Mean-Square Stability of Scalar Linear SDEs

Mean-Square Decay: Drift Competes with Noise

Consider a state whose deterministic part contracts while its random fluctuations continually add variability. For a scalar linear stochastic differential equation, both effects can be quantified exactly: the drift controls the deterministic tendency, while the diffusion contributes an additional term to the evolution of the second moment. The stability question is whether the expected squared state tends to zero as time increases.

Throughout, work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Let \(a,\sigma\in\mathbb{R}\) be constants and let \(x_0\in\mathbb{R}\) be deterministic. We use \(B_t\), rather than a different letter, for the driving Brownian motion. The drift coefficient is \(aX_t\), and the diffusion coefficient is \(\sigma X_t\).

The equation and its integral form are \[ dX_t=aX_t\,dt+\sigma X_t\,dB_t,\qquad X_0=x_0, \] \[ X_t=x_0+\int_0^t aX_s\,ds+\int_0^t \sigma X_s\,dB_s. \] The Brownian integrand in the Itô integral is adapted and progressively measurable. For each finite \(T\), it belongs to the class \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T(\sigma X_s)^2\,ds<\infty\); this integrability will follow from the explicit solution below. The differential multiplication rules used in Itô's formula are \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\).

Explicit Solution and Its Verification

Proof. Define \(f(t,x)=x_0\exp((a-\tfrac12\sigma^2)t+\sigma x)\) and set \(X_t=f(t,B_t)\). Its derivatives are \(f_t=(a-\tfrac12\sigma^2)f\), \(f_x=\sigma f\), and \(f_{xx}=\sigma^2f\). Itô's formula, including its second-derivative correction, gives \[ dX_t=f_t(t,B_t)\,dt+f_x(t,B_t)\,dB_t+\tfrac12 f_{xx}(t,B_t)\,dt =aX_t\,dt+\sigma X_t\,dB_t. \] Also \(X_0=x_0\). To check integrability, the normal moment-generating function gives \[ \mathbb{E}[X_t^2]=x_0^2 e^{(2a-\sigma^2)t}\mathbb{E}[e^{2\sigma B_t}] =x_0^2e^{(2a-\sigma^2)t}e^{2\sigma^2t} =x_0^2e^{(2a+\sigma^2)t}. \] Thus \(\mathbb{E}\int_0^T(\sigma X_s)^2\,ds=\sigma^2x_0^2\int_0^T e^{(2a+\sigma^2)s}\,ds<\infty\) for every finite \(T\); this includes \(\sigma=0\) or \(x_0=0\), when the integral is zero.

For uniqueness, let \(Y_t\) be any solution driven by the same Brownian motion and define \(G_t=\exp((-a+\tfrac12\sigma^2)t-\sigma B_t)\). Applying Itô's formula to \(G_t=g(t,B_t)\), with \(g_t=(-a+\tfrac12\sigma^2)g\), \(g_x=-\sigma g\), and \(g_{xx}=\sigma^2g\), gives \(dG_t=(-a+\sigma^2)G_t\,dt-\sigma G_t\,dB_t\). The product rule gives \[ d(G_tY_t)=G_t\,dY_t+Y_t\,dG_t+d[G,Y]_t. \] The diffusion coefficients of \(G\) and \(Y\) are \(-\sigma G\) and \(\sigma Y\), so \(d[G,Y]_t=-\sigma^2G_tY_t\,dt\). Substitution cancels the drift terms and the Brownian terms, yielding \(d(G_tY_t)=0\). Hence \(G_tY_t=G_0Y_0=x_0\), and \(Y_t=x_0/G_t\), exactly the displayed solution. The calculation is valid after localization on finite time intervals, so it also applies to solutions initially specified only locally. \(\square\)

The Exact Second-Moment Criterion

Proof. The explicit solution gives \[ |X_t|^2=x_0^2e^{(2a-\sigma^2)t}e^{2\sigma B_t}. \] Since \(B_t\) has distribution \(N(0,t)\), its moment-generating function satisfies \(\mathbb{E}[e^{uB_t}]=e^{u^2t/2}\) for every real \(u\). Using \(u=2\sigma\) yields \(\mathbb{E}[e^{2\sigma B_t}]=e^{2\sigma^2t}\). Thus \(\mathbb{E}[|X_t|^2]=x_0^2e^{(2a-\sigma^2)t}e^{2\sigma^2t}=x_0^2e^{(2a+\sigma^2)t}\). For \(x_0\ne0\), the factor \(x_0^2\) is strictly positive. An exponential \(e^{ct}\) tends to zero as \(t\to\infty\) exactly when \(c<0\); if \(c=0\), it is constantly \(1\), and if \(c>0\), it diverges. Taking \(c=2a+\sigma^2\) proves both directions of the criterion. If \(x_0=0\), the explicit solution is zero at every time, proving the stated exception. The criterion for every deterministic initial value follows because it must in particular hold for a nonzero initial value. \(\square\)

The threshold can also be read as a competition between the deterministic contraction rate and a noise contribution. The drift alone would suggest decay when \(a<0\), but the second moment contains the larger exponent \(2a+\sigma^2\). The noise therefore requires a sufficiently negative drift: \(a<-\sigma^2/2\). When \(\sigma=0\), this reduces to \(a<0\), the familiar deterministic condition.

A Common Error: Using the Ordinary Chain Rule

A tempting but incorrect calculation is to apply the ordinary chain rule to \(X_t^2\): \(d(X_t^2)=2X_t\,dX_t\). This drops the quadratic-variation contribution. For \(f(x)=x^2\), the derivatives are \(f'(x)=2x\) and \(f''(x)=2\). Itô's formula therefore gives \[ d(X_t^2)=2X_t\,dX_t+\tfrac12(2)\,d[X]_t =(2a+\sigma^2)X_t^2\,dt+2\sigma X_t^2\,dB_t, \] because \(d[X]_t=\sigma^2X_t^2\,dt\). The additional \(\sigma^2X_t^2dt\) is essential.

For the boundary example, the incorrect chain rule would use \(2a=-\tfrac14\) as the second-moment exponent and predict decay. The correct Itô correction adds \(\sigma^2=\tfrac14\), giving \(2a+\sigma^2=0\), so the second moment stays at \(9\). The error is precisely the failure to include \(dB_t\,dB_t=dt\).

There is also a useful distinction between this result and a pathwise statement. Mean-square stability asks whether an expectation tends to zero; it does not assert that every trajectory decreases monotonically. A second-moment calculation alone does not establish almost-sure stability. The exact criterion here is only for the mean-square question defined above.