When Mean-Square and Almost-Sure Stability Differ

Two Ways to Measure Long-Time Stability

Consider a scalar process whose drift pulls toward zero while its diffusion continually perturbs it. A typical sample path may approach zero even though the second moment grows. The reason is that a second moment gives disproportionate weight to large values: rare paths with unusually large Brownian fluctuations can determine the expectation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). For fixed real parameters \(a,b\) and deterministic initial value \(x_0\), consider the SDE

Here \(aX_t\) is the drift coefficient and \(bX_t\) is the diffusion coefficient. The Ito integral is understood for progressively measurable integrands \(H_s\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\) on each finite time interval. In particular, the solution below has this property. The coefficients are globally Lipschitz, since \(|ax-ay|=|a||x-y|\) and \(|bx-by|=|b||x-y|\), and satisfy linear growth, since \(a^2x^2+b^2x^2\le(a^2+b^2)(1+x^2)\). Thus the standard global Lipschitz existence-and-pathwise-uniqueness theorem applies. The explicit solution will also be verified directly.

The Explicit Solution and the Two Exponents

Set \(k=a-\tfrac12b^2\) and define \(g(t,z)=x_0e^{kt+bz}\). Ito's formula applies to \(g(t,B_t)\): its derivatives are \(g_t=kg\), \(g_z=bg\), and \(g_{zz}=b^2g\), so the second-order correction is \(\tfrac12g_{zz}\,dt=\tfrac12b^2g\,dt\). Therefore

It follows that the solution is

On every finite interval this is an admissible Ito integrand: the normal moment formula gives \(\mathbb{E}[X_s^2]=x_0^2e^{(2a+b^2)s}\), and hence \(\mathbb{E}\int_0^T b^2X_s^2\,ds<\infty\). The formula also shows that the solution is strong: it is constructed as an adapted function of the given Brownian motion. The standard global Lipschitz theorem gives pathwise uniqueness, meaning solutions driven by the same Brownian motion with the same initial value agree almost surely.

For \(x_0\ne0\), take logarithms of the absolute value. The standard Brownian strong law \(B_t/t\to0\) almost surely yields

This is the almost-sure growth exponent. For the second moment, the normal moment-generating formula \(\mathbb{E}[e^{\lambda B_t}]=e^{\lambda^2t/2}\) gives

The exponent \(2a+b^2\) controls mean-square behavior. The two exponents differ because expectation weights the large values of \(X_t^2\), while the almost-sure exponent describes the long-run rate along a typical path.

Proof. First suppose \(b\ne0\). If \(a-\tfrac12b^2<0\), the displayed logarithmic limit is negative, so \(|X_t|\to0\) almost surely. If \(a-\tfrac12b^2>0\), it is positive, so \(|X_t|\) grows without bound almost surely. At equality, \(a=\tfrac12b^2\), the solution is \(X_t=x_0e^{bB_t}\). Brownian motion has almost surely \(\limsup_{t\to\infty}B_t=+\infty\) and \(\liminf_{t\to\infty}B_t=-\infty\), the standard Brownian oscillation result; hence \(X_t\) does not converge to zero. This proves the almost-sure criterion, including its boundary. For mean-square stability, the exact formula is \(x_0^2e^{(2a+b^2)t}\): it tends to zero exactly when \(2a+b^2<0\), is constant and positive when \(2a+b^2=0\), and grows when \(2a+b^2>0\). Finally, if \(b=0\), then \(X_t=x_0e^{at}\), and both stability definitions reduce to \(a<0\). The interval \([-\tfrac12b^2,\tfrac12b^2)\) lies inside the almost-surely stable region but outside the mean-square stable region, proving the stated comparison. \(\square\)

The Brownian strong law and oscillation result used here are standard Brownian-motion results; their proofs are not repeated in this tutorial. In particular, the equality case for the almost-sure criterion cannot be settled merely by observing that the logarithmic exponent is zero: oscillation is needed to rule out convergence to zero.

Worked Parameter Regimes

Why Rare Fluctuations Matter

The second-moment formula can be understood by changing the weighting used in the expectation, without changing the probability measure of the SDE. For \(b\ne0\), the density of \(B_t\) is proportional to \(e^{-z^2/(2t)}\). In the expectation of \(X_t^2\), the factor \(e^{2bB_t}\) reweights this density. Completing the square gives the exact identity

After normalization, the reweighted Gaussian is centered at \(2bt\), not at zero. Thus the second moment is especially sensitive to paths with Brownian motion of order \(t\), even though \(B_t/t\to0\) on almost every path. For the boundary example, \(a=-\tfrac12\) and \(b=1\), the factor outside \(e^{2bB_t}\) in \(X_t^2=e^{-2t+2B_t}\) is \(e^{-2t}\). The reweighting contributes \(e^{2t}\), exactly canceling it and leaving \(\mathbb{E}[X_t^2]=1\). Typical paths still decay, but these less typical large fluctuations prevent the second moment from decaying.

This calculation is an explanation of the expectation, not a claim that a particular path has positive probability of following the line \(B_t=2bt\). The density identity is exact at each fixed \(t\); it identifies which region of Brownian values receives extra weight in computing the second moment.

Common Pitfall: Inferring Moment Stability from Typical Paths

A tempting but incorrect calculation is to use the almost-sure exponent \(a-\tfrac12b^2\) to decide whether \(\mathbb{E}[X_t^2]\) decays. For \(a=-\tfrac14\), \(b=1\), that exponent is \(-\tfrac34\), so this mistake would predict mean-square stability. The correction is to square the explicit solution and then take its expectation:

With \(x_0=2\) and the stated parameters, this gives \(4e^{t/2}\), which grows. The missing term in the mistaken argument comes from the normal moment \(\mathbb{E}[e^{2bB_t}]=e^{2b^2t}\); replacing a random exponential by its typical pathwise rate loses precisely the contribution that can control the expectation.

For \(b\ne0\), the full comparison is therefore: both notions of stability hold for \(a<-\tfrac12b^2\); only almost-sure stability holds for \(-\tfrac12b^2\le a<\tfrac12b^2\); and neither holds for \(a\ge\tfrac12b^2\). At the lower boundary the second moment is constant, while at the upper boundary Brownian oscillations prevent almost-sure convergence to zero. When \(b=0\), the noise disappears and the two criteria coincide.