Moment Estimates for Ornstein-Uhlenbeck and Geometric Brownian Motion

What the Coefficients Tell Us About Moments

Two familiar diffusion models illustrate different long-time moment behavior. The Ornstein–Uhlenbeck process has a restoring drift: it pulls the state toward a fixed level, while its diffusion continually adds random variation. Geometric Brownian motion has drift and diffusion proportional to its current value, so its moments evolve exponentially. Computing the moments makes the roles of these coefficients precise.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). All expectations below are under \(\mathbb{P}\), not under a risk-neutral measure \(\mathbb{Q}\). Every integral against \(B_t\) is an Itô integral: its integrand is progressively measurable and belongs to \(\mathcal{L}^2\) on each finite time interval, meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). The stochastic integrals used below are square-integrable martingales, so they have mean zero and conditional expectation equal to their current value. We use the standard Itô isometry for such integrals.

Both SDEs are interpreted in the Itô sense. The quadratic variation is \([B]_t=t\), giving the multiplication rules \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\). In particular, the Itô correction is not part of the ordinary chain rule. For example, \(\int_0^t H_s\circ dB_s=\int_0^t H_s\,dB_s+\tfrac12[H,B]_t\) when the Stratonovich integral is defined and the covariation exists. The explicit solutions here are adapted functions of the given Brownian motion, hence strong solutions; no claims about weak uniqueness or pathwise uniqueness are needed.

Ornstein–Uhlenbeck Moments

Take constants \(k>0\), \(\eta\in\mathbb{R}\), \(m\in\mathbb{R}\), and deterministic \(X_0=x\). The Ornstein–Uhlenbeck equation is \(dX_t=-k(X_t-m)\,dt+\eta\,dB_t\). Its drift is \(-k(X_t-m)\), which points toward \(m\), and its diffusion coefficient is the constant \(\eta\).

Proof. Apply Itô’s formula to \(f(t,z)=e^{kt}(z-m)\). Its derivatives are \(f_t=k e^{kt}(z-m)\), \(f_z=e^{kt}\), and \(f_{zz}=0\); thus the \(\tfrac12 f_{zz}\,dt\) correction is zero. Substituting the SDE gives \(d(e^{kt}(X_t-m))=\eta e^{kt}\,dB_t\). Integrating from \(0\) to \(t\) yields \(X_t=m+(x-m)e^{-kt}+\eta\int_0^t e^{-k(t-s)}\,dB_s\). The integrand \(s\mapsto\eta e^{-k(t-s)}\) is deterministic and has finite squared integral on \([0,t]\). The Itô integral therefore has mean zero, and the Itô isometry gives its variance as \(\eta^2\int_0^t e^{-2k(t-s)}\,ds=\frac{\eta^2}{2k}(1-e^{-2kt})\). Adding the square of the mean to the variance gives the second moment. These computations also establish finite second moments on every finite interval. \(\square\)

The formula shows that the mean approaches \(m\), while the variance approaches \(\eta^2/(2k)\). Thus the second moment stays bounded over all \(t\ge0\). This conclusion depends on \(k>0\); without a restoring drift, the displayed limit for the variance does not apply.

Checking the OU Second-Moment Equation Directly

Itô’s formula also provides a differential check on the second-moment expression. For \(g(z)=z^2\), the derivatives are \(g'(z)=2z\) and \(g''(z)=2\), so the correction term is \(\tfrac12g''(X_t)\eta^2dt=\eta^2dt\). Consequently,

Because the explicit solution has finite second moments on finite intervals, \(\mathbb{E}\int_0^T(2\eta X_s)^2\,ds<\infty\). Hence the stochastic integral in this identity is a square-integrable martingale and its expectation is zero. Taking expectations gives \(\frac{d}{dt}\mathbb{E}[X_t^2]=-2k\mathbb{E}[X_t^2]+2km\mathbb{E}[X_t]+\eta^2\). This equation is a check, not a replacement for the explicit derivation: inserting the already calculated mean and second moment satisfies it and the initial condition \(\mathbb{E}[X_0^2]=x^2\).

Geometric Brownian Motion Moments

Let \(S_0=s>0\) be deterministic, and let \(\alpha,\sigma\in\mathbb{R}\). Geometric Brownian motion satisfies \(dS_t=\alpha S_t\,dt+\sigma S_t\,dB_t\). Here \(\alpha S_t\) is the drift coefficient and \(\sigma S_t\) is the diffusion coefficient. The first moment depends on \(\alpha\); the second moment depends on both coefficients.

Proof. Define \(F(t,z)=s\exp((\alpha-\tfrac12\sigma^2)t+\sigma z)\). Its derivatives are \(F_t=(\alpha-\tfrac12\sigma^2)F\), \(F_z=\sigma F\), and \(F_{zz}=\sigma^2F\). Itô’s formula, including its correction, gives \(dF(t,B_t)=F_tdt+F_zdB_t+\tfrac12F_{zz}dt=\alpha Fdt+\sigma FdB_t\). Also \(F(0,0)=s\), so this explicit adapted process satisfies the SDE. Since \(B_t\) is normal with mean zero and variance \(t\), its moment-generating function gives \(\mathbb{E}[e^{cB_t}]=e^{c^2t/2}\) for every real \(c\). Therefore \(\mathbb{E}[S_t]=s e^{(\alpha-\sigma^2/2)t}e^{\sigma^2t/2}=se^{\alpha t}\), and \(\mathbb{E}[S_t^2]=s^2e^{(2\alpha-\sigma^2)t}e^{(2\sigma)^2t/2}=s^2e^{(2\alpha+\sigma^2)t}\). These finite moments on every finite interval ensure that the integrands \(\sigma S_t\) and \(2\sigma S_t^2\), when used in the moment checks below, are square-integrable on finite time intervals; for the latter use the same Gaussian moment formula with the fourth moment of \(S_t\). Thus the corresponding Itô integrals are square-integrable martingales with zero mean. \(\square\)

The factor \(\sigma^2\) in the second-moment exponent arises from quadratic variation. To see it directly, apply Itô’s formula to \(g(z)=z^2\): \(g'(z)=2z\), \(g''(z)=2\), and the correction is \(\tfrac12(2)\sigma^2S_t^2dt\). It follows that \(d(S_t^2)=(2\alpha+\sigma^2)S_t^2dt+2\sigma S_t^2dB_t\). Taking expectations is justified by the integrability just noted, and gives the differential equation \(M_2'=(2\alpha+\sigma^2)M_2\), with \(M_2(0)=s^2\). Its solution agrees with the formula in the theorem.

Moment Boundedness and a Common Pitfall

Proof. For OU, \(0<e^{-kt}\le1\), so \(|\mathbb{E}[X_t]|\le |m|+|x-m|\). Also, the variance is at most \(\eta^2/(2k)\), and \(|m+(x-m)e^{-kt}|\le |m|+|x-m|\). Since the second moment is variance plus squared mean, it is bounded by \(\eta^2/(2k)+(|m|+|x-m|)^2\), uniformly in \(t\). For geometric Brownian motion, the first moment is \(s e^{\alpha t}\) with \(s>0\). If \(\alpha\le0\), this is at most \(s\); if \(\alpha>0\), it grows without bound as \(t\) increases. The second moment is \(s^2e^{(2\alpha+\sigma^2)t}\), and the same argument applied to its exponent proves the stated condition. \(\square\)

A common mistake is to use the square of the mean as the second moment of geometric Brownian motion. That calculation would give \((\mathbb{E}[S_t])^2=s^2e^{2\alpha t}\), omitting the diffusion contribution. The correction is visible by applying Itô’s formula to \(S_t^2\): the term \(\tfrac12g''(S_t)\sigma^2S_t^2dt=\sigma^2S_t^2dt\) changes the moment exponent to \(2\alpha+\sigma^2\). The square of the mean is only one part of the second moment, since \(\mathbb{E}[S_t^2]=\operatorname{Var}(S_t)+(\mathbb{E}[S_t])^2\).

These formulas are useful because moment behavior can be read directly from the coefficients. In OU, the restoring drift limits the effect of accumulated noise and produces a finite long-time variance. In geometric Brownian motion, drift and diffusion combine in the second-moment exponent, so a decreasing mean alone does not guarantee a decreasing second moment. Moment boundedness is a statement about expectations; it is not, by itself, a claim about pathwise behavior.