Time Grids and Brownian Increments for SDE Discretization

Time Grids as the Skeleton of a Simulation

A numerical path approximation replaces a continuously indexed process by values at a finite set of times. The time grid determines when those values are recorded; Brownian increments supply the random changes between consecutive grid points. Getting their distribution and scaling right is essential: a normal draw with the wrong variance gives the wrong stochastic forcing, even if the rest of a numerical method is implemented correctly.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) relative to the filtration. In particular, \(B_0=0\), its paths are continuous, and for \(s<t\), the increment \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). A finite time grid on \([0,T]\), where \(T>0\), is a strictly increasing sequence \(0=t_0<t_1<\cdots<t_N=T\). Its step lengths are \(h_i=t_{i+1}-t_i>0\), for \(i=0,\ldots,N-1\), and its mesh size is \(|\pi|=\max_{0\le i<N}h_i\). Equal step lengths give a uniform grid; unequal step lengths give a nonuniform grid.

For context, an SDE with drift \(\mu(t,x)\), diffusion coefficient \(\sigma(t,x)\), and initial value \(X_0\) is interpreted in integral form as \(X_t=X_0+\int_0^t\mu(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\). The first coefficient governs the finite-variation drift contribution; the second multiplies the Brownian noise. For the Ito integral on \([0,T]\), its integrand \(H_s=\sigma(s,X_s)\) must be progressively measurable and satisfy \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); that is, \(H\) belongs to \(\mathcal{L}^2\). The integral is then defined in the Ito sense. A numerical path method evaluates its approximate state at grid points using the time steps and random inputs; the particular update rule is a separate choice.

The Distribution of the Grid Increments

Proof. Fix an index \(i\). The defining increment property of Brownian motion, applied with \(s=t_i\) and \(t=t_{i+1}\), gives \(\Delta B_i\sim N(0,t_{i+1}-t_i)=N(0,h_i)\) and says that \(\Delta B_i\) is independent of \(\mathcal{F}_{t_i}\). For \(j<i\), the increment \(\Delta B_j\) is measurable with respect to \(\mathcal{F}_{t_i}\), since both of its endpoint values occur no later than \(t_i\). Thus \(\Delta B_i\) is independent of the collection of earlier increments. Applying this argument successively shows that all the increments are independent. Finally, scaling a standard normal variable by \(\sqrt{h_i}\) gives a normal variable with mean \(0\) and variance \(h_i\), so the stated representation has the required joint distribution. \(\square\)

This theorem gives a direct simulation recipe: generate independent standard normal draws \(Z_i\), multiply each by the square root of its own step length, and accumulate the resulting increments. The square root is important because a Brownian increment has variance \(h_i\), not \(h_i^2\) or \(1\). On a nonuniform grid, each increment therefore needs its own scaling factor.

Reconstructing the Brownian Values at Grid Points

Proof. Each \(\Delta \widetilde B_i\) is normal with mean \(0\) and variance \(h_i\), and the increments are independent because the \(Z_i\) are independent. The Brownian increments on the same grid have exactly these distributions and are also independent by the theorem. Thus the two increment vectors have the same joint distribution. Each grid-value vector is obtained from its increment vector by the same deterministic operation, namely taking successive partial sums, with initial value \(0\). Applying this operation to identically distributed vectors gives identically distributed grid-value vectors, as claimed. When \(N=0\), there are no increments and both vectors consist only of the value \(0\); the statement holds in that degenerate case as well. \(\square\)

The proposition is precise about what the simulation recipe reproduces: the joint law of the Brownian values at the chosen grid points. It does not generate the Brownian path between those points. Connecting simulated values by straight lines is a plotting convention, not a claim that Brownian motion moves linearly between grid times.

How Grid Increments Drive an Approximate SDE Path

A grid-based path approximation is built recursively. Once the numerical state at \(t_i\) is known, a chosen update rule uses the step information and the random input \(\Delta B_i\) to produce the next state. Abstractly, one may write \(\widehat X_{i+1}=\Psi_i(\widehat X_i,\Delta B_i)\), where \(\Psi_i\) is a specified deterministic update map that may also depend on \(t_i\), \(h_i\), and the SDE coefficients. This notation isolates the role of the random input without specifying a particular numerical method. The same correctly generated increments can be supplied to any such method on the same grid.

What the Grid Does Not Tell You

Although the increments determine the simulated values at grid times, they do not determine the path between those times. For example, on one interval \([0,h]\), Brownian motion has \(B_h\sim N(0,h)\). Its midpoint is not generally the straight-line interpolation \(B_h/2\). To see the difference, set \(Y=B_{h/2}-B_h/2\). Brownian increments on the two half-intervals are independent \(N(0,h/2)\) variables. Writing them as \(U=B_{h/2}\) and \(V=B_h-B_{h/2}\), we have \(Y=(U-V)/2\) and \(B_h=U+V\). Both are jointly normal, their covariance is \(\mathbb{E}[YB_h]=\tfrac12(\mathbb{E}[U^2]-\mathbb{E}[V^2])=0\), and \(\mathbb{E}[Y^2]=\tfrac14(h/2+h/2)=h/4\). Jointly normal variables with zero covariance are independent, so conditional on \(B_h\), the midpoint has distribution \(N(B_h/2,h/4)\). Its conditional variance is positive when \(h>0\); the straight line is therefore only an interpolation, not the actual Brownian bridge between the endpoints.

Common Pitfalls and Practical Checks

A sound implementation should check the time grid and the random inputs separately. Verify that the grid is strictly increasing, that all step lengths are positive, and that their sum is \(T\). For each step, use its own \(h_i\) and form \(\sqrt{h_i}Z_i\). The \(Z_i\) should be independent standard normal draws; reusing one draw at every step does not produce independent Brownian increments. Finally, reconstruct grid values by cumulative summation and retain the increments if the numerical update needs them.

A finer grid creates more recorded values, but it does not by itself prescribe a particular path between those values or guarantee a specified accuracy for an SDE approximation. The accuracy depends on the update method and the SDE, not just on the random-number scaling. The increments establish the right Brownian forcing on the chosen grid; analysis of how well a numerical method approximates the continuous-time solution is a separate question.