  We establish an explicit expression for the conditional Laplace transform of
the integrated Volterra Wishart process in terms of a certain resolvent of the
covariance function. The core ingredient is the derivation of the conditional
Laplace transform of general Gaussian processes in terms of Fredholm's
determinant and resolvent. Furthermore , we link the characteristic exponents
to a system of non-standard infinite dimensional matrix Riccati equations. This
leads to a second representation of the Laplace transform for a special case of
convolution kernel. In practice, we show that both representations can be
approximated by either closed form solutions of conventional Wishart
distributions or finite dimensional matrix Riccati equations stemming from
conventional linear-quadratic models. This allows fast pricing in a variety of
highly flexible models, ranging from bond pricing in quadratic short rate
models with rich autocorrelation structures, long range dependence and possible
default risk, to pricing basket options with covariance risk in multivariate
rough volatility models.
