Hermite Martingales
### HERMITE MARTINGALES

(To appear in the *Séminaire de Probabilités*)

Following up on work of A. Plucinska [A stochastic characterization of Hermite
polynomials, *J. Math. Sci.***89** (1998) 1541-1544]
we examine the class of Brownian martingales of the form
a(t)h(B(t)/b(t)) where B(.) is a standard one-dimensional Brownian motion.
Apart from trivial cases, if a(t)h(B(t)/b(t)) is a martingale then a(t)=t^{p/2},
b(t)=t^{1/2}, and
h''-xh'+ph=0 (Hermite equation). Moreover, if the Brownian motion B(.) satisfies
P[B(0) = 0]
<1, then p must be a non-negative integer and h must be proportional to the Hermite polynomial of degree p.

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version of June 27, 2000