Moment Recursions From Ito's Formula

From Powers of the Process to Moments

For geometric Brownian motion, the distribution of \(S_t\) gives its moments directly. Ito's formula offers a different route: apply it to a power \(S_t^n\), take expectations, and obtain a differential equation for the \(n\)th moment. This method makes the contribution of the drift and diffusion visible, and gives a recursion that can be checked against the distributional calculation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\) adapted to the filtration. Let \(s>0\) and \(\mu,\sigma\in\mathbb{R}\) be constants, and let \(S_t\) solve the geometric Brownian motion equation \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\), with \(S_0=s\). Here \(\mu S_t\) is the drift coefficient and \(\sigma S_t\) is the diffusion coefficient. The Ito integral is understood for progressively measurable integrands \(H\) in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_t^2\,dt<\infty\) on each finite time interval \([0,T]\).

The explicit solution is \(S_t=s\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\). To check this expression with Ito's formula, set \(g(t,x)=s\exp((\mu-\tfrac12\sigma^2)t+\sigma x)\). Its derivatives are \(g_t=(\mu-\tfrac12\sigma^2)g\), \(g_x=\sigma g\), and \(g_{xx}=\sigma^2g\). The time-dependent Ito formula contributes the correction \(\tfrac12g_{xx}\,dt\), so \(dg(t,B_t)=\mu g(t,B_t)\,dt+\sigma g(t,B_t)\,dB_t\), as required.

The Moment Differential Equation

Proof. First take \(n\ge2\) and \(f(x)=x^n\). Its derivatives are \(f'(x)=nx^{n-1}\) and \(f''(x)=n(n-1)x^{n-2}\). The quadratic variation of \(S\) is \(d[S]_t=\sigma^2S_t^2\,dt\), since its Brownian coefficient is \(\sigma S_t\) and \(dB_t\,dB_t=dt\). Ito's formula, including its second-derivative correction, gives

For \(n=1\), the same equation follows directly from the defining SDE, because \(c_1=\mu\). For \(n=0\), \(S_t^0=1\), so its differential is zero and the equation holds with \(c_0=0\).

It remains to justify taking expectations. The explicit solution and the Gaussian moment-generating function give, for every \(q\ge0\) and finite \(T\), \(\mathbb{E}[S_t^q]=s^q\exp((q\mu+\tfrac12q(q-1)\sigma^2)t)\) for \(0\le t\le T\). In particular, this expectation is bounded on \([0,T]\). Thus \(H_t=n\sigma S_t^n\) is progressively measurable and \(\mathbb{E}\int_0^T H_t^2\,dt=n^2\sigma^2\int_0^T\mathbb{E}[S_t^{2n}]\,dt<\infty\). The Ito integral of \(H\) is therefore a square-integrable martingale: its increments have conditional expectation zero, by the martingale property of the Ito integral for \(\mathcal{L}^2\) integrands. Taking expectations in the integrated equation yields \(m_n(t)=s^n+c_n\int_0^t m_n(u)\,du\). This integral equation implies that \(m_n\) is absolutely continuous and satisfies \(m_n'=c_nm_n\) almost everywhere. Multiplication by \(e^{-c_nt}\) gives \(\frac{d}{dt}(e^{-c_nt}m_n(t))=0\) almost everywhere, so \(m_n(t)=s^ne^{c_nt}\). This also holds for \(n=0\), for which \(m_0=1\). \(\square\)

The differential equation is the moment recursion in time: each moment evolves according to its own constant coefficient \(c_n\). The diffusion contributes \(\tfrac12n(n-1)\sigma^2\), not merely a term proportional to \(n\). For \(n=0\), the moment is the constant one; for \(n=1\), the diffusion correction vanishes, so the mean evolves according to the drift alone.

Checking Against the Lognormal Moment Formula

Proof. The explicit solution gives \(S_t^n=s^n\exp(n(\mu-\tfrac12\sigma^2)t+n\sigma B_t)\). Since \(B_t\) has distribution \(N(0,t)\), its moment-generating function gives \(\mathbb{E}[e^{aB_t}]=e^{a^2t/2}\) for every real \(a\). Taking \(a=n\sigma\) therefore yields \(\mathbb{E}[S_t^n]=s^n\exp((n\mu-\tfrac12n\sigma^2+\tfrac12n^2\sigma^2)t)\). The exponent coefficient simplifies to \(n\mu+\tfrac12n(n-1)\sigma^2=c_n\), exactly the coefficient in the differential equation. When \(n=0\), both expressions equal \(1\), so the identity includes that case as well. \(\square\)

This check is independent of solving the moment equation: it starts from the explicit solution and the Gaussian moment-generating function. It confirms both the factor \(n(n-1)\) and the sign of the diffusion contribution. The result also gives a useful relation between adjacent moments. For \(n\ge1\), division of the positive moment formulas gives \(m_n(t)/m_{n-1}(t)=s\exp((\mu+(n-1)\sigma^2)t)\). This relation is a consequence of the explicit formula, not an additional assumption about the process.

Common Mistake: Omitting the Ito Correction

A common incorrect calculation treats the power as though the ordinary chain rule applied. For the square, that would give \(d(S_t^2)=2S_t\,dS_t\), and hence the false moment equation \(m_2'=2\mu m_2\). In the second-moment example above, this predicts \(m_2(t)=e^{2t/5}\), omitting the contribution from the diffusion.

The correction is forced by the second derivative in Ito's formula: \(f''(x)=2\), so its contribution is \(\tfrac12 f''(S_t)d[S]_t=\sigma^2S_t^2dt\). The correct coefficient is \(2\mu+\sigma^2=\tfrac{19}{25}\), not \(2\mu=\tfrac25\). The lognormal check independently exposes the error: its exponent for the second moment is \(2\mu+\sigma^2\). The multiplication rules \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\) encode precisely why this second-order term remains.

The moment equation is useful whenever expectations of powers are needed without repeatedly integrating a density. It also clarifies which effects control each moment: the drift contributes \(n\mu\), while the curvature of \(x^n\), combined with the quadratic variation, contributes \(\tfrac12n(n-1)\sigma^2\). This calculation concerns unconditional moments at a single time; it does not assert a conditional-moment or transition-law formula.