Constructive Probability, Foundations and Numerical Analysis for mathematical finance and machine learning

Abstract

This monograph sets out a constructive theory of stochastic processes in the continuum, in the sense of constructive field theory: the continuum is not postulated but reached as the limit of a sequence of lattice approximations, with the renormalisation the limit requires exhibited rather than assumed, and the limit controlled by estimates that can be checked. The primitive object is not a sigma-algebra but de Finetti's prevision, the price of finitely many gambles at a fixed resolution, which coherence turns into a conditional expectation. The theorem at the centre of the book, called plainly the fundamental theorem, is at once de Finetti's fundamental theorem of probability, the fundamental theorem of Bayesian statistics, and the one-period fundamental theorem of asset pricing; the financial vocabulary is a convenience only. The applications are two, mathematical finance and machine learning, and they are one mathematics read in opposite directions: an assessment is a price in the one and a prediction in the other, and both are computed by one adjoint recursion, the adjoint of a propagation in the first case and backpropagation in the second. Carried through, the two directions give sharp rates of convergence rather than asymptotic orders: for the moments of Ito diffusions, for the value function of Feynman-Kac semigroups, and for the measure changes of the fundamental theorem, the last as a limit theorem for reweighted generators, under a condition keeping the reweighting well behaved. A measure change is thus calculated, not postulated: a lattice approximation with Farkas' lemma computes it, completeness makes it unique, and the passage to the continuum is earned by the regularity of what is computed. Sixty-seven Python scripts regenerate every numerical table, and much of the finite-dimensional core of the argument has been verified in Lean 4.