  We introduce a theory of non-commutative $L^{p}$ spaces suitable for
non-commutative probability in a non-tracial setting and use it to develop
stochastic analysis of Grassmann-valued processes, including martingale
inequalities, stochastic integrals with respect to Grassmann It\^o processes,
Girsanov's formula and a weak formulation of Grassmann SDEs. We apply this new
setting to the construction of several unbounded random variables including a
Grassmann analog of the $\Phi^{4}_{2}$ Euclidean QFT in a bounded region and
weak solution to singular SPDEs in the spirit of the early work of Jona-Lasinio
and Mitter on the stochastic quantisation of $\Phi^{4}_{2}$.
