Topics in Probability, Parametric Estimation and Stochastic Calculus

TL;DR

This paper explores probability theory fundamentals, emphasizing geometric aspects and stochastic calculus, with applications to parametric estimation, concentration inequalities, and financial models like Black-Scholes.

Contribution

It uniquely integrates geometric probability perspectives with stochastic calculus, extending traditional expositions and illustrating applications in finance and measure concentration.

Findings

Derivation of Gaussian concentration inequality

Application of Feynman-Kac formula to heat kernels

Demonstration of Black-Scholes strategy in finance

Abstract

We begin our journey by recalling the fundamentals of Probability Theory that underlie one of its most significant applications to real-world problems: Parametric Estimation. Throughout the text, we systematically develop this theme by presenting and discussing the main tools it encompasses (concentration inequalities, limit theorems, confidence intervals, maximum likelihood, least squares, and hypothesis testing) always with an eye toward both their theoretical underpinnings and practical relevance. While our approach follows the broad contours of conventional expositions, we depart from tradition by consistently exploring the geometric aspects of probability, particularly the invariance properties of normally distributed random vectors. This geometric perspective is taken further in an extended appendix, where we introduce the rudiments of Brownian motion and the corresponding stochastic calculus, culminating in It\^o's celebrated change-of-variables formula. To highlight its scope and elegance, we present some of its most striking applications: the sharp Gaussian concentration inequality (a central example of the "concentration of measure phenomenon"), the Feynman-Kac formula (used to derive a path integral representation for the Laplacian heat kernel), and, as a concluding delicacy, the Black-Scholes strategy in Finance.