Gaussian Exponential Martingales for Deterministic Integrands

From a Gaussian Integral to a Martingale

Exponential processes of Brownian integrals appear in change-of-measure arguments and in the construction of stochastic models. Their exponential form alone does not guarantee that a local martingale is a true martingale. For a deterministic integrand, however, the integral is Gaussian, so its distribution and conditional expectations can be computed directly. This gives a complete martingale proof without using Novikov's criterion.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) with respect to the filtration: in particular, \(B_t-B_s\) is independent of \(\mathcal{F}_s\) when \(s<t\). Fix a finite horizon \(T\), and let \(\theta:[0,T]\to\mathbb{R}\) be deterministic and measurable with \(\int_0^T\theta_s^2\,ds<\infty\). Then \(\theta\) is progressively measurable, and it belongs to \(\mathcal{L}^2\), the class of progressively measurable processes \(H\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Thus the Ito integral below is well-defined.

The Gaussian Law and Conditional Distribution

Proof. First suppose \(\theta\) is a deterministic step function on \([0,t]\). Its integral is a finite sum of constants times Brownian increments. Those increments are independent centered normal variables, so the sum is centered normal, and its variance is the sum of the squared constants times the interval lengths, namely \(A_t\). On \([s,t]\), the same argument gives variance \(A_t-A_s\); all increments used lie after \(s\), so this integral is independent of \(\mathcal{F}_s\).

For general deterministic \(\theta\in L^2([0,T])\), choose deterministic step functions converging to \(\theta\) in \(L^2([0,T])\). By the Ito isometry, their integrals converge to \(M_t\) in \(L^2(\mathbb{P})\), hence in probability. Their variances converge to \(A_t\). Their characteristic functions therefore converge to \(u\mapsto\exp(-u^2A_t/2)\), the characteristic function of \(N(0,A_t)\); convergence in probability also gives convergence of characteristic functions to that of \(M_t\). This proves the asserted Gaussian law. Applying the same approximation to the integral over \([s,t]\) proves its Gaussian law. Independence from \(\mathcal{F}_s\) passes to the limit: for any bounded \(\mathcal{F}_s\)-measurable random variable \(Y\), convergence in probability of the integrals implies convergence of expectations against bounded continuous functions, preserving the factorization that expresses independence. Finally, \(\log Z_t=M_t-A_t/2\), so the stated distribution follows, including the degenerate case \(A_t=0\). \(\square\)

The same calculation gives useful moments. For every real \(p\), the Gaussian moment-generating function yields \(\mathbb{E}[Z_t^p]=\exp\big(\tfrac12(p^2-p)A_t\big)\). In particular, \(Z_t\) is integrable, \(\mathbb{E}[Z_t]=1\), and \(\mathbb{E}[Z_t^2]=e^{A_t}\). The conditional distribution is just as explicit: given \(\mathcal{F}_s\), the ratio \(Z_t/Z_s\) is independent of \(\mathcal{F}_s\) and has the distribution of \(\exp(G-\tfrac12(A_t-A_s))\), where \(G\sim N(0,A_t-A_s)\).

Worked Distributions

Why the Exponential Is a True Martingale

Proof. The identity \(M_t=M_s+\int_s^t\theta_u\,dB_u\) and additivity of the deterministic integral \(A_t=A_s+(A_t-A_s)\) give \(Z_t=Z_s\exp(\int_s^t\theta_u\,dB_u-\tfrac12(A_t-A_s))\). The second factor is independent of \(\mathcal{F}_s\), and the Gaussian law proved above gives its expectation as \(\exp(-\tfrac12(A_t-A_s))\exp(\tfrac12(A_t-A_s))=1\). Because \(Z_s\) is \(\mathcal{F}_s\)-measurable and integrable, conditional expectation yields \(\mathbb{E}[Z_t\mid\mathcal{F}_s]=Z_s\). The moment formula gives \(\mathbb{E}[|Z_t|]=1<\infty\) for each \(t\), so this is a martingale, not merely a local martingale. The conditional-expectation identity also covers \(s=t\), and if \(A_t=0\), then \(Z_t=1\) almost surely. \(\square\)

Ito's formula independently identifies the stochastic differential and shows where the correction comes from. For \(f(m,a)=e^{m-a/2}\), the partial derivatives are \(f_m=f\), \(f_a=-\tfrac12f\), and \(f_{mm}=f\). Since \(dM_t=\theta_t\,dB_t\), \(d[M]_t=\theta_t^2\,dt\), and \(dA_t=\theta_t^2\,dt\), Ito's formula, including its second-derivative correction, gives

The cancellation uses \(dB_t\,dB_t=dt\), while \(dB_t\,dt=0\) and \(dt\,dt=0\). This Ito calculation by itself identifies a local-martingale form; the conditional Gaussian argument above is what proves the true martingale property. In fact, the stochastic integral in the differential is square-integrable here: since \(\mathbb{E}[Z_s^2]=e^{A_s}\le e^{A_T}\),

Thus its integrand belongs to \(\mathcal{L}^2\); in particular the Ito integral is a true martingale by the Ito isometry and the conditional-mean-zero property of square-integrable Ito integrals. The direct conditional calculation remains the explicit verification of the desired identity.

A Common Pitfall: Dropping the Correction

For \(\theta_s=1\) on \([0,1]\), the incorrect expression \(e^{B_1}\) has mean \(e^{1/2}\), not \(1\): the normal moment-generating function gives \(\mathbb{E}[e^{B_1}]=e^{1/2}\). The correct exponential is \(Z_1=e^{B_1-1/2}\), whose mean is \(e^{-1/2}e^{1/2}=1\). The term \(-\tfrac12A_t\) is not optional; it cancels the Gaussian exponential moment. It is also precisely the drift cancellation in Ito's formula. Replacing Ito integration with a Stratonovich integral or applying the ordinary chain rule would change the calculation.

For deterministic \(\theta\), the Gaussian argument establishes a true martingale on every finite horizon on which \(\theta\) is square-integrable. This conclusion depends on the deterministic-integrand assumption: it is the independent Gaussian increment calculation that directly controls the conditional expectation. For general random integrands, such a calculation is not available, and a local-martingale calculation alone does not establish a true martingale. Novikov's criterion gives a sufficient condition for a broader setting.