Probability Measures, Stochastic Processes, and the Martingales¶

Job: develop the probability space / Kolmogorov extension / Wiener measure / Brownian, the stochastic process / Markov / martingale, the Itô integral / formula / SDE / quadratic variation, the infinite-dim / rough paths / SPDE, and the law / characteristic / large deviations — with the Wiener measure via Kolmogorov / the Itô integral as the worked example.

Core¶

Probability measures and the Kolmogorov extension¶

Probability / Kolmogorov / Wiener	Description
Probability space $(\Omega,\mathcal{F},\mathbb{P})$	the measure-theoretic probability ($\mathbb{P}(\Omega)=1$)
Kolmogorov extension	the consistent finite-dimensional distributions $\Rightarrow$ the measure on the product $\prod X_i$ (the Polish)
Wiener / Brownian	the Kolmogorov the consistent Gaussians $\Rightarrow$ the Wiener measure on $C([0,\infty))$ (the Brownian motion, the Kolmogorov continuity)

Stochastic processes and the Markov / the martingale¶

Stochastic / Markov / martingale	Description
Stochastic process $(X_t)$	$X_t:\Omega\to E$ (the process, the index $T$)
Markov process	$\mathbb{P}(X_{t+1}\in A
Martingale	$\mathbb{E}[X_{t+1}

Itô calculus and the stochastic integration¶

Itô / formula / SDE / quadratic variation	Description
Itô integral $\int f\,dW$	the stochastic integral against the Brownian (the adapted $f$, the $L^2$-isometry)
Itô formula	$df=\partial_t f\,dt+\partial_x f\,dW+\frac12\partial_{xx}f\,dt$ (the $\langle dW\,dW\rangle=dt$ the quadratic variation)
SDE	$dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t$ (the existence / the uniqueness, the SPDE / foundations)

The stochastic processes and the SPDE / the rough paths¶

Infinite-dim / rough paths / SPDE	Description
Infinite-dim	$X_t\in H$ the Hilbert-valued (the cylindrical Wiener, the SPDE / foundations)
Rough paths (Lyons)	the rough path / the signature (the enhanced Brownian, the rough paths / foundations)
Stochastic analysis / SPDE	the stochastic analysis the SPDE (the Itô / the rough paths, the infinite-dim)

The measure-theoretic probability and the foundations¶

Law / characteristic / large deviations	Description
Law / distribution	$\mu_X=\mathbb{P}\circ X^{-1}$ (the pushforward, the law of $X$)
Characteristic / Fourier	$\hat\mu(\xi)=\int e^{i\xi x}\,d\mu$ (the characteristic function, the fourier-analysis)
Large deviations	the Cramér / the Sanov (the information theory the rate function)

See it¶

Wiener measure via the Kolmogorov and the Itô integral. The Wiener measure: the Kolmogorov extension (the consistent finite-dim Gaussians $\mathbb{P}(W_{t_1}\in A_1,\dots,W_{t_n}\in A_n)$ the Gaussian, the $\Rightarrow$ the measure on $\mathbb{R}^{[0,\infty)}$, the Kolmogorov), the Kolmogorov continuity (the $\mathbb{E}|W_t-W_s|^\alpha\le C|t-s|^{\alpha/2}$ the Hölder, the $\Rightarrow$ the $C([0,\infty))$ the continuous modification, the Wiener measure on $C$). The Itô integral: $\int f\,dW$ (the adapted $f$, the $L^2$-isometry $\mathbb{E}(\int f\,dW)^2=\int\mathbb{E}f^2\,dt$, the Itô). The Itô formula: $d(W_t^2)=2W_t\,dW_t+dt$ (the $\langle dW\,dW\rangle=dt$ the quadratic variation, the Itô the non-classical). The key point: the Wiener measure the Kolmogorov (the consistent Gaussians / the continuity), the Itô the integral / the formula — the measure-theoretic probability / the stochastic the Brownian.

Check¶

Show that the Kolmogorov extension + the Kolmogorov continuity give the Wiener measure on $C([0,\infty))$, and explain the consistency.

The Kolmogorov extension: the consistent finite-dimensional distributions $\mathbb{P}(W_{t_1}\in A_1,\dots,W_{t_n}\in A_n)$ (the Gaussian: the $W_{t_i}-W_{t_{i-1}}$ the independent Gaussians $\mathcal{N}(0,t_i-t_{i-1})$, the consistent the Kolmogorov). The Kolmogorov extension theorem: the consistent finite-dim distributions $\Rightarrow$ the measure on the product $\mathbb{R}^{[0,\infty)}$ (the Polish, the topological). The Kolmogorov continuity: $\mathbb{E}|W_t-W_s|^\alpha\le C|t-s|^{\alpha/2}$ (the Gaussian the moments, the $\alpha=2+\delta$ the Hölder), the $\Rightarrow$ the continuous modification (the Kolmogorov continuity theorem, the $C([0,\infty))$ the continuous paths). The Wiener measure: the measure on $\mathbb{R}^{[0,\infty)}$ restricted to $C([0,\infty))$ (the continuous modification, the Wiener measure on $C$). The consistency: the finite-dim the Gaussian (the independent increments, the consistent), the Kolmogorov the extension (the product measure), the continuity the $C$ (the Wiener).

Explain the Itô formula $d(W_t^2)=2W_t\,dW_t+dt$ and the quadratic variation, and contrast with the classical chain rule.

The Itô formula: for $f(t,W_t)=W_t^2$, the $df=\partial_t f\,dt+\partial_x f\,dW+\frac12\partial_{xx}f\,dt$. The $\partial_t f=0$ (no $t$-dependence), $\partial_x f=2W_t$, $\partial_{xx}f=2$. So $d(W_t^2)=2W_t\,dW_t+\frac12\cdot2\,dt=2W_t\,dW_t+dt$ (the Itô formula). The $dt$ the quadratic variation: the $\langle dW\,dW\rangle=dt$ (the Brownian the quadratic variation the $dt$, the $(dW)^2=dt$ the Itô the non-classical). The classical chain rule: $d(W_t^2)=2W_t\,dW_t$ (the classical, the $(dW)^2=0$, the no quadratic variation). The contrast: the Itô the $+dt$ (the quadratic variation, the non-classical), the classical the $0$ (the no quadratic variation). The Itô the non-classical (the $\langle dW\,dW\rangle=dt$ the Brownian the non-smooth, the quadratic variation the $dt$), the classical the smooth (the $(dW)^2=0$ the smooth, the no quadratic variation).

Pitfalls¶

The Kolmogorov the extension (the Polish the topological, the consistency) — the consistent finite-dim the product measure, the Polish the topological (the Kolmogorov the extension theorem the Polish).
The Itô the integral (the adapted, the $L^2$-isometry, the Itô formula the non-classical the $\langle dW\,dW\rangle=dt$) — the adapted the $f$, the $L^2$ the isometry, the Itô the non-classical (the quadratic variation the $dt$).
The rough paths the Lyons (the enhanced Brownian, the signature, the rough paths) — the rough path the signature (the enhanced Brownian, the iterated integrals, the Lyons).
The large deviations the Cramér / the Sanov (the information theory the rate function) — the Cramér the $\mathbb{P}(X_n\approx x)\sim e^{-nI(x)}$ the rate, the Sanov the empirical measure.
The Wiener measure the Kolmogorov (the consistent Gaussians / the continuity), the Itô the integral / the formula — the measure-theoretic the probability, the stochastic the Brownian.

Refs¶

Kolmogorov, Foundations of the Theory of Probability (1933); the extension theorem
Itô, "Stochastic integral" (1944); the Itô formula / the quadratic variation
Revuz–Yor, Continuous Martingales and Brownian Motion; the Wiener measure / the stochastic
Lyons, "Differential equations driven by rough signals" (1998); the rough paths / the signature

Next¶

MT index · Foundations · Geometric Measure Theory · Ergodic & Recurrence · Harmonic & Singular Integrals · Stochastic processes · SPDE · Probability