Linearization Near an Allocation Target

Why Linearize Around an Allocation Target?

An adaptive allocation rule can be designed so that its allocation proportions move toward a desired interior target. Near that target, the key question is not just whether the drift vanishes, but how the drift responds to a small displacement. A displacement that produces a restoring drift tends to shrink; one that produces a drift in the same direction can grow. The Jacobian records these local responses in all state directions at once.

The state may contain allocation proportions alone, or proportions together with estimates used to calculate allocation probabilities. Write the state as \(Z_n\in\mathbb{R}^d\), and suppose its recursion has the stochastic-approximation form

Here \(a_{n+1}>0\) is a step size, \(h\) is the conditional mean drift, and \(\xi_{n+1}\) is the random increment after subtracting that drift. In the standard martingale-noise formulation, \(\xi_{n+1}\) is measurable at step \(n+1\) and \(E(\xi_{n+1}\mid\mathcal{F}_n)=0\), where \(\mathcal{F}_n\) contains the information available through step \(n\). A usual convergence setting has \(a_n\to0\), \(\sum_n a_n=\infty\), and \(\sum_n a_n^2<\infty\), along with suitable moment and boundedness conditions. These conditions support stochastic-approximation arguments; the Jacobian calculation by itself does not prove convergence.

Let \(z^*\) be an interior target equilibrium, meaning \(h(z^*)=0\), and assume \(h\) is differentiable in a neighborhood of \(z^*\). For allocation shares constrained to sum to one, use \(d-1\) independent coordinates, or another valid local coordinate system, before taking derivatives. This avoids treating a constrained state as though it could move freely outside the allocation simplex.

Deriving the Local Recursion

Set \(e_n=Z_n-z^*\), the displacement from the target. Differentiability gives the first-order expansion

The remainder \(r(e)\) collects terms smaller than first order. Substituting this expansion into the recursion yields

Ignoring the smaller-order remainder and the random increment gives the deterministic local model \(e_{n+1}\approx (I+a_{n+1}J)e_n\). The associated drift differential equation is \(\dot e=Je\). Its solutions are governed by the eigenvalues of \(J\): a mode with a negative real part decays in this differential equation, whereas a mode with a positive real part grows.

This is a local statement about the mean drift and its associated differential equation. It is not, on its own, a guarantee that a noisy allocation recursion converges: the step sizes, noise, boundedness, and the path’s ability to reach and remain near the equilibrium also matter. Further, eigenvalues of \(J\) classify the differential equation, not every discrete update with an arbitrary fixed step size. For a fixed step \(a\), the linearized deterministic update has matrix \(I+aJ\); its eigenvalues are \(1+a\lambda\), where \(\lambda\) ranges over the eigenvalues of \(J\). Discrete-time contraction requires their absolute values to be less than one.

Several Directions: Eigenvalues and Transient Behavior

In multiple dimensions, the Jacobian can couple allocation or estimator coordinates. Eigenvalues still identify the long-run behavior of linear modes, but the components need not move independently in the original coordinates. Complex eigenvalues describe combinations of rotation and contraction or expansion: their real parts determine exponential growth or decay in the drift differential equation.

Noise and Local Fluctuations

The local stochastic recursion contains both the restoring term \(Je_n\) and the noise term \(\xi_{n+1}\). In a stable direction, the drift tends to pull deviations back, while random increments continually perturb them. The Jacobian therefore helps explain the shape and persistence of fluctuations: noise projected onto a weakly restoring direction can produce more persistent deviations than noise projected onto a strongly restoring direction. If the noise has little or no projection onto a direction, its first-order contribution in that direction is correspondingly small.

The following constant-step calculation isolates this balance. It is not a limit theorem for a decreasing-step allocation process. It shows why drift and noise must be considered together when interpreting variation near a target.

When the First-Order Test Is Inconclusive

An eigenvalue with zero real part requires care. If another eigenvalue has positive real part, the equilibrium is already unstable; the zero-real-part direction does not undo that conclusion. For example, \(J=\operatorname{diag}(1,0)\) has a growing direction from its eigenvalue \(1\), so the equilibrium is unstable regardless of higher-order behavior along the zero-eigenvalue direction.

The genuinely inconclusive case is when there are no eigenvalues with positive real part and at least one eigenvalue with zero real part. Consider the one-dimensional drift \(h(y)=-y^3\) at \(y=0\). Its derivative there is zero, yet \(\dot y=-y^3\) moves toward zero from either side. Changing the drift to \(h(y)=y^3\) leaves the same zero Jacobian but makes nonzero solutions move away from zero. Thus the Jacobian alone cannot distinguish these cases.

A separate warning concerns boundaries. At an interior target, small displacements in the chosen local coordinates are permitted in all directions. At a boundary target, a displacement may violate the allocation constraints, and a Jacobian on unconstrained coordinates may not describe the relevant one-sided behavior. Boundary targets require methods that respect the constraint; the interior linearization should not be applied without checking its domain.

Diagnostics and Alternatives

Before interpreting eigenvalues, check that the target is actually an equilibrium, that the state coordinates respect the allocation constraints, and that the drift is differentiable near the target. If derivatives are estimated numerically, compare them across smaller perturbations: large changes can signal that the local region is too wide, the drift is nonsmooth, or numerical error dominates. Simulated paths can reveal unexpected transients or constraint violations, but a finite collection of paths cannot establish local stability.

A Jacobian test is compact and informative when its assumptions hold, but it is not the only stability tool. Lyapunov criteria can establish local stability through a function that decreases along trajectories, including settings where a direct eigenvalue argument is awkward. The earlier material on Lyapunov criteria treats that approach. Numerical simulation is useful for diagnostics but is not a proof; direct eigenvalue analysis gives a local linear classification but does not certify behavior far from the target.