note: poisson limit

The Poisson Process from Discrete to Continuous Time

In financial markets, we often model rare events, like large price moves or credit defaults, using continuous-time models. But these emerge from discrete processes when we take limits.

Problem Setup
Consider a speculative asset where each trading minute has probability \(p = 0.1\) of experiencing a mini-flash crash, like a sudden price drop of 0.5%. Events are independent across minutes, with negligible chance of two crashes in the same minute. Let \(N_t\) be crash events by time \(t\) minutes, \(X_i\) be time between crashes, and \(T_n\) be time of the \(n\)-th crash.

Discrete-Time Model
\(N_t\) counts crashes in \(t\) independent Bernoulli trials, so \(N_t \sim \text{Binomial}(t, 0.1)\). The interarrival time \(X_i\) follows \(\text{Geometric}(0.1)\), representing minutes until next crash. The \(n\)-th crash time \(T_n = X_1 + \cdots + X_n\) gives \(T_n \sim \text{NegativeBinomial}(n, 0.1)\). The fundamental relationship \(\{N_t \geq n\} \iff \{T_n \leq t\}\) captures that having at least \(n\) crashes by time \(t\) means the \(n\)-th crash occurs before \(t\).

Shrinking Time Intervals
Now monitor price in tiny intervals of \(\tau\) minutes. Crash probability per \(\tau\)-interval is \(\tau p = 0.1\tau\). Number of intervals in \(t\) minutes is \(t/\tau\), so \(N_t \sim \text{Binomial}(t/\tau, 0.1\tau)\). Interarrival time in \(\tau\)-units is \(\text{Geometric}(0.1\tau)\), and \(T_n\) in \(\tau\)-units is \(\text{NegativeBinomial}(n, 0.1\tau)\). The relationship \(\{N_t \geq n\} \iff \{T_n \leq t\}\) persists.

Continuous-Time Limit

As \(\tau \to 0\), we observe three elegant transitions supported by concrete mathematical limits:

Binomial \(\to\) Poisson

The binomial PMF \(\binom{m}{k}(\tau p)^k(1-\tau p)^{m-k}\) with \(m = t/\tau\) has mean \(m\tau p = tp\) fixed. Taking \(\tau \to 0\): \[ \binom{t/\tau}{k}(\tau p)^k(1-\tau p)^{t/\tau - k} \to \frac{(tp)^k}{k!}e^{-tp} \] This is the Poisson PMF with rate \(\lambda t = 0.1t\).

Geometric \(\to\) Exponential

The geometric survival function \(P(X > t) = (1-\tau p)^{t/\tau}\) for \(t\) multiple of \(\tau\) satisfies: \[ \lim_{\tau\to 0} (1-\tau p)^{t/\tau} = e^{-pt} \] which is the survival function of \(\text{Exponential}(p)\). Alternatively, the geometric MGF \((pe^s)/(1-(1-p)e^s)\) with \(p=\tau\lambda\) converges to the exponential MGF \(\lambda/(\lambda-s)\) as \(\tau\to 0\).

Negative Binomial \(\to\) Gamma

Since \(T_n = \sum_{i=1}^n X_i\) where \(X_i \sim \text{Geometric}(\tau p)\), and geometric \(\to\) exponential, we have: \[ T_n \to \sum_{i=1}^n \text{Exponential}(p) \sim \text{Gamma}(n, p) \] The negative binomial MGF \([pe^s/(1-(1-p)e^s)]^n\) with \(p=\tau\lambda\) converges to the gamma MGF \([\lambda/(\lambda-s)]^n\). This is the classic case of a sum of i.i.d. geometric variables converging to a sum of exponentials, yielding the gamma distribution.

Each limit preserves the essential structure wih the Poisson limit theorem for rare events, the memoryless property from geometric to exponential, and the additive property from negative binomial to gamma.

Financial Interpretation
High-frequency traders see this limit in practice. Monitoring prices continuously, crash arrivals follow \(\text{Poisson}(\lambda t)\) with \(\lambda = 0.1\) per minute. Time between crashes becomes \(\text{Exponential}(0.1)\), and time until the \(n\)-th systemic shock follows \(\text{Gamma}(n, 0.1)\). This framework lies under many continuous-time finance models, from jump processes in option pricing to default timing in credit risk.