Exponential Martingales and Geometric Brownian Moments

From Ito's Correction to an Exponential Martingale

Exponentials turn Brownian motion into positive processes that often appear to be martingales. The decisive point is the correction in Ito's formula: the quadratic-variation term cancels the deterministic drift placed in the exponent. That cancellation produces a local martingale, but a separate argument is needed to establish a true martingale. For constant coefficients, conditional Gaussian expectations give that argument directly.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\) relative to the filtration. For a fixed finite horizon \(T\), an Ito integral \(\int_0^t H_s\,dB_s\) is used with a progressively measurable integrand \(H\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); this is the class \(\mathcal{L}^2\). An integral whose integrand is only locally square-integrable is understood after stopping at times that make it square-integrable.

For the general expression, the Ito integral is defined locally when its integrand is locally square-integrable; in the constant-kernel results below, the relevant integrands are in \(\mathcal{L}^2\) on every finite horizon. A useful sufficient condition for the general stochastic exponential to be a true martingale on \([0,T]\) is Novikov's condition: \(\mathbb{E}\exp\!\left(\tfrac12\int_0^T\theta_s^2\,ds\right)<\infty\). Novikov's theorem is a standard result; its proof belongs with the later material on Girsanov's theorem. The condition is sufficient, not necessary. We will verify the constant-kernel case directly instead.

The Constant-Kernel Exponential

Proof. Take \(f(t,x)=\exp(\theta x-\tfrac12\theta^2t)\). Its derivatives are \(f_t=-\tfrac12\theta^2f\), \(f_x=\theta f\), and \(f_{xx}=\theta^2f\). Ito's formula, including its second-derivative correction, gives

Thus Ito's formula gives a local martingale. For the true martingale property, use the independent Brownian increment \(B_t-B_s\), which is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). The moment-generating function of this normal variable yields

This conditional expectation is well-defined: the same Gaussian calculation gives \(\mathbb{E}[Z_t]=1\), so \(Z_t\) is integrable. For the second moment, \(Z_t^2=\exp(2\theta B_t-\theta^2t)\), and the normal moment-generating function gives \(\mathbb{E}[Z_t^2]=e^{2\theta^2t-\theta^2t}=e^{\theta^2t}\). When \(\theta=0\), the formula itself gives \(Z_t=1\), and all the claims still hold. \(\square\)

Geometric Brownian Motion and Its Moments

Let \(S_0\ge0\), and let \(\mu,\sigma\in\mathbb{R}\) be constants. The geometric Brownian motion (GBM) with drift \(\mu\) and diffusion coefficient \(\sigma\) is the process solving \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\), with initial value \(S_0\). Here the drift is the coefficient of \(dt\), and the diffusion coefficient is the coefficient of \(dB_t\). Its explicit candidate is

Proof. First suppose \(S_0>0\), and apply Ito's formula to \(f(t,x)=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma x)\). The derivatives are \(f_t=(\mu-\tfrac12\sigma^2)f\), \(f_x=\sigma f\), and \(f_{xx}=\sigma^2f\). Hence

The Gaussian moment-generating function gives, for each real \(p\), \(\mathbb{E}[S_t^p]=S_0^p\exp((p\mu+\tfrac12p(p-1)\sigma^2)t)\) when \(S_0>0\). In particular, \(p=1\) and \(p=2\) give the stated moments. For \(0\le s\le t\), independence of \(B_t-B_s\) from \(\mathcal{F}_s\) gives \(\mathbb{E}[S_t\mid\mathcal{F}_s]=S_s e^{\mu(t-s)}\). The conditional expectation exists because the first moment is finite. If \(\mu=0\), this equals \(S_s\), proving the martingale property. If \(\mu\ne0\) and \(S_0>0\), then for any \(t>s\), \(e^{\mu(t-s)}\ne1\), so \(S_s e^{\mu(t-s)}\ne S_s\) almost surely; the martingale condition fails. If \(S_0=0\), the explicit solution is identically zero, including when \(\sigma=0\), and its conditional expectations are zero. \(\square\)

The moment formulas can also be checked using Ito's formula on powers of the process. For \(f(x)=x^2\), the derivatives are \(f'(x)=2x\) and \(f''(x)=2\); the correction term is therefore \(\tfrac12 f''(S_t)(dS_t)^2=\sigma^2S_t^2\,dt\). The resulting identity is

Its stochastic integral has zero mean: the integrand \(2\sigma S_t^2\) is progressively measurable and in \(\mathcal{L}^2\) on \([0,T]\), since \(\mathbb{E}\int_0^T4\sigma^2S_s^4\,ds<\infty\) by the Gaussian fourth-moment formula above. The Ito isometry then gives a square-integrable martingale with zero expectation. Taking expectations in the integral identity yields the moment equation corresponding to the displayed second-moment formula. This provides an Ito-based check independent of directly evaluating the second moment at the final time.

When Is the Process a Martingale?

The condition \(\mu=0\) concerns the process under the stated probability measure \(\mathbb{P}\). It is not a statement about a risk-neutral measure \(\mathbb{Q}\), nor does it assert that a discounted asset price is a martingale. If a constant interest rate \(r\) is introduced, then the conditional expectation of the discounted process is \(\mathbb{E}[e^{-rt}S_t\mid\mathcal{F}_s]=e^{-rs}S_s e^{(\mu-r)(t-s)}\). For \(S_0>0\), this is a martingale under \(\mathbb{P}\) exactly when \(\mu=r\). A change from \(\mathbb{P}\) to \(\mathbb{Q}\) is a separate operation and is not needed for these moment calculations.

Common Pitfall: Omitting the Ito Correction

A tempting but incorrect calculation is to differentiate \(S_0e^{\mu t+\sigma B_t}\) and conclude that it solves \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\). The missing term is the second derivative correction: for \(f(t,x)=S_0e^{\mu t+\sigma x}\), \(f_t=\mu f\), \(f_x=\sigma f\), and \(f_{xx}=\sigma^2f\), so Ito's formula gives drift \((\mu+\tfrac12\sigma^2)f\), not \(\mu f\). The correct exponent must subtract \(\tfrac12\sigma^2t\).

A second error is to infer that every positive local martingale is a true martingale. The local-martingale conclusion from Ito's formula alone does not establish integrability and the conditional-expectation identity. In the constant-kernel exponential, those properties were verified explicitly using the normal moment-generating function. For more general kernels, conditions such as Novikov's condition provide a sufficient route, but they are not automatic.