On the Markov property of some Brownian martingales

Let h n be the (probabilists’) Hermite polynomial of degree n . Let H n ( z , a ) = a n / 2 h n ( z / a ) and H n ( z , 0 ) = z n . It is well-known that H n ( B t , t ) is a martingale for every n . In this paper, we show that for n ≥ 3 , H n ( B t , t ) is not Markovian. We then give a brief discussion on mimicking H n ( B t , t ) in the sense of constructing martingales whose marginal distributions match those of H n ( B t , t ) .