Stochastic Calculus
1 Overview
Stochastic calculus is the branch of mathematics that extends classical calculus to stochastic (random) processes. Where classical calculus deals with deterministic functions and their derivatives/integrals, stochastic calculus handles processes whose evolution involves randomness. The key insight is that stochastic processes like Brownian motion have infinite variation, requiring new integration and differentiation rules—particularly the Itô integral and Itô formula.
2 Fundamental Areas
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- [[measure-theoretic-probability]] - Probability spaces, filtrations, adapted processes, information flow
- Brownian Motion - The canonical stochastic process (continuous-time random walk)
- Martingale Theory - Fair-game processes fundamental to stochastic modeling
- Stopping Times - Random times adapted to the filtration
Quadratic Variation — How stochastic processes differ from classical ones. Brownian motion has infinite total variation but finite quadratic variation, requiring new calculus rules.
Itô Integral — The stochastic version of the Riemann integral. Integration with respect to Brownian motion and other martingales. Properties (linearity, martingale property, Itô isometry).
Itô Calculus (Change of Variables) — The stochastic generalization of the chain rule
- Itô Process and Itô Formula (1D and multivariate) — how functions of stochastic processes evolve
- Stochastic Integration By Parts — product rules for stochastic processes
- Quadratic covariation — the extra terms that appear in stochastic calculus
Stochastic Differential Equations — Definition and theory
- Existence and uniqueness of solutions (Picard-Lindelöf conditions: growth and Lipschitz)
- Strong vs weak solutions
- Markov property and time-homogeneity of solutions
SDE Solution Methods & Examples — Specific important processes
- Geometric Brownian Motion — Exponential growth with random noise (stock prices)
- Ornstein-Uhlenbeck Process — Mean-reverting processes (interest rates, commodities)
- Brownian Bridge — Brownian motion conditioned to return to a fixed point
- Linear SDEs and their explicit solutions
Diffusion Processes — Time-homogeneous SDEs and their properties
- Itô diffusions: definition and key properties (flow property, Markov property)
- Continuous sample paths and regularity
Infinitesimal Generator — Operator associated with a diffusion
- Generator and semi-group theory
- [[dynkins-formula]] — relating expectations to the generator (martingale characterization)
- Kolmogorov backward and forward equations
Advanced Results — Tools for solving and transforming SDEs
- Girsanov Theorem — Change of measure and drift transformation
- Feynman-Kac Formula — Connection between SDEs and PDEs
- Fokker-Planck Equation / Kolmogorov Equations — Evolution of probability densities
- Martingale Representation Theorem — Every martingale is an Itô integral