Stochastic Processes¶
Job: map filtrations, martingales, Markov/BM structure, and Itô calculus as the probabilistic engine for continuous-time modeling and SPDE.
Core¶
A stochastic process is a family \((X_t)_{t\in T}\) of r.v.s on \((\Omega,\mathcal{F},\mathbb{P})\), usually \(T=\mathbb{R}_+\) or \(\mathbb{N}\), with values in a Polish space. The research objects are path regularity, adaptedness to filtrations, and weak/strong solutions of SDEs.
| Layer | Content |
|---|---|
| Filtrations & stopping | information flow; optional sampling |
| Martingales | fair games; Doob theory; convergence |
| Brownian motion | continuous martingale with quadratic variation \(t\) |
| Markov / Feller | transition semigroups; generators |
| Itô calculus | stochastic integrals; SDEs; Itô formula |
| SPDE interface | mild solutions, noise regularity |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | filtrations, stopping times, martingales, BM, Markov, Itô setup, SDE Lipschitz existence, generators |
| Martingales | OST, convergence, Doob inequalities, applications |
Dependency order¶
- Probability foundations (cond. expectation, modes of convergence)
- Measure theory (integration, product measures)
- This section: filtrations \(\to\) martingales \(\to\) BM \(\to\) Itô \(\to\) SDE
- SPDE for infinite-dimensional noise
- PDE / functional analysis for generators and mild solutions
Where it appears¶
- Mathematical finance (arbitrage \(\Leftrightarrow\) martingale measures)
- Filtering / control (Kalman–Bucy; HJB with generators)
- Interacting particle systems \(\to\) McKean–Vlasov / SPDE limits
- Stochastic quantization and Euclidean QFT (pointer via SPDE)
See it¶
Lévy characterization. Continuous local martingale \(M\) with \(\langle M\rangle_t=t\) is Brownian motion. This turns “identify quadratic variation” into “prove \(M\) is BM” — standard in SDE uniqueness proofs and Dambis–Dubins–Schwarz time-change.
Check¶
State the difference between a strong and weak solution of an SDE \(dX=b\,dt+\sigma\,dW\).
Strong: \(X\) adapted to a given filtration of a given \(W\), pathwise construction. Weak: existence of some probability space carrying a BM \(W\) and an adapted \(X\) satisfying the integral equation in law. Weak uniqueness \(\neq\) strong uniqueness (e.g. Tanaka equation).
Why does progressive measurability matter for \(\int H\,dW\)?
Progressive (or predictable) integrands make the integral a well-defined adapted process with martingale properties; mere adaptedness + joint measurability is not enough for the standard construction without progressive modification.
Pitfalls¶
- Version vs process: path properties (continuity, cadlag) are properties of a modification; always fix a regular version when quoting pathwise theorems.
- Filtration completion / right-continuity: usual conditions \((\mathcal{F}_{t+}=\mathcal{F}_t\), completion) are not free — they matter for debut theorems and optional sampling.
- Local martingale \(\neq\) martingale: UI or \(L^1\)-boundedness needed to upgrade; finance “bubble” examples are local but not true martingales.
Refs¶
Karatzas–Shreve, Brownian Motion and Stochastic Calculus; Revuz–Yor; Øksendal; Protter, Stochastic Integration and Differential Equations.
Next¶
Foundations · Martingales · Probability · SPDE · PDE · Functional analysis