Jump Processes - Generalizing Stochastic Integrals...
Transcript of Jump Processes - Generalizing Stochastic Integrals...
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JUMP PROCESSESGENERALIZING STOCHASTIC INTEGRALS WITH JUMPS
Tyler Hofmeister
University of CalgaryMathematical and Computational Finance Laboratory
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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
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General Method
Define aStochasticProcess
Adjust theProcess to aMartingale
Define aStochasticIntegral
Ito’s Formulaand Generator
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POISSON PROCESSES
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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
�2016/05/18
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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
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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
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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]
�2016/05/18
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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
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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!