Simulating Brownian Paths on a Time Grid

From Brownian Increments to a Simulated Grid Path

Brownian motion is specified by its increments: over an interval of length \(\Delta t\), the increment is normal with mean zero and variance \(\Delta t\), and increments over disjoint intervals are independent. This gives a direct way to simulate the values of a Brownian path at finitely many times. It does not produce the entire continuous path between those times; joining simulated values by straight lines is an additional approximation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Thus \(B_0=0\), its paths are continuous, and for \(0\le s<t\), \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). Fix a time horizon \(T>0\) and an integer \(n\ge1\). Use the equally spaced grid \(t_k=k\Delta t\), where \(\Delta t=T/n\) and \(k=0,\ldots,n\).

The superscript “sim” distinguishes the generated increments from increments of a particular Brownian motion on the original probability space. In an implementation, a pseudorandom-number generator produces the normal draws; setting its seed makes a run reproducible. The mathematical target is independent \(N(0,1)\) draws. The accuracy of a finite computer generator is an implementation issue, so diagnostics assess its output rather than establish exact independence or exact normality.

Why the Grid Algorithm Has the Right Law

Proof. Multiplying a \(N(0,1)\) random variable by \(\sqrt{\Delta t}\) gives a normal random variable with mean \(0\) and variance \((\sqrt{\Delta t})^2=\Delta t\). Applying separate deterministic scalings to independent variables preserves independence, so the simulated increments are independent and each has distribution \(N(0,\Delta t)\). Brownian motion has independent increments, and \(B_{t_k}-B_{t_{k-1}}\) has distribution \(N(0,t_k-t_{k-1})=N(0,\Delta t)\). Therefore the vector of simulated increments has the same joint distribution as the vector of Brownian increments. The map from increments \(x_1,\ldots,x_n\) to grid values is the deterministic map that takes successive partial sums, with initial value zero. Applying this same map to two vectors with the same distribution gives vectors with the same distribution. This proves the asserted equality in law, including the initial value \(\widehat B_0=B_0=0\). \(\square\)

In particular, for each \(k\), \(\widehat B_k\) has distribution \(N(0,t_k)\), since it is a sum of \(k\) independent normal increments with total variance \(k\Delta t=t_k\). The equality in law is a statement about the grid values only. A linear interpolation between them is generally not a Brownian path: its segments are straight, whereas Brownian paths are almost surely nowhere differentiable.

Empirical Diagnostics Across Repeated Paths

One path checks the recursion but says little about whether many generated paths have the intended distribution. For diagnostics, generate \(N\) paths, each with \(n\) increments. At a fixed grid index \(k\), denote the simulated values across paths by \(\widehat B_k^{(1)},\ldots,\widehat B_k^{(N)}\). Their empirical mean and population-form empirical variance are \(\overline B_k=N^{-1}\sum_{j=1}^N\widehat B_k^{(j)}\) and \(V_k=N^{-1}\sum_{j=1}^N(\widehat B_k^{(j)}-\overline B_k)^2\). When \(N\ge2\), the sample variance uses denominator \(N-1\) instead. At time \(t_k\), the theoretical mean is zero and the theoretical variance is \(t_k\); neither empirical statistic is required to equal its target for a finite run.

Proof. The theorem shows that the values at time \(t_k\) are independent across paths, each with mean zero and variance \(t_k\). Linearity of expectation therefore gives \(\mathbb{E}[\overline B_k]=0\). For the variance calculation, expand the sum of squared deviations: \(\sum_j(\widehat B_k^{(j)}-\overline B_k)^2=\sum_j(\widehat B_k^{(j)})^2-N\overline B_k^2\). The expectation of the first term is \(Nt_k\). Independence and zero means give \(\mathbb{E}[\overline B_k^2]=\operatorname{Var}(\overline B_k)=t_k/N\), so the expected second term is also \(t_k\). Hence the expected sum of squared deviations is \((N-1)t_k\). Dividing by \(N\) proves the formula for \(\mathbb{E}[V_k]\). If \(N\ge2\), dividing instead by \(N-1\) proves \(\mathbb{E}[S_k^2]=t_k\). When \(N=1\), \(V_k=0\) by its definition and the sample variance with denominator \(N-1\) is undefined; the stated sample-variance claim is therefore restricted to \(N\ge2\). \(\square\)

Checking Endpoint Distributions

Matching a mean and variance alone does not establish that a generated endpoint has a normal distribution. A basic additional diagnostic compares the empirical distribution with the theoretical normal distribution. At time \(T>0\), the standardized endpoint \(B_T/\sqrt{T}\) has distribution \(N(0,1)\). For a threshold \(x\), compare the fraction of standardized simulated endpoints at most \(x\) with \(\Phi(x)\), where \(\Phi\) is the standard normal cumulative distribution function. The membership test here is non-strict: it counts values satisfying \(\widehat B_T^{(j)}/\sqrt{T}\le x\). In a continuous theoretical distribution, the probability of equality at a fixed threshold is zero, but a finite computed sample can contain a value equal to the threshold.

Implementation and Common Pitfalls

A simulation can be implemented with two nested loops: the outer loop runs over paths, and the inner loop generates a fresh standard normal draw and adds its scaled value to the current path value. Save the initial zero and each updated value if the whole grid path is needed. For several diagnostics, retain the endpoints and, when checking increments, retain the increments as well. At grid index \(k\), the increment distribution should be \(N(0,\Delta t)\), while the endpoint distribution should be \(N(0,t_k)\). Comparing an increment to the endpoint variance is a common scaling error.

Do not confuse the two variance conventions. The population-form empirical variance divides by \(N\), whereas the sample variance divides by \(N-1\) and is defined only when \(N\ge2\). Also, a close empirical mean or variance is a useful diagnostic, not a proof of correct simulation. Conversely, a modest discrepancy in a small run is not automatically a defect: the theoretical statements describe distributions and expectations over repeated runs, not exact finite-sample equalities.

The grid algorithm is valuable because it makes Brownian finite-dimensional distributions computationally accessible while keeping the scale of every increment explicit. Once the grid values are generated, further numerical questions can be studied from those discrete observations. The distinction between grid data and the continuous process remains important: a simulation stores finitely many values, and any claims about behavior between grid points require an additional approximation or analysis.