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### Transcript of Doubly Stochastic Poisson aswish/Doubly_ آ  Doubly Stochastic Poisson processes are...

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

λ(t) = lim h→0

P[Nt+h − Nt = 1 | σ(Nu)u

• 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

0 g(t − s)dNs

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

Jonathan Chávez (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

0 λsds

is a martingale.

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

O-U process: dλt = κ

( γλJ κ E[�]− λt

) dt + γdĴt

Jump-diffusion: dλt = κ(θ +

γλJ κ E[�]− λt)dt + η

√ λtdWt + γdĴt

Jonathan Chávez (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 that µ(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 0

g(s−)dNs = Nt∑ τk

g(τ−k ),

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

0 g(s−)dN̂s =

Nt∑ τk

g(τ−k )− ∫ t

0 gsλsds

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

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