Doubly Stochastic Poisson processes are generalizations of Compound Poisson processes, in thesense that the intensity of the (simple) counting process, Nt , is stochastic.The (random) intensity function is defined as

λ(t) = limh→0

P[Nt+h − Nt = 1 | σ(Nu)u<t ]h = lim

h→0

E[Nt+h − Nt | σ(Nu)u<t ]h

The Doubly Stochastic Poisson Process, or Cox process satisfy:

� P [Nt − Ns = n | Fs ∨ σ(λu)u∈[s,t]]

= exp(−∫ t

s λudu) (∫ t

sλudu)n

n!

� P [Nt − Ns = n | Fs ] = E[

exp(−∫ t

s λudu) (∫ t

sλudu)n

n!

∣∣∣∣ Fs

]Where does the randomness of the intensity function may come from?What is the difference in the two equations above?

Jonathan Chavez (University of Calgary) Doubly Stochastic Poisson Processes 18/05/2016 1 / 5

Some examples of possible (random) intensity functions are:Feller process:

dλt = κ(θ − λt)dt + η√λtdWt

O-U process:dλt = −κλtdt + γdJt

Jump-diffusion:dλt = κ(θ − λt)dt + η

√λtdWt + γdJt

Hawkes process:

dλt =∫ t

0g(t − s)dNs

where J is an independent compound Poisson process with intensity λJ and i.i.d. jump size ε ∼ Fand ε > 0 a.s.

Jonathan Chavez (University of Calgary) Doubly Stochastic Poisson Processes 18/05/2016 2 / 5

Compensator (Doob-Meyer decomposition) of a Doubly Stochastic process Nt is∫ t

0 λsds. That is,

Nt −∫ t

0λsds

is a martingale.

If J is the compensated version of J , that is, J = J − E[ε]JλJ t, then we can rewrite the O-U andjump-diffusion intensity processes above as:

O-U process:dλt = κ

(γλJκE[ε]− λt

)dt + γdJt

Jump-diffusion:dλt = κ(θ + γλJ

κE[ε]− λt)dt + η

√λtdWt + γdJt

Jonathan Chavez (University of Calgary) Doubly Stochastic Poisson Processes 18/05/2016 3 / 5

Let µ be a discrete measure with support on {a1, a2, . . . , aN} with N ≤ ∞. Assume thatµ(aj) = hj ∈ R for each j ∈ {1, 2, . . . ,N}. What is the definition of

∫A f (x)dµ(x)?

Can a counting process be seen as a discrete measure?

Thus, we have (it is NOT a definition) that for a doubly stochastic PP, Nt , with intensity λt ,

∫ t

0g(s−)dNs =

Nt∑τk

g(τ−k ),

where τk are the jumping times of N on (0, t). Also, for Nt the compensated doubly stochastic PP,(why?) ∫ t

0g(s−)dNs =

Nt∑τk

g(τ−k )−∫ t

0gsλsds

Jonathan Chavez (University of Calgary) Doubly Stochastic Poisson Processes 18/05/2016 4 / 5

Jonathan Chavez (University of Calgary) Doubly Stochastic Poisson Processes 18/05/2016 5 / 5