  Path dependence is omnipresent in many disciplines such as engineering,
system theory and finance. It reflects the influence of the past on the future,
often expressed through functionals. However, non-Markovian problems are often
infinite-dimensional, thus challenging from a conceptual and computational
perspective. In this work, we shed light on expansions of functionals. First,
we treat static expansions made around paths of fixed length and propose a
generalization of the Wiener series$-$the intrinsic value expansion (IVE). In
the dynamic case, we revisit the functional Taylor expansion (FTE). The latter
connects the functional It\^o calculus with the signature to quantify the
effect in a functional when a "perturbation" path is concatenated with the
source path. In particular, the FTE elegantly separates the functional from
future trajectories. The notions of real analyticity and radius of convergence
are also extended to the path space. We discuss other dynamic expansions
arising from Hilbert projections and the Wiener chaos, and finally show
financial applications of the FTE to the pricing and hedging of exotic
contingent claims.
