46 research outputs found
Functional Expansions
Path dependence is omnipresent in social science, engineering, 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
seriesthe 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.Comment: 39 pages, 7 figure