JUMP PROCESSESGENERALIZING STOCHASTIC INTEGRALS WITH JUMPS
Tyler Hofmeister
University of CalgaryMathematical and Computational Finance Laboratory
Overview
1. General Method
2. Poisson Processes
3. Diffusion and Single Jumps
4. Compound Poisson Process
5. Jump-Diffusion
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GENERAL METHOD
General Method
Define aStochasticProcess
Adjust theProcess to aMartingale
Define aStochasticIntegral
Ito’s Formulaand Generator
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POISSON PROCESSES
Definition
Definition: Poisson Process
A Poisson process N � {Nt}0≤t≤T ∈ Z+, with intensity λ, is a
stochastic process with the following properties
(i) N0 � 0 almost surely,(ii) Nt − N0 has a Poisson distribution with parameter λt.(iii) N has independent increments, so (s , t)∩ (v , u) � ∅ implies
Nt − Ns is independent of Nv − Nu .(iv) N has stationary increments, so Ns+t − Ns follows the same
distribution as Nt for all s , t > 0.
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Poisson Process Example
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Poisson Process: Properties
Properties
(i) E [Nt] � λt
(ii) Var [Nt] � λt
(iii) The time between jumps of N are independent and followan exponential distribution.
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Compensated Poisson Process
Proposition: Compensated Poisson Process
The compensated Poisson process N �
{Nt
}0≤t≤T
whereNt � Nt − λt is a martingale with respect to it’s generated fil-tration F .
Proof.
E [Nt+s − λ(t + s)|Ft] � E [Nt+s − Ns + Ns − λ(t + s)|Ft]� E [Nt − λt + Ns − λs |Ft]� Nt − λt
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Compensated Poisson Process Example
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Stochastic Integral
Definition: Stochastic Integral with respect to aCompensated Poisson Process
Let g be an Ft-adapted process, where Ft is the natural filtra-tion generated by Poisson process N . Define stochastic integralY � {Yt}0≤t≤T of g with respect to N as
Yt �
∫ t
0gs−dNs �
Nt∑k�1
gτ−k −∫ t
0gsλds
where {τ1 , τ2 , . . .} is the collection of times when N jumps.
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Ito’s Formula for Poisson Processes
Theorem: Ito’s Formula for Poisson Processes
Suppose Y is the stochastic integral given previously. LetZ � {Zt}0≤t≤T with Zt � f (t ,Yt) for some function f , oncedifferentiable in t. Then
dZt � (∂t f (t ,Yt) − λgt∂y f (t ,Yt))dt
+�
f (t ,Yt− + gt−) − f (t ,Yt−)� dNt
� {∂t f (t ,Yt) + λ([ f (t ,Yt− + gt−) − f (t ,Yt−)]− gt∂y f (t ,Yt))}dt
+ [ f (t ,Yt− + gt−) − f (t ,Yt−)]dNt
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Infinitesimal Generator
Recall that the generator Lt of a process Xt acts on twicedifferentiable functions f as
Lt f (x) � limh↓0
E[ f (Xt+h |Xt � x)] − f (x)h
which is a generalization of a derivative of a function which canbe applied to stochastic processes.
The generator of stochastic integral Y from a Poisson processacts as
L Yt f (y) � λ �[ f (y + gt) − f (y)] − gt∂y f (y)�
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DIFFUSION AND SINGLE JUMPS
Sum of Stochastic Integrals
Using the framework developed previously for StochasticIntegrals with respect to diffusion and jumps, we sum thesetwo as follows.
Yt �
∫ t
0fs ds +
∫ t
0gs dWs +
∫ t
0hs−dNs ,
where f , g , h are Ft adapted processes, and filtration F is thenatural one generated by both the Brownian motion W andPoisson process N , which are mutually independent.
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Ito’s Formula for Single Jumps and Diffusion
Theorem: Ito’s Formula for Single Jumps and Diffusion
Suppose Y is the stochastic integral given previously. LetZ � {Zt}0≤t≤T with Zt � l(t ,Yt) for some function l, oncedifferentiable in t and twice differentiable in y. Then
dZt � (∂t + ft∂y +12 g2
t ∂y y − λht∂y)l(t ,Yt))dt
+ gt∂y l(t ,Yt)dWt + [l(t ,Yt− + ht−) − l(t ,Yt−)] dNt
���∂t + ft∂y +
12 g2
t ∂y y�
l(t ,Yt)+λ([l(t ,Yt− + ht−) − l(t ,Yt−)] − ht∂y l(t ,Yt)) dt
+ gt∂y l(t ,Yt)dWt + [l(t ,Yt− + ht−) − l(t ,Yt−)]dNt
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Generator
The generator of Y acts as
L Yt l(y) � ft∂y l(y)+ 1
2 g2t ∂y y l(y)+λ �[l(y + ht) − l(y)] − ht∂y l(y)�
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COMPOUND POISSON PROCESS
Definition
Definition: Compound Poisson Processes
Let N be a Poisson process with intensity λ and {ε1 , ε2 , . . .} bea set of independent identically distributed random variableswith distribution function F and E[ε] < +∞. A compoundPoisson process J � { Jt}0≤t≤T is given by
Jt �
Nt∑k�1
εk , t ≥ 0
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Compound Poisson Process Example
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Compound Poisson Process: Properties
Properties
(i) E [Jt] � λtE[ε](ii) Var [Jt] � λtE
�ε2�
(iii) As with the standard Poisson process, the inter-arrivaltimes are independent and exponentially distributed.
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Compensated Compound Poisson Process
Proposition:
The compensated compound Poisson process J �
{Jt
}0≤t≤T
where Jt � Jt − E[ε]λt is a martingale.
Proof.
E[Jt+s |Ft
]� E
[Σ
Nt+sk�1 εk − λ(t + s)E[ε]|Ft
]
� E[Σ
Ntk�1εk + Σ
Nt+sk�Nt+1 − λ(t + s)E[ε]|Ft
]
� ΣNtk�1 − λtE[ε]
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Compensated Compound Poisson Process
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Corresponding Stochastic Integral
Let F be the natural filtration generated by J. We define thestochastic integral Y � {Yt}0≤t≤T of an F -adapted process gwith respect to the compensated compound Poisson process Jas
Yt �
∫ t
0gs−d Js �
∑s≤t
gs−∆Js −
∫ t
0gsλE[ε]ds
where ∆Js � Js − Js−
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JUMP-DIFFUSION
Sum of Stochastic Integral
Let f , g , and h be F -adapted stochastic processes where F isthe natural filtration generated by an independent Brownianmotion W and J. We define the stochastic integral Y as
Yt �
∫ t
0fs ds +
∫ t
0gs dWs +
∫ t
0hs−d Js
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Ito’s Formula for Jump-Diffusion
Theorem: Ito’s Formula for Jump-Diffusion
Suppose Y is the stochastic integral given previously. LetZ � {Zt}0≤t≤T with Zt � l(t ,Yt) for some function l, oncedifferentiable in t and twice differentiable in y. ThendZt � (∂t + ft∂y +
12 g2
t ∂y y − λE[ε]ht∂y)l(t ,Yt))dt
+ gt∂y l(t ,Yt)dWt +�l(t ,Yt− + εNt ht−) − l(t ,Yt−)� dNt
���∂t + ft∂y +
12 g2
t ∂y y�
l(t ,Yt)+λ(E[l(t ,Yt− + ht−) − l(t ,Yt−)] − E[εt]ht∂y l(t ,Yt)) dt
+ gt∂y l(t ,Yt)dWt + [l(t ,Yt− + εNt ht−) − l(t ,Yt−)]dNt
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Generator
The generator of Y acts as
L Yt l(y) � ft∂y l(y) + 1
2 g2t ∂y y l(y)
+ λ�E[l(t , y + εht) − l(t , y)] − E[ε]ht∂y l(t ,Yt)�
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References
Alvaro Cartea, Sebastian Jaimungal, and Jose PenalvaAlgorithmic and High-Frequency TradingCambridge University Press, 2015
Nicolas PrivaultNotes on Stochastic FinanceNanyang Technological University
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Thank you!