Numerically Testing Ito's Formula for Geometric Brownian Motion

What the Numerical Test Checks

Geometric Brownian motion gives a direct computational test of Ito's formula: its solution is explicit, while its differential contains the correction that distinguishes stochastic calculus from ordinary calculus. On one simulated Brownian path, we can compute the exact process values at grid points and compare their changes with left-point sums for the drift and diffusion terms. As the grid is refined, the residual in that differential identity should shrink.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\), adapted to the filtration. Fix \(S_0>0\), a drift coefficient \(\mu\in\mathbb{R}\), a diffusion coefficient \(\sigma\in\mathbb{R}\), and a finite horizon \(T>0\). The drift describes the \(dt\) term; the diffusion coefficient multiplies the Brownian \(dB_t\) term. The model is

For a stochastic integral \(\int_0^t H_s\,dB_s\), the integrand is understood to be adapted and progressively measurable, with \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\) when a square-integrable Ito integral is used. In this model, the integrand is \(H_s=\sigma S_s\); the explicit solution below has finite second moments on every finite time interval, so it meets this condition. The Ito multiplication rules are \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\).

The Exponential Solution and Its Correction

Proof. Define \(f(t,x)=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma x)\) and set \(S_t=f(t,B_t)\). Its derivatives are \(f_t=(\mu-\tfrac12\sigma^2)f\), \(f_x=\sigma f\), and \(f_{xx}=\sigma^2 f\). Ito's formula, including its second-derivative correction, gives

Also \(B_0=0\), so \(S_{t=0}=S_0\). This verifies the stated solution directly. The coefficient \(-\tfrac12\sigma^2\) in the exponent cancels the \(\tfrac12 f_{xx}\) contribution to leave drift \(\mu S_t\). Without it, the drift would instead be \((\mu+\tfrac12\sigma^2)S_t\). The coefficients are globally Lipschitz in \(S\) and have linear growth: \(|\mu x|+|\sigma x|\le(|\mu|+|\sigma|)(1+|x|)\). Thus the standard existence-and-uniqueness theorem for Lipschitz SDEs applies; the explicit verification also supplies a solution. \(\square\)

For comparison with numerical output, the solution has \(\mathbb{E}[S_t]=S_0e^{\mu t}\). Indeed, the normal moment-generating formula gives \(\mathbb{E}[e^{\sigma B_t}]=e^{\sigma^2t/2}\), and multiplication by the deterministic exponential in the solution gives this mean. This is an independent check on the drift correction; it does not mean that a single simulated path follows its expectation.

Grid Simulation and the Residual

Take a uniform grid \(t_i=ih\), \(i=0,\ldots,n\), with \(h=T/n\). To simulate Brownian motion on this grid, draw independent standard normal variables \(Z_i\) and set \(\Delta B_i=\sqrt{h}Z_i\), \(B_{t_i}=\sum_{j=1}^{i}\Delta B_j\). For a fixed set of simulated increments, calculate exact process values from

Then form the left-point approximation to the integrated Ito differential. Its drift sum is \(\sum_{i=0}^{n-1}\mu S_{t_i}h\), and its diffusion sum is \(\sum_{i=0}^{n-1}\sigma S_{t_i}\Delta B_i\). Define the terminal residual by subtracting both sums from the exact change:

This is a test of the integral equation, not a claim that each finite-grid increment equals its first-order approximation. The exponential solution supplies exact grid values, while the sums approximate the drift and Ito integrals. An Euler update, \(S_{t_{i+1}}\approx S_{t_i}+\mu S_{t_i}h+\sigma S_{t_i}\Delta B_i\), is a different operation: it constructs an approximate process rather than evaluating the exact solution at the grid points.

Why the Residual Shrinks

Proof. The integrated Ito equation gives \(S_T-S_0=\int_0^T\mu S_s\,ds+\int_0^T\sigma S_s\,dB_s\). Consequently \(R_n\) is the sum of the error in approximating the ordinary integral by its left-point sum and the error in approximating the Ito integral by its left-point sum. Write \(\eta(s)=t_i\) when \(t_i\le s<t_{i+1}\). The drift error is \(\mu\int_0^T(S_s-S_{\eta(s)})\,ds\); by Cauchy–Schwarz its squared expectation is at most \(\mu^2T\int_0^T\mathbb{E}|S_s-S_{\eta(s)}|^2\,ds\). The Ito isometry gives the diffusion error's squared expectation as \(\sigma^2\int_0^T\mathbb{E}|S_s-S_{\eta(s)}|^2\,ds\).

To bound the common integral, let \(u=\eta(s)\) and \(\delta=s-u\), so \(0\le\delta\le h\). The explicit solution yields \(S_s=S_uY\), where \(Y=\exp((\mu-\tfrac12\sigma^2)\delta+\sigma(B_s-B_u))\). The increment \(B_s-B_u\) is independent of \(\mathcal{F}_u\), and normal moment calculations give \(\mathbb{E}[Y]=e^{\mu\delta}\) and \(\mathbb{E}[Y^2]=e^{(2\mu+\sigma^2)\delta}\). Also \(\mathbb{E}[S_u^2]=S_0^2e^{(2\mu+\sigma^2)u}\). Therefore

The right side is a continuous function of \((u,\delta)\) on the compact set \(0\le u\le T,\ 0\le\delta\le h\), and its bracket is zero at \(\delta=0\). More quantitatively, the mean value theorem applied to the exponentials bounds the absolute value of the bracket by \(C\delta\) for \(0\le\delta\le T\), for a finite constant \(C\) depending only on \(\mu,\sigma,T\). The prefactor is bounded for \(u\in[0,T]\). Hence \(\mathbb{E}|S_s-S_{\eta(s)}|^2\le C_1h\) for a finite constant \(C_1\), and its integral over \([0,T]\) tends to zero. Both errors thus tend to zero in \(L^2\); the squared norm of their sum is bounded by twice the sum of their squared norms. This proves \(R_n\to0\) in \(L^2(\mathbb{P})\). \(\square\)

The stochastic sums here are Ito integrals of left-step adapted integrands. The integrand \(\sigma S_{\eta(s)}\) is progressively measurable and square-integrable on the finite horizon. The Ito isometry controls its approximation error; in particular, the Ito integral is square-integrable. For such an integrand its integral is a martingale: it is integrable by the isometry, and its conditional expectation increment over \([u,t]\) given \(\mathcal{F}_u\) is zero, first for adapted step integrands by independent Brownian increments and then for square-integrable integrands by the Ito-isometry limit. The residual convergence is a statement in mean square, not a guarantee that every finite simulated path has a small residual.

Interpreting the Test and Avoiding a Common Error

For a computational study, keep the simulated Brownian path fixed while refining the grid when possible, and compute the exact solution at each grid point from the cumulative Brownian values. Record \(|R_n|\) or \(R_n^2\) for each refinement. Across independently simulated paths, the residual is random, so a single sequence need not decrease monotonically. The proposition predicts convergence in mean square as the step shrinks, not a strict decrease at every refinement for every outcome.

A common mistake is to use \(S_0e^{\mu t+\sigma B_t}\) as the exact solution. Applying Ito's formula to \(g(t,x)=S_0e^{\mu t+\sigma x}\) gives \(g_t=\mu g\), \(g_x=\sigma g\), and \(g_{xx}=\sigma^2g\), so its drift is \((\mu+\tfrac12\sigma^2)g\), not \(\mu g\). For example, with \(\mu=0\) and \(\sigma=1\), the uncorrected exponential has drift \(\tfrac12g\,dt\); the correct solution is \(S_0e^{B_t-t/2}\). The correction is a consequence of \(dB_t\,dB_t=dt\), whereas the ordinary chain rule has no such term. A Stratonovich integral uses the ordinary chain rule, but converting a Stratonovich equation to Ito form requires a correction; it is not interchangeable with the Ito differential tested here.

Finally, distinguish two useful numerical objects. The exact grid transition evaluates the known exponential solution using the simulated Brownian increment. The Euler update uses only the first-order drift and diffusion increments and generally differs from that exact value at finite \(h\). Comparing the Euler update directly to the exact transition measures a discretization error; the residual defined above tests whether the integrated Ito differential is recovered by left-point sums. Keeping these checks separate makes the role of the Ito correction visible rather than hiding it inside a numerical update.