Kalle Törmänen

Kalle Törmänen completed his doctoral studies in the Faculty of Physics at Saint Petersburg State University, where his research examined the mathematical structures shared by quantum mechanics and the theory of stochastic processes. The proximity of Helsinki to St. Petersburg, and the long history of scientific exchange across the Finnish-Russian border, shaped a research environment that treated theoretical physics and probability theory as branches of the same analytical tradition. His dissertation developed a rigorous functional-analytic treatment of Feynman path integrals — the formulation in which quantum mechanical amplitudes are expressed as integrals over spaces of continuous trajectories weighted by the exponential of the classical action — and established conditions under which these path integrals admit a precise measure-theoretic interpretation. The central result of the thesis was a derivation of the Feynman–Kac formula directly from the operator-theoretic structure of quantum mechanics: under imaginary-time continuation, the unitary evolution operator of quantum mechanics becomes a semigroup generated by a Schrödinger-type operator, and the path integral representation of this semigroup is the expectation of a functional of Brownian motion. Törmänen showed that this correspondence extends to a broad class of Hamiltonians with singular or state-dependent potentials — precisely the class arising in the study of diffusion processes in mathematical finance.

Following his doctorate, Törmänen developed the formal analogy between the quantum Hamiltonian and the Hamiltonian of the Hamilton–Jacobi–Bellman equation in stochastic optimal control. Under the substitution that replaces the imaginary time of quantum mechanics with real time and Planck's constant with the diffusion coefficient, the Schrödinger equation becomes a parabolic PDE of Black–Scholes type, and the path integral representation of its solution becomes the risk-neutral pricing formula for contingent claims. He exploited this correspondence to import spectral methods from quantum mechanics — eigenfunction expansions of Schrödinger operators, resolvent estimates, and analytic perturbation theory — into the study of short-rate models and stochastic control problems in finance. His research in this period established new analytical results for short-rate models with state-dependent volatility, characterising the term structure of interest rates as a spectral problem for a Schrödinger-type differential operator on the half-line, and deriving asymptotic expansions for bond prices that go beyond the affine class without sacrificing analytical tractability.

At IADU, Törmänen applies these frameworks to central bank policy problems — forward guidance under stochastic short-rate dynamics, optimal intervention thresholds as free boundaries of singular control problems calibrated to ECB and Riksbank policy episodes — and to mean field game models of interbank lending markets, where each institution optimises its reserve position against the aggregate liquidity distribution. The resulting coupled HJB–Fokker–Planck systems are calibrated to EONIA and SOFR clearing data and used to derive quantitative benchmarks for macroprudential capital buffer design. His work within the Institute is distinguished by its insistence on grounding financial mathematics in the structural principles of theoretical physics — a perspective that systematically reveals the spectral, variational, and path-integral architecture underlying models that are typically approached through purely probabilistic methods.