How to Test an Exponential Brownian Process for the Martingale Property

For a standard Brownian motion

W

in its natural filtration and real constants

a,b

, the process

M_t=e^{aW_t+bt}

is a martingale exactly when

b=-a^2/2

. Factor the exponential into its value at the conditioning time and a term involving the independent future increment. Then use the moment-generating function of a normal random variable to test whether the expected multiplier is one.

Factor out what is already known at time s

For $s<t$ , rewrite the candidate as

M_t=M_s\exp\!\left(a(W_t-W_s)+b(t-s)\right).

The past determines

M_s

, and

W_t-W_s

is independent of that past with variance

t-s

. If

Z

has a centred Gaussian law with variance

v

\mathbb E[e^{aZ}]=e^{a^2v/2}

. Thus

\mathbb E[M_t\mid\mathcal F_s] =M_s e^{(b+a^2/2)(t-s)}.

The process is adapted because its value only uses the Brownian value at time

t

. Its finite-time expectation is finite by the Gaussian moment formula. With adaptation and integrability checked, the conditional equality decides the classification. A positive exponent coefficient gives a submartingale, and a negative coefficient gives a supermartingale.

Itô's lemma supplies an independent check. It gives

dM_t=(b+a^2/2)M_t\,dt+aM_t\,dW_t

. For constant

a,b

on a finite horizon, the diffusion is square integrable because the exponential of a Gaussian has finite second moment. That fact makes the drift-free Itô integral a true martingale.

The proof needs adaptedness and integrability before the conditional calculation counts. The exponential is adapted to the Brownian filtration. Its first moment is finite at every finite time by the normal moment formula. Then take the past-measurable factor outside the conditional expectation. The remaining factor has mean

e^{(b+a^2/2)(t-s)}

. Comparing this positive multiplier with one proves the martingale, submartingale or supermartingale result directly. A zero drift calculation alone would only give a local statement until integrability is checked.

Test two choices of drift with the same noise

For an original exercise, compare $M_t=e^{0.7W_t-0.245t}$ with $N_t=e^{0.7W_t+0.10t}$ . Since $0.7^2/2=0.245$ , $M$ is a martingale. The second process is a submartingale because its mean multiplier from year 1 to year 2 is $e^{0.345}\approx1.412$ . At a time when $N_1=3$ , the conditional mean of $N_2$ is about $4.24$ . The number $0.10$ in the exponent therefore understates the growth of the expected process.

For $M$ in this example, the first moment stays at $1$ while the second moment grows as $e^{0.49t}$ . This follows from applying the same Gaussian moment formula with loading $1.4$ to $M_t^2$ . A flat conditional mean is compatible with increasing dispersion over time.

How to Test an Exponential Brownian Process for the Martingale Property

Factor out what is already known at time s

Test two choices of drift with the same noise

Turn a fair random walk into a biased one

References and next steps