Stochastic process Study Guide

📖 Core Concepts Stochastic process – A family $\{Xt\}{t\in T}$ of random variables indexed by a set $T$ (usually time). Index set – The “time” labels; countable (discrete‑time) or uncountable (continuous‑time). State space – Common range of each $Xt$ (e.g., $\mathbb Z$, $\mathbb R$, $\mathbb R^d$). Sample function (realization) – One concrete path $t\mapsto Xt(\omega)$ obtained by fixing the outcome $\omega$. Increment – Difference between two times: $\Delta X{s,t}=Xt-Xs$. Filtration $\{\mathcal Ft\}$ – Growing sigma‑algebras that encode all information available up to time $t$. 📌 Must Remember Discrete‑time vs. continuous‑time – Countable vs. uncountable index set. Bernoulli process – i.i.d. $Bi\sim\text{Bernoulli}(p)$, values $0$ or $1$. Simple random walk – $Sn=S{n-1}+Yn$, $Yn\in\{+1,-1\}$ with $P(Yn=+1)=p$. Wiener (Brownian) motion – $W0=0$, continuous paths, $Wt-Ws\sim N(0,t-s)$, independent & stationary increments. Poisson process – $N(t)$ counts events; $N(t)\sim\text{Poisson}(\lambda t)$, independent increments. Markov property – $P(X{t{n+1}}\in A\mid X{tn},\dots,X{t0})=P(X{t{n+1}}\in A\mid X{tn})$. Martingale – $E[Mt\mid\mathcal Fs]=Ms$ for all $s<t$. Lévy process – Stationary independent increments, $X0=0$ (includes Wiener & Poisson). Finite‑dimensional distribution – Joint law of $(X{t1},\dots,X{tn})$. 🔄 Key Processes Constructing a Bernoulli process → generate i.i.d. $Bi\sim\text{Bernoulli}(p)$. Simple random walk step → $S{n}=S{n-1}+Yn$, update $Yn$ each trial. Simulating Wiener motion (Euler discretisation) Set $W0=0$. For small $\Delta t$, $W{t+\Delta t}=Wt+\sqrt{\Delta t}\,Z$, $Z\sim N(0,1)$. Poisson counting → for each small interval $\Delta t$, add 1 with prob. $\lambda\Delta t$, else 0 (Poisson thinning). Martingale verification → check $E[X{t+1}\mid\mathcal Ft]=Xt$ (e.g., fair gambler’s game). 🔍 Key Comparisons Discrete‑time vs. Continuous‑time Discrete: $t\in\mathbb N$, paths are sequences. Continuous: $t\in[0,\infty)$, paths can be continuous (Brownian) or have jumps (Poisson). Markov chain vs. Markov process Chain: discrete state space or discrete time. Process: may have continuous state space or continuous time. Martingale vs. Sub‑/Super‑martingale Martingale: $E[M{t}\mid\mathcal Fs]=Ms$. Sub‑martingale: $E[M{t}\mid\mathcal Fs]\ge Ms$. Super‑martingale: $E[M{t}\mid\mathcal Fs]\le Ms$. Lévy process vs. General stochastic process Lévy: stationary and independent increments, starts at 0. General: may lack one or both properties. ⚠️ Common Misunderstandings “Stationary” ≠ “Independent” – A stationary process has time‑invariant distributions, but increments can still be dependent. Brownian motion is not a deterministic straight line – Its paths are almost surely nowhere differentiable. Martingale does NOT mean “no variance” – Martingales can have large fluctuations; only the conditional mean is preserved. Poisson process increments are not always integer‑valued differences – The count itself is integer, but the increment $N(t)-N(s)$ is also integer‑valued (difference of counts). 🧠 Mental Models / Intuition Random walk = “drunkard’s steps” – Each step forgets the past; the position after many steps behaves like a scaled normal distribution (central limit). Brownian motion = “continuous drunkard” – Infinitely many infinitesimal steps → smooth, but highly jagged paths. Filtration = “knowledge horizon” – At time $t$ you only know events up to $t$, nothing about the future. Martingale = “fair betting game” – No strategy can systematically increase expected wealth. 🚩 Exceptions & Edge Cases Different modifications of the same process – Poisson process can be defined with right‑continuous or left‑continuous sample paths; they share finite‑dimensional distributions but are distinct processes. Non‑unique construction – Same finite‑dimensional distributions may correspond to multiple processes unless separability/regularity is imposed. Stationarity of higher‑order moments – A weakly stationary process has constant mean & autocovariance; full stationarity requires invariance of all finite‑dimensional distributions. 📍 When to Use Which Modeling counts of events → Poisson process (or renewal process if inter‑arrival distribution ≠ exponential). Modeling continuous price paths → Geometric Brownian motion (Black‑Scholes) or Lévy jump models if jumps are essential. Analyzing long‑run behavior → Markov chain ergodicity and stationary distribution. Proving limit theorems → Martingale convergence theorems (bounded $L^1$ or $L^2$). Describing spatial randomness → Random field notation $X{\mathbf s}$ for $\mathbf s\in\mathbb R^d$. 👀 Patterns to Recognize Independent stationary increments → immediately signals a Lévy process (check for Poisson or Wiener). Conditional expectation equal to current value → spot a martingale. Transition probabilities depending only on current state → Markov property. Poisson counts with mean $\lambda t$ → linear growth of expectation; variance equals mean. Gaussian finite‑dimensional distributions → Gaussian process. 🗂️ Exam Traps “Stationary process” vs. “process with stationary increments” – The latter (e.g., Wiener) is not stationary because the marginal distribution changes with time. Confusing a Markov chain with any Markov process – a chain must have a discrete index set or discrete state space; continuous‑time chains require a generator matrix, not just a transition matrix. Assuming every Lévy process is a pure jump process – Wiener motion is a Lévy process with continuous paths. Believing a martingale must have zero drift – In continuous time, a drift term can be cancelled under a risk‑neutral measure; the martingale property is about conditional expectation, not the deterministic trend. Mixing up “independent increments” with “independent random variables” – Independence is only required for non‑overlapping increments, not for the whole collection. --- Use this guide to quiz yourself: write the definition, draw a sample path, and decide which family a new process belongs to before the exam!