JUMP PROCESSES GENERALIZING STOCHASTIC INTEGRALS WITH JUMPS Tyler Hofmeister University of Calgary Mathematical and Computational Finance Laboratory
Overview 1. General Method 2. Poisson Processes 3. Diffusion and Single Jumps 4. Compound Poisson Process 5. Jump-Diffusion Page 1
GENERAL METHOD
General Method Define a Stochastic Process Adjust the Process to a Martingale Define a Stochastic Integral Itô s Formula and Generator Page 3
POISSON PROCESSES
Definition Definition: Poisson Process A Poisson process N {N t } 0 t T Z +, with intensity λ, is a stochastic process with the following properties (i) N 0 0 almost surely, (ii) N t N 0 has a Poisson distribution with parameter λt. (iii) N has independent increments, so (s, t) (v, u) implies N t N s is independent of N v N u. (iv) N has stationary increments, so N s+t N s follows the same distribution as N t for all s, t > 0. Page 5
Poisson Process Example Page 6
Poisson Process: Properties Properties (i) E [N t ] λt (ii) Var [N t ] λt (iii) The time between jumps of N are independent and follow an exponential distribution. Page 7
Compensated Poisson Process Proposition: Compensated Poisson Process The compensated Poisson process N { Nt }0 t T where N t N t λt is a martingale with respect to it s generated filtration F. Proof. E [N t+s λ(t + s) F t ] E [N t+s N s + N s λ(t + s) F t ] E [N t λt + N s λs F t ] N t λt Page 8
Compensated Poisson Process Example Page 9
Stochastic Integral Definition: Stochastic Integral with respect to a Compensated Poisson Process Let g be an F t -adapted process, where F t is the natural filtration generated by Poisson process N. Define stochastic integral Y {Y t } 0 t T of g with respect to N as Y t t 0 g s d Ns N t k 1 g τ k t g s λds where {τ 1, τ 2,...} is the collection of times when N jumps. 0 Page 10
Itô s Formula for Poisson Processes Theorem: Itô s Formula for Poisson Processes Suppose Y is the stochastic integral given previously. Let Z {Z t } 0 t T with Z t f (t, Y t ) for some function f, once differentiable in t. Then dz t ( t f (t, Y t ) λg t y f (t, Y t ))dt + f (t, Y t + g t ) f (t, Y t ) dn t { t f (t, Y t ) + λ([ f (t, Y t + g t ) f (t, Y t )] g t y f (t, Y t ))}dt + [ f (t, Y t + g t ) f (t, Y t )]d Nt Page 11
Infinitesimal Generator Recall that the generator L t of a process X t acts on twice differentiable functions f as L t f (x) lim h 0 E[ f (X t+h X t x)] f (x) h which is a generalization of a derivative of a function which can be applied to stochastic processes. The generator of stochastic integral Y from a Poisson process acts as Lt Y f (y) λ [ f (y + g t ) f (y)] g t y f (y) Page 12
DIFFUSION AND SINGLE JUMPS
Sum of Stochastic Integrals Using the framework developed previously for Stochastic Integrals with respect to diffusion and jumps, we sum these two as follows. Y t t 0 t t f s ds + g s dw s + h s d Ns, 0 0 where f, g, h are F t adapted processes, and filtration F is the natural one generated by both the Brownian motion W and Poisson process N, which are mutually independent. Page 14
Ito s Formula for Single Jumps and Diffusion Theorem: Itô s Formula for Single Jumps and Diffusion Suppose Y is the stochastic integral given previously. Let Z {Z t } 0 t T with Z t l(t, Y t ) for some function l, once differentiable in t and twice differentiable in y. Then dz t ( t + f t y + 1 2 g2 t y y λh t y )l(t, Y t ))dt + g t y l(t, Y t )dw t + [l(t, Y t + h t ) l(t, Y t )] dn t t + f t y + 1 2 g2 t y y l(t, Yt ) +λ([l(t, Y t + h t ) l(t, Y t )] h t y l(t, Y t )) dt + g t y l(t, Y t )dw t + [l(t, Y t + h t ) l(t, Y t )]d Nt Page 15
Generator The generator of Y acts as L Y t l(y) f t y l(y)+ 1 2 g2 t y yl(y)+λ [l(y + h t ) l(y)] h t y l(y) Page 16
COMPOUND POISSON PROCESS
Definition Definition: Compound Poisson Processes Let N be a Poisson process with intensity λ and {ε 1, ε 2,...} be a set of independent identically distributed random variables with distribution function F and E[ε] < +. A compound Poisson process J {J t } 0 t T is given by N t J t ε k, t 0 k 1 Page 18
Compound Poisson Process Example Page 19
Compound Poisson Process: Properties Properties (i) E [J t ] λte[ε] (ii) Var [J t ] λte ε 2 (iii) As with the standard Poisson process, the inter-arrival times are independent and exponentially distributed. Page 20
Compensated Compound Poisson Process Proposition: The compensated compound Poisson process Ĵ { Ĵ t } where Ĵ t J t E[ε]λt is a martingale. Proof. E [ Ĵ t+s F t ] E [ Σ N t+s k 1 ε k λ(t + s)e[ε] F t ] E [ Σ N t k 1 ε k + Σ N t+s k N t +1 λ(t + s)e[ε] F t Σ N t k 1 λte[ε] 0 t T ] Page 21
Compensated Compound Poisson Process Page 22
Corresponding Stochastic Integral Let F be the natural filtration generated by Ĵ. We define the stochastic integral Y {Y t } 0 t T of an F -adapted process g with respect to the compensated compound Poisson process Ĵ as Y t t where J s J s J s 0 g s dĵ s s t t g s J s g s λe[ε]ds 0 Page 23
JUMP-DIFFUSION
Sum of Stochastic Integral Let f, g, and h be F -adapted stochastic processes where F is the natural filtration generated by an independent Brownian motion W and Ĵ. We define the stochastic integral Y as Y t t 0 t t f s ds + g s dw s + h s dĵ s 0 0 Page 25
Ito s Formula for Jump-Diffusion Theorem: Itô s Formula for Jump-Diffusion Suppose Y is the stochastic integral given previously. Let Z {Z t } 0 t T with Z t l(t, Y t ) for some function l, once differentiable in t and twice differentiable in y. Then dz t ( t + f t y + 1 2 g2 t y y λe[ε]h t y )l(t, Y t ))dt + g t y l(t, Y t )dw t + l(t, Y t + ε Nt h t ) l(t, Y t ) dn t t + f t y + 1 2 g2 t y y l(t, Yt ) +λ(e[l(t, Y t + h t ) l(t, Y t )] E[ε t ]h t y l(t, Y t )) dt + g t y l(t, Y t )dw t + [l(t, Y t + ε Nt h t ) l(t, Y t )]d ˆN t Page 26
Generator The generator of Y acts as L Y t l(y) f t y l(y) + 1 2 g2 t y yl(y) + λ E[l(t, y + εh t ) l(t, y)] E[ε]h t y l(t, Y t ) Page 27
References Alvaro Cartea, Sebastian Jaimungal, and Jose Penalva Algorithmic and High-Frequency Trading Cambridge University Press, 2015 Nicolas Privault Notes on Stochastic Finance Nanyang Technological University Page 28
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