Week 1:Motivation, rationale& concept of path integrals; Elements of probability distributions, Moments &Cumulants& their generating functions; Binomial & Gaussian distributions, Laplace &Stirling formula.
Week 2:Gaussian integrals of matrix functions; The Central Limit Theorem; Elementary theory of stochastic processes. Chapman Kolmogorov equation, Master equation, Kramer Moyal equation, Fokker Planck equation; Random Walks & Brownian motion.
Week 3:Path integral solution of diffusion equation; Feynman Kac formula; Autocorrelators; Langevin equation; Path integral solution of Schrodinger equation &Langevin equation.
Planck vs Langevin equations; Determinism vs Classical & Quantum
randomness; Basic theory of QM path integral; Introduction to the QM machinery
& notational conventions; Equivalence of Schrodinger & Heisenberg
pictures; Single particle non-relativistic QM path integral.
Week 5:QM Harmonic Oscillator path integral; Equivalence between Schrodinger & path integral approaches; Correlation functions; Relevenace of functional derivatives.
Week 6:Relativistic path integral; Saddle point approach to path integral; Interpretation of QM path integral; Causality violation; Need for QFT.
Week 7:0-Dimensional Field Theory, Correlation Functions, Generating Functionals, Field interactions, Perturbative expansions, Convergence issues, Propagators, Feynman Rules, Schwinger-Dyson Equation.
from Feynman Diagrams, Loop expansions, Limiting behavior, Saddle point
approximation, Effective field,
Renormalization, 1-D field theory, Feynman Rules & SDE, propagator, SDE in
terms of propagators, Explicit expression for propagator, Generalization to
D-dimensions, Continuum limits.
Week 9:Mode space computations, divergences, loop integrals, Generatingfunctionals, correlators and allied results in D- Dimensional Euclidean space; Field theory in Minkowski space, Scalar field generating functional.
Week 10:Scalar field propagator & correlators, Fourier integrals, Causality, Interacting field in Minkowski space, Important identities, Derivation of SDE,Fermionic fields & their path integral quantization.
Week 11:Maxwell equations & gauge field quantization, their path integral. Fluctuation properties offinancial assets, Brownian motion & Ito’s lemma; Lognormal distribution & stock price distribution, Formulation of & solution to the Fokker Planck equation for stock price model.
Week 12:Basic theory of options; Profit diagrams; Put-Call parity; Option pricing, the Binomial &Black Scholes Models;Risk Neutral valuation; American options; Derivation of and solution to the Black-Scholes PDE; Path integral approach to pricing of path independent options; Path integral valauation of other cash flow structures e.g. zero coupon bonds etc..