  For a non-zero parameter $q$, we define Hermite trace polynomials, which are
multivariate polynomials indexed by permutations. We prove several
combinatorial properties for them, such as expansions and product formulas. The
linear functional determined by these trace polynomials is a state for $q =
\frac{1}{N}$ for $N$ a non-zero integer. For such $q$, Hermite trace
polynomials of different degrees are orthogonal. The product formulas extend to
the closure with respect to the state. The state can be identified with the
expectation induced by the $N \times N$ Hermitian Brownian motion. Hermite
trace polynomials are martingales for this Brownian motion, while the elements
in the closure can be interpreted as stochastic integrals with respect to it.
Using the grading on the algebra, we prove several chaos decompositions for
such integrals, as well as analyze corresponding creation and annihilation
operators. In the univariate, pure trace polynomial case, trace Hermite
polynomials can be identified with the Hermite polynomials of matrix argument.
