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So, for problem 1b I noticed many of you had used a lot of words for your explanations. This is not quite rigorous enough in my opinion. I wish more people would do it properly with mathematics, instead of using a lot of words. This is not economics.
Stochastic Processes teaching assistant
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Wenyuan Li, Igor V. Ovchinnikov, Honglin Chen, Zhe Wang
The “supersymmetric theory of stochastics” (STS) programme I’ve mentioned before (introduction here) now includes a starting point for applications to neuroscience.
The authors use simulations on a neuromorphic chip and in vitro current-clamp recordings of brain slices to validate the generic rudimentary phase diagram for stochastic systems predicted by STS. This phase diagram divides the parameter space (including, crucially, the noise intensity) of a large class of stochastic systems into 1) a “T-phase” characterized by thermal equilibrium and ergodic dynamics, 2) a “C-phase” characterized by chaotic dynamics (which corresponds to ordinary deterministic chaos in the noise-free case), and 3) an intermediate “N-phase” of noise-induced chaos, which includes systems displaying self-organized criticality (with all its hallmarks, such power-law power spectra). The N-phase is characterized by the spontaneous breakdown of a supersymmetry (the relevant supersymmetry operator is the exterior derivative on the exterior algebra of differential forms on the system’s phase space), common to a large class of stochastic differential equations, in the system’s ground state. It is to this ”noise-induced chaos” phase that healthy conscious brains seem to belong.
A non-paywalled early version, which includes the simulations but not the in vitro recordings, is available here.
Richard Jordan, David Kinderlehrer, and Felix Otto
Abstract. The Fokker–Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker–Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker–Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker–Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.
Édgar Roldán, Izaak Neri, Meik Dörpinghaus, Heinrich Meyr, Frank Jülicher
We show that the steady-state entropy production rate of a stochastic process is inversely proportional to the minimal time needed to decide on the direction of the arrow of time. Here we apply Wald's sequential probability ratio test to optimally decide on the direction of time's arrow in stationary Markov processes. Furthermore the steady state entropy production rate can be estimated using mean first-passage times of suitable physical variables. We derive a first-passage time fluctuation theorem which implies that the decision time distributions for correct and wrong decisions are equal. Our results are illustrated by numerical simulations of two simple examples of nonequilibrium processes.
We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to:
--Write arbitrage as curvature of a principal fibre bundle.
--Parameterize arbitrage strategies by its holonomy.
--Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.
--Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance.
--Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where:
-->Arbitrage is allowed but minimized.
-->Arbitrage is not allowed.
--Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio Dynamics to be satisfied.
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I plan to waste your time by considering the history of nineteenth century work on Brownian motion in unnecessary detail. ... Studying the development of a topic in science can be instructive. One realizes what an essentially comic activity scientific investigation is (good as well as bad).
Edward Nelson, Dynamical Theories of Brownian Motion
Mathematician who tamed randomness wins Abel Prize
Michel Talagrand laid mathematical groundwork that has allowed others to tackle problems involving random processes. A mathematician who developed formulas to make random processes more predictable and helped to solve an iconic model of complex phenomena has won the 2024 Abel Prize, one of the field’s most coveted awards. Michel Talagrand received the prize for his “contributions to probability…
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After more than 10 years writing, I decide to add some music to my stuff. Not a profesional at all, not even musician. It was created with help of a mobile app.
