Doubly Stochastic Poisson Processespeople.ucalgary.ca/~aswish/Doubly_StochasticPP.pdf · Doubly...
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Doubly Stochastic Poisson processes are generalizations of Compound Poisson processes, in the sense that the intensity of the (simple) counting process, N t , is stochastic. The (random) intensity function is defined as λ(t ) = lim h→0 P[N t +h - N t =1 | σ(N u ) u<t ] h = lim h→0 E[N t +h - N t | σ(N u ) u<t ] h The Doubly Stochastic Poisson Process, or Cox process satisfy: P [N t - N s = n |F s ∨ σ(λ u ) u∈[s ,t ] = exp - t s λ u du ( t s λ u du) n n! P [N t - N s = n |F s ]= E exp - t s λ u du ( t s λ u du) n n! F s Where does the randomness of the intensity function may come from? What is the difference in the two equations above? Jonathan Ch´ avez (University of Calgary) Doubly Stochastic Poisson Processes 18/05/2016 1/5
Transcript of Doubly Stochastic Poisson Processespeople.ucalgary.ca/~aswish/Doubly_StochasticPP.pdf · Doubly...
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
