LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Models of market jumps are wrong by degrees, and the degree matters. A Poisson count with a fixed average, here 20 events in a 30-minute window, can describe a calm rate. It cannot describe a selloff in which one shock raises the odds of the next. A Hawkes process makes the intensity self-exciting. Put that intensity inside a jump diffusion and the lecture's simulated returns cluster, show still fatter tails, and imply a three-standard-deviation event about every 36 or 37 days instead of the roughly 370 days a fitted normal suggests.
Why it matters
Rare events do not arrive on a schedule. A Poisson random variable counts them over a window you choose: trades in 30 minutes, or gaps over a year. Lambda is the average, and the maximum-likelihood estimate is the sample average. Draw repeatedly from a Poisson mass with mean 20 and the histogram fills in that mass by the law of large numbers. Each draw is a whole window. A Poisson process is what you want if you need the arrivals inside the window. Increments are Poisson, the rate is scaled by time, and at 30 minutes the terminal distribution matches the original count. One frame might show about 25 trades, another about 18. The path is the new information. The assumption that hurts is a constant lambda from the open through a quiet afternoon.
A time-inhomogeneous Poisson process lets the rate be a function of time. The lecture's animation is quiet, then switches at time 50. The rate goes from 0.2 to 0.8 in the units of the picture, and arrivals speed up. You can set one rate for 9:30 and another for 1:00. That is more realistic and still misses contagion. The rate knows the clock. It does not know that something just happened.
The idea in plain words
The boat-on-a-lake image is the lecture's account of clustering and of mean reversion in volatility. A calm lake, one boat, a little chop, then more boats, then a rough surface, then a calm lake again. Energy was required to leave the baseline, and the system can return, or the new level can become normal. A Hawkes intensity has a baseline, called mu, and a triggering kernel. Past events push the current rate up, and the push decays. On a handful of paths the rate spikes, breeds more spikes, and falls back. The counting process accelerates during the burst and then looks ordinary again. Estimation is described as difficult and the computation as heavy, because the intensity is path dependent. The lecture calls the process non-Markov and says you can sometimes enlarge the state until a Markov representation appears. That construction is left for later, as it is for some rough-volatility models.
How to use it
Use a constant-rate Poisson model when the window really is homogeneous, and check that claim against the open and the afternoon before you trust the probabilities. Use a time change in the rate when the clock is the main effect. Use a Hawkes kernel when the economic story is contagion: a gap down, then risk reduction, then more selling, then a fade to the baseline. The probabilities you read off the model, including the three-sigma wait, move with that choice. The lecture's comparison is the reason to prefer the self-exciting jump model over the plain one when clustering is part of what you need the tail probability to reflect.
What usually goes wrong
Fitting a normal, seeing kurtosis, and shopping for a single fatter distribution treats the histogram as a fixed law. The lecture's point is that the distribution moves, and the Hawkes paths are one way to show the movement. A plain jump diffusion fixes the tail relative to a normal and still sprinkles jumps independently, so a risk system built on it will not expect bursts.
Questions people ask
What is lambda equal to 20?
The average event count in the window, in the example a 30-minute trade count. A Poisson process with that rate ends at the same distribution as drawing the 30-minute count directly.
How is the inhomogeneous example different from Hawkes?
The rate jumps from 0.2 to 0.8 at time 50 because the clock says so. A Hawkes rate jumps because past events push it, through a baseline plus a triggering kernel, and then decays.
What wait times does the lecture compare?
A normal fit to the simulated returns puts a three-sigma event at about 370 days. The jump models put it near 36 and 37 days. The Hawkes version also clusters the shocks.