Polynomial diffusions and applications in finance

We thank Mykhaylo Shkolnikov for suggesting a way to improve an earlier version of this result.

For geometric Brownian motion, there is a more fundamental reason to expect that uniqueness cannot be proved via the moment problem: it is well known that the lognormal distribution is not determined by its moments; see Heyde [29]. It thus becomes natural to pose the following question: Can one find a process  \(Y\) , essentially different from geometric Brownian motion, such that all joint moments of all finite-dimensional marginal distributions, coincide with those of geometric Brownian motion? We have not been able to exhibit such a process. Note that any such \(Y\) must possess a continuous version. Indeed, the known formulas for the moments of the lognormal distribution imply that for each \(T\ge0\), there is a constant \(c=c(T)\) such that \({\mathbb {E}}[(Y_{t}-Y_{s})^{4}] \le c(t-s)^{2}\) for all \(s\le t\le T, |t-s|\le1\), whence Kolmogorov’s continuity lemma implies that \(Y\) has a continuous version; see Rogers and Williams [42, Theorem I.25.2].

Note that unlike many other results in that paper, Proposition 2 in Bakry and Émery [4] does not require \(\widehat{\mathcal {G}}\) to leave \(C^{\infty}_{c}(E_{0})\) invariant, and is thus applicable in our setting.

Details regarding stochastic calculus on stochastic intervals are available in Maisonneuve [36]; see also Mayerhofer et al. [37], Carr et al. [7], Larsson and Ruf [34].

A matrix \(A\) is called strictly diagonally dominant if \(|A_{ii}|>\sum_{j\ne i}|A_{ij}|\) for all \(i\); see Horn and Johnson [30, Definition 6.1.9].