t-Δt,x) ≈ u(t,x) + Δt·L[u](t,x).
4) Hitting probability formula
For constant drift and diffusion between boundaries a and b, the upper-bound hitting probability solves the backward ODE:
μh′(x) + 0.5σ²h″(x) = 0,
with h(a)=0 and h(b)=1.
If μ = 0, then
h(x) = (x-a)/(b-a).
If μ ≠ 0, then
h(x) = [1-e^{-2μ(x-a)/σ²}] / [1-e^{-2μ(b-a)/σ²}].
How to Use This Calculator
- Choose the mode that matches your task: generator evaluation, finite-difference stepping, or boundary hitting probability.
- Enter drift and diffusion values. Keep diffusion nonnegative, and positive for hitting-probability work.
- Fill in the mode-specific inputs, such as polynomial coefficients, neighboring grid values, or boundary positions.
- Press Calculate. The result section appears above the form, directly below the page header.
- Review the result table, inspect the Plotly graph, and export the output as CSV or PDF.
FAQs
1) What is the backward Kolmogorov equation?
It is a partial differential equation that tracks how a future-dependent quantity changes when viewed from the current state. Instead of evolving a density forward, it evolves expected function values backward from a terminal condition.
2) What is the interpretation of the Kolmogorov backward equation?
It describes how the current state influences a later payoff, probability, or expected cost. The generator summarizes local drift and randomness, and the backward equation tells how those local effects shape earlier-time values.
3) Why does the calculator evaluate the generator?
The generator is the key spatial operator inside the backward equation. Once you know it, you can approximate time changes, build finite-difference schemes, and solve many expectation or hitting-probability problems.
4) What is hitting probability for a neutral process in the backward equation?
In a neutral process, the drift is zero. Then the upper-bound hitting probability becomes linear: h(x) = (x-a)/(b-a). This is the classic backward-equation solution with boundary values 0 and 1.
5) Why does drift change the hitting probability?
Drift tilts the process toward one side. Positive drift usually increases the chance of reaching the upper boundary first, while negative drift usually favors the lower boundary. The backward ODE captures that directional bias.
6) When should I use finite-difference mode?
Use it when you already have values on a spatial grid and want a practical numerical update. It is helpful for quick PDE checks, educational demonstrations, or prototype schemes before building a larger solver.
7) Does this page solve every backward equation exactly?
No. It provides structured calculations for common one-dimensional settings. Exact closed forms exist only for selected models, while general problems usually require numerical schemes, boundary conditions, and a chosen terminal payoff.
8) What assumptions should I check before trusting the result?
Confirm that your drift, diffusion, boundary setup, and payoff description match the formula being used. Also check units, sign conventions, grid spacing, and whether a one-dimensional diffusion model is appropriate for your problem.