On the Markov property of some Brownian martingales

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