Interpolating Density Values at Boundary Locations

Why the Boundary’s Position Within a Cell Matters

In a sequential calculation, the density at an interim look is retained only over the states that have not crossed a stopping boundary. A numerical grid represents that density at discrete locations, but the boundary need not coincide with one of them. Treating a boundary-cut interval as wholly surviving or wholly stopped can therefore change the survivor mass passed to the next look. The size and direction of that change depend on where the boundary falls within the interval and on the density there.

The earlier material on representing interim looks as numerical states and on the survivor-density recursion supplies the setting: at a look, a subdensity describes probability mass among paths that have not stopped. Here the focus is narrower. We compare snapping a boundary to a nearby grid point with estimating the mass up to the actual boundary by linearly interpolating the density.

A subdensity need not integrate to one: probability mass may already have been removed at earlier looks. If \(x\) is measured in standardized-statistic units, for example, \(g(x)\) has units of probability per standardized-statistic unit, and its integral is dimensionless probability mass. The calculations below assume the grid values represent the same subdensity at the current look, are nonnegative, and are sufficiently local that a straight-line approximation across a cut interval is a reasonable numerical approximation.

Interpolating and Integrating a Cut Interval

Consider adjacent grid nodes \(x_k\) and \(x_{k+1}\), with spacing \(h=x_{k+1}-x_k\), and density values \(g_k=g(x_k)\) and \(g_{k+1}=g(x_{k+1})\). Suppose an upper stopping boundary \(b\) lies between them. Write \(t=(b-x_k)/h\), so \(0<t<1\). Linear interpolation estimates the boundary density as a weighted average of the two node values.

For the continuation region below the upper boundary, the cut portion runs from \(x_k\) to \(b\). Under the piecewise-linear density approximation, its area is a trapezoid: its width is \(th\), and its endpoint heights are \(g_k\) and \(\widehat g(b)\). This gives the partial-interval mass.

The full interval from \(x_k\) to \(x_{k+1}\), by the same trapezoidal rule, has mass \(M_{\mathrm{full}}=h(g_k+g_{k+1})/2\). Thus the linear method keeps complete intervals on the continuation side, adds the appropriate partial area, and excludes the portion beyond the boundary. The formula is a direct integration of the line joining the two density values; it is not an assumption that the unknown true density is exactly linear.

For comparison, define the nearest-node version of nearest-cell snapping precisely: represent each interval by its grid nodes, and snap \(b\) to whichever of \(x_k\) and \(x_{k+1}\) is closer. If \(t\leq 1/2\), the snapped boundary is \(x_k\) and none of the cut interval is retained. If \(t>1/2\), it is \(x_{k+1}\) and the whole interval is retained. This convention makes the comparison reproducible; implementations using cell edges or another tie rule should state that convention, because it can change the result.

The direction statement follows because the interpolated density is nonnegative throughout the interval: the partial area and the omitted remainder are both nonnegative. This establishes the sign of the snapping error, not which method is closer to the true, continuous-density mass. That depends on how well the grid values and the piecewise-linear shape approximate the actual density.

Worked Calculations

Using the Approximation in a Survivor-Mass Update

At each look, the survivor mass is the integral of the current subdensity over the continuation region. In a grid calculation, each full interval and each boundary-cut interval must be handled consistently. The following steps describe the local boundary calculation; they do not replace the transition-density calculation that produced the grid values.

For a two-sided continuation region \(\ell<x<u\), there may be two cut intervals, one at each end. The lower partial interval is integrated from \(\ell\) up to the next node; the upper partial interval is integrated from the preceding node up to \(u\). Complete intervals between them are included once. If a boundary coincides with a node, the corresponding partial interval has zero width or is a complete interval, so no special extrapolation is needed.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to calculate both methods at a few representative cut intervals. A large difference warns that snapping is materially affecting the mass at the current grid spacing. Also inspect the interpolated boundary density: it must lie between the two endpoint values, and it must be nonnegative when both endpoint values are nonnegative. For a two-sided region, check that the integrated survivor mass is nonnegative and does not exceed the pre-boundary mass.

These checks can expose implementation errors, but they do not prove that interpolation is accurate. A straight line can poorly approximate a curved density, especially where the density changes rapidly. Interpolation can also be misleading if an interval straddles a discontinuity—for example, if a previous truncation makes one side of the interval zero while the other side has positive density. In that case, values on opposite sides do not describe one smooth segment, so interpolating across them can invent density in a region where the model says none exists. Keep truncation locations explicit and do not interpolate across known jumps.

Linear interpolation is generally preferable to snapping when the grid values vary smoothly and the boundary lies inside the grid. It remains an approximation: errors in the grid density values, transition calculation, or integration over other intervals are not corrected by handling the boundary more carefully. Conversely, nearest-cell snapping can be adequate when its possible mass discrepancy is negligible for the calculation’s purpose, but that should be checked rather than assumed.

A common mistake is to say, “The boundary is closer to the upper node, so I should keep the whole interval; that is the mass below the boundary.” Closeness is only the snapping rule, not an integration argument. It may retain states beyond the boundary and overstate survivor mass, as in the two-sided example. The correction is to distinguish the approximation from the target: the target integrates only over the true continuation region, and linear interpolation estimates the cut portion of that region.

The impact can carry forward. Survivor mass at one look supplies the mass available for later transitions, so an error at a cut boundary may affect subsequent crossing probabilities. But a local discrepancy does not by itself determine the final error: later transition and boundary calculations also matter. Report the boundary treatment and interpret computed probabilities with the numerical approximation in view.