Admistrivia

Continuous markov models

Pure birth process

Review of poisson

• P(X(t+h) - X(t) = 1|X(t) = x) = lambda h + o(h)
• P(X(t+h) - X(t) = 0|X(t) = x) = 1 - lambda h + o(h)
• X(0) = 0.

Nice property: we can easilly adjust lambda dynamically in this formalization. this generates continuous time markov processes.

(Pure) Birth Processes

• P(X(t+h) - X(t) = 1|X(t) = x) = lambda_x h + o(h)
• P(X(t+h) - X(t) = 0|X(t) = x) = 1 - lambda_x h + o(h)
• P(X(t+h) - X(t) < 0|X(t) = x) = 0.
• X(0) = 0.

• NOTE: if we can solve this, we know "everything."
• We just have to shift the lambda's

Forward equations?

• P_n'(t) = - lambda_nP_n(t) + lambda_n-1P_n-1(t)
• Special case for n = 0: P₀'(t) = - lambda_nP_n(t)
• Derive via o() notation and take limit

Sojourn times

• Times between births called sojourn times
• S_k is time between kth and (k+1)st birth
• Each is exponential

Finiteness

• ES_k = 1/lambda_k
• Sum 1/lambda_k = expected time till explosion
• if infinite--math very difficult.
• Model goes to infinite in finite time

Yule Process

• Explicit case: lambda_k = beta * k.
• NOTE: X(0) = 1 for conviencence
• P_n(t) = exp(-beta t)(1 - exp(-beta t))^n-1
• Check that it is correct.