Compute Ito sums from sampled Brownian grids. Check left point terms, variation, and exportable summaries. Build intuition using formulas, tables, examples, and practical steps.
Use the left point Ito sum with a sampled path. The integrand model is X_t = alpha + beta*t + gamma*W_t + delta*t*W_t.
This example uses alpha = 0, beta = 0, gamma = 1, delta = 0. Then X_t = W_t.
| Step | t_i | W_i | t_{i+1} | W_{i+1} | Delta W | X(t_i,W_i) | Term |
|---|---|---|---|---|---|---|---|
| 1 | 0.00 | 0.00 | 0.25 | 0.20 | 0.20 | 0.00 | 0.0000 |
| 2 | 0.25 | 0.20 | 0.50 | -0.10 | -0.30 | 0.20 | -0.0600 |
| 3 | 0.50 | -0.10 | 0.75 | 0.35 | 0.45 | -0.10 | -0.0450 |
| 4 | 0.75 | 0.35 | 1.00 | 0.10 | -0.25 | 0.35 | -0.0875 |
| Total Ito estimate | -0.1925 | ||||||
The calculator uses the left point Ito approximation:
Integral ≈ Σ X(t_i, W_i) × (W_{i+1} - W_i)
with
X_t = alpha + beta*t + gamma*W_t + delta*t*W_t
It also reports:
Delta W_i = W_{i+1} - W_i
Quadratic variation ≈ Σ (Delta W_i)^2
Ito isometry proxy ≈ Σ X_i^2 × Delta t_i
For the special case X_t = W_t, the exact identity is:
Integral W_t dW_t = 1/2 × (W(T)^2 - W(0)^2 - (T - 0))
This Ito integral calculator estimates stochastic integrals from sampled path data. It is designed for Brownian style paths on a discrete grid. The tool applies the left point rule. That rule is central in stochastic calculus. It separates Ito integration from ordinary Riemann or Stratonovich style sums.
The calculator supports a practical integrand model. You can use alpha, beta, gamma, and delta to define X_t = alpha + beta*t + gamma*W_t + delta*t*W_t. This covers constants, time trends, path dependence, and a mixed interaction term. It is useful for teaching, checking notes, and testing small examples.
In Ito calculus, the integrand is evaluated at the left endpoint. That choice changes the value of the stochastic integral. It also creates the correction terms seen in Ito formula. This page makes that rule visible. Each interval shows the sampled integrand, the Brownian increment, and the contribution to the total sum.
The result area reports the Ito integral estimate, the quadratic variation estimate, and an Ito isometry proxy. These outputs help you judge path roughness and scale. For common special cases, the page also shows an exact identity check. That is useful when you set X_t = 1 or X_t = W_t.
Use this tool in stochastic processes, mathematical finance, diffusion modeling, and classroom work. It is especially helpful when you already have a sampled path from notes, simulation, or data export. You can quickly test how the integral changes when the path or the coefficients change.
This calculator gives a discrete approximation from the grid you provide. A finer grid often gives a better estimate. Still, the value depends on the sampled path. That is normal in stochastic integration. Use the interval table, example table, and export options to document each experiment clearly.
It computes a discrete Ito integral estimate from a sampled time grid and sampled Brownian path values. It also reports interval terms, quadratic variation, and an isometry style proxy.
Left endpoints define the Ito integral in discrete form. That choice reflects nonanticipation. The integrand uses only current information, not future path values.
Yes. You can separate numbers with commas, spaces, semicolons, or line breaks. The time list and path list must contain the same number of values.
The page stops the calculation and shows an error. A valid Ito sum needs a strictly increasing time grid so each interval has a positive length.
It is the sum of squared path increments. For Brownian motion on a fine grid, this value often tracks the time horizon. It helps check whether the sampled path behaves as expected.
It is the discrete sum of X_i squared times Delta t_i. It is often compared with the integral variance structure predicted by Ito isometry.
No. It gives a grid based approximation. Exact values appear only in special identities, such as X_t = 1 or X_t = W_t, and only when the path data fits that setup.
Yes. You can download a CSV file for data work or a PDF file for sharing. Both exports include the summary and interval breakdown.
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.