C Itô Calculus, Brownian Motion, and Ergodicity

This appendix collects the stochastic-calculus background needed for Chapters Chapter 7--Chapter 8. For the longer, example-driven exposition see Section 8.2; for a full textbook treatment, see Shreve (2004).

C.1Brownian motion¶

A standard Brownian motion $B_t$ has independent Gaussian increments $B_t - B_s \sim \mathcal{N}(0, t-s)$, continuous paths almost surely, and quadratic variation $\langle B\rangle_t = t$. Equivalently, the simple-random-walk scaling limit $X_{t+\Delta t} = X_t + \sqrt{\Delta t}\,\varepsilon_t$ converges in distribution to $B_t$ as $\Delta t \to 0$ (Donsker).

C.2Itô’s lemma¶

For $X_t$ following $dX_t = \mu\,dt + \sigma\,dB_t$ and $f \in C^2$:

The differential algebra is $dt\cdot dt = dt\cdot dB_t = 0$, $dB_t \cdot dB_t = dt$. (Throughout the script, the stochastic integral is interpreted in the Itô sense; the Stratonovich convention would replace the drift correction $\tfrac{1}{2}f''(X_t)\sigma^2$ by 0.)

C.3Ergodicity in one paragraph¶

A Markov process with stationary distribution $\pi$ is ergodic if the time-average along any sample path converges almost surely to the spatial average against $\pi$: $\tfrac{1}{T}\int_0^T \varphi(X_t)\,dt \xrightarrow{T\to\infty} \int \varphi\,d\pi$ for every bounded measurable $\varphi$. In economic models with bounded state spaces and aperiodic dynamics, ergodicity is what justifies replacing population integrals by simulation-time averages in DEQN training (Chapter Chapter 2, Section 2.3).