Part VIII — Stochastic Processes

Part VII let the sample size run on independent draws and found that every theorem it proved supplies a limit and schedules nothing. This part indexes by time, and the first thing that happens is that the sample size stops being a number you can count. Once observations are dependent, \(n\) rows of data are worth some smaller number of independent ones, the ratio is a property of the model rather than of the file, and every standard error, every \(t\)-statistic and every confidence interval downstream inherits it. What is gained in exchange is the ability to say something about when — how long a state persists, how long until a barrier is touched, how much of a track record is above water — and each of those quantities turns out to have a closed form that is nothing like what the eye expects.

The dependencies run in file order with two things worth knowing. Random Processes supplies the stationarity, ergodicity and effective-sample-size vocabulary that every later page uses, so it reads first; and the two continuous-time pages are a pair, since Geometric Brownian Motion is the exponential of the process Brownian Motion constructs and assumes its reflection principle rather than re-deriving it. Otherwise the part splits into three independent blocks that can be read in any order: arrival processes in Bernoulli Processes, Poisson Processes and Renewal Processes, state processes in Markov Chains, Continuous-Time Markov Chains and Hidden Markov Models, and sum processes in 08 through Random Walks. Where this part stops is worth stating too: the Itô integral, Itô's lemma and the Black–Scholes argument are Stochastic Calculus and are assumed rather than built, so page 08 states quadratic variation and attributes it; nothing here is simulated for the purpose of estimating an integral and no resampling scheme is constructed, which is Part IX; no dependence model is fitted, selected or tested, which is Time Series and Part XIII; the state is never continuous, so a Kalman or particle filter is Particle and Kalman Filters; no test, critical region or multiplicity correction is built, which are Part XII and Part XV; and the applications these results are aimed at — hitting times, regime detection, microstructure — are Part XVIII.

One failure runs through the part and it has a single shape. Every model here buys its tractability by fixing the form of the memory — none at all in the arrival processes, memory through the present in the chains, none in the increments of the walks — and in every case the fitted model reproduces the marginal distribution while getting the dependence wrong, so the error is invisible in exactly the diagnostic anyone would run. A GARCH series is uncorrelated to four decimals at every lag out to fifty and carries \(3.7\%\) of its nominal sample size for a variance. A clustered arrival pattern is equidispersed at the daily horizon and three times over-dispersed at the minute. A Markov chain fitted to regimes that age matches the mean duration exactly and overstates the six-month tail by a factor of \(7{,}876\). A Baum–Welch fit to pure noise returns two persistent states with double-digit durations. A ninety-one-day regime observed quarterly reports two hundred and forty-three days. A strategy reviewed weekly as its history accumulates looks significant \(35\%\) of the time with no edge at all. In each case the arithmetic is correct, the estimator is consistent, and the quantity the model was asked to certify is one it never claimed — and in each case the null distribution that would have caught it costs twenty lines of simulation that nobody runs.

Topics

Topic	Focus
Random Processes	A process as one draw from a space of functions, strict against covariance stationarity and the positive-semidefiniteness that characterises an autocovariance, mean-ergodicity as the Cesàro average of the autocovariances vanishing, two processes with identical marginals whose sample means converge to different things, the effective sample size \(n/(1+2\sum\rho_k)\) matching \((1-\rho)/(1+\rho)\) to three decimals, and a series clean in the level and hopeless in the square
Bernoulli Processes	Three equivalent descriptions and the Pascal law derived by counting and by duality, the fresh-start property and the stopping times it permits, a ten-trade winning run appearing in \(0.4581\) of zero-edge track records, the conditional win-rate estimator biased below one half on a fair coin, the split-stream correlation \(-p/(2-p)\), and two streak intuitions failing in opposite directions
Poisson Processes	Three definitions collapsing into one assumption about memorylessness, arrival times uniform given the count with every rate cancelling, thinning exact off the grid where it was correlated on it, a clustered pattern equidispersed at one horizon and three times over-dispersed at another, the multinomial null \(1-1/b\) nobody applies, and a compound sum whose mean survives a stopping rule while its variance and its tail do not
Renewal Processes	The fresh start surviving at arrivals and nowhere else, the long-run rate depending on the mean gap alone, the gap straddling a random instant at \(\mu(1+c^{2})\) and twice the truth at exponential dispersion, renewal-reward tolerating any dependence between a trade's reward and its duration, a mean-of-ratios annualization overstating by a factor of \(6.26\), and twenty-five trades a year carrying a standard deviation of \(5.03\)
Markov Chains	The Markov property as a claim about the state rather than about the process, multi-step probabilities as matrix powers with the two-state chain in closed form, the gap between two starting points equal to \(\lambda_2^{\,n}\) to four decimals, a persistence estimate whose standard error at one year exceeds the exit probability it resolves, five percent of one-year fits reporting a regime that never ends, and a forced-geometric sojourn overstating a six-month tail by \(7{,}876\) times
Continuous-Time Markov Chains	Memorylessness forcing exponential holding times by the Cauchy functional equation, the generator as a derivative and the transition function as its exponential, periodicity evaporating with the grid that caused it, a daily matrix with no hourly version and a logm that returns complex numbers rather than complaining, and a ninety-one-day regime read quarterly reporting two hundred and forty-three days
Hidden Markov Models	One factorization that every algorithm manipulates, a forward recursion agreeing with brute-force enumeration of \(65{,}536\) paths to fourteen decimals, filtering as sequential Bayes and smoothing as a quarter of the strategy, a smoothed backtest beating perfect knowledge of the hidden state, EM's monotone ascent guaranteeing nothing about the answer, and two persistent regimes fitted to white noise that only the likelihood gain rejects
Brownian Motion	Four properties and Donsker making the step law a question about speed rather than about the limit, scaling forcing paths with no derivative and a deterministic quadratic variation, the reflection principle turning a maximum into a marginal, an expected drawdown of \(\sigma\sqrt{2T/\pi}\) that a zero-edge strategy is guaranteed, new highs on a set of times of measure zero, and a stop hit with probability one whose mean waiting time is not a number
Geometric Brownian Motion	The drag term as the concavity of the logarithm rather than a cost, mean and median separating until seven investors in ten fall below the average, a drift standard error identical across an eight-thousandfold increase in sampling frequency, a volatility standard error collapsing like \(1/\sqrt{2n}\) over the same range, and leverage raising the mean to \(21{,}815\) while the median returns to \(1.005\)
Martingales	A filtration as point-in-time discipline given a name, every sizing rule and stop as a predictable stake that cannot change an expectation, a doubling system winning \(99.9947\%\) of its sessions for exactly nothing, optional stopping failing by a full unit through a vanishing probability of an unbounded loss, Wald's identity as the reason a declared loss limit distorts nothing, and monthly review turning a \(5\%\) test into a \(30\%\) one
Random Walks	Stirling's central binomial coefficient governing return probabilities and forcing recurrence, a certain return with an infinite expected waiting time, ruin exponential in capital so that ninety percent of the table is lost a third of the time at \(p=0.49\), the arcsine law making a balanced record four and a half times rarer than a lopsided one, and the variance ratio as the effective sample size under a second name