LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A Poisson distribution counts rare events in a window: trades, defaults, operational losses. With a fixed rate, the counts in a Poisson process have exponential waiting times and independent, stationary increments. A rate fitted to last period can call next period's counts almost impossible. A Hawkes process lets the rate jump when events cluster, which is the lecture's model of a selloff feeding itself.
Why it matters
The lecture's running example uses lambda equal to 4, the mean count in the window. If the window is 100 milliseconds, that is four trades on average. Mass rises toward four and then falls quickly. The support is the counting numbers, including zero, with no fractional events, even though a fitted lambda can be fractional. Zero events are allowed. There is no finite cap, but the far tail is described as indistinguishable from zero for practical purposes.
Four assumptions do the work. Events are independent, so one arrival does not make the next more likely. The rate is constant. Two events do not land at the same instant. Counts are whole numbers. Finance violates independence when activity clusters. A Hawkes process is the lecture's later answer to that. A constant rate is the stationarity assumption, and it is the one that makes a fitted probability fragile.
The idea in plain words
The maximum-likelihood estimate of lambda is the average count you observed. Count jumps in a stock over five-day blocks, average them, and you have a calibrated Poisson model for the next week: the chance of no jump, one jump, two, and so on. More historical jumps shift the fitted curve out. Fewer pull it toward zero. The lecture's warning is that the fit only matches the past you fed it. In the mismatch picture, the model says the probability of exceeding the smallest frequency in the next sample is under 0.01 percent, while the new sample sits on a curve far to the right. A structural break is one way that happens.
Waiting times are exponential and continuous. A density height is not a probability. The memoryless property says that having already waited five minutes, the chance of waiting two more equals the chance of waiting more than two minutes from a fresh start. A Poisson count is not memoryless. Having already seen three trades changes the chance of three more.
A Poisson process is a counting process. It starts at zero. The number of events in an interval of length t is Poisson with mean lambda times t. Increments are independent and stationary: a one-day increment has the same law whether it is the interval from day 9 to 10 or from day 10 to 11. Simulate to time 100 and a cross-section of many paths matches the theoretical Poisson histogram at that time, by the law of large numbers. The gaps between jumps line up with the exponential law, again by construction.
How to use it
A constant lambda is a poor description of a restaurant, or of a trading day. Morning and evening, or the open and a quiet afternoon, want different rates. A time-inhomogeneous Poisson process makes lambda a function of time, so the count distribution changes shape, not only scale. The lecture's game example makes lambda a function of the bid-ask spread instead. As the spread widens, the chance of nobody trading reaches over 40 percent in the picture. Tighten it and the mass shifts toward more arrivals.
What usually goes wrong
Treating a fitted lambda as next period's rate is the 0.01 percent mistake. Independence is the wrong assumption in a panic, which is why a plain Poisson process cannot produce the self-exciting bursts the Hawkes picture is drawn to show. Stationary increments are a modeling choice, not a fact about the open versus the afternoon.
Questions people ask
What does lambda equal to 4 mean?
On average, four events in the window you chose. For trades, the lecture's window is 100 milliseconds. The mass is highest around that mean and thin in the far tail.
How can a model call real data almost impossible?
By using last period's lambda. In the lecture, that model put under a 0.01 percent probability on counts beyond the smallest frequency the next period actually produced.
What does a Hawkes process add?
The rate becomes self-exciting. A shock raises the chance of more shocks for a while, then the intensity falls back, which is the clustering a constant-rate Poisson process leaves out.