1 Poisson Process is an exponential random variable if it is with density 955e 955t t 0 t To construct a Poisson process we begin with a sequence of independent expo nential random variables all with the same mean 1 The arrival times are de64 ID: 25219 Download Pdf

273K - views

Published byalexa-scheidler

1 Poisson Process is an exponential random variable if it is with density 955e 955t t 0 t To construct a Poisson process we begin with a sequence of independent expo nential random variables all with the same mean 1 The arrival times are de64

Download Pdf

Download Pdf - The PPT/PDF document "SEEM Advanced Models in Financial Engi..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

SEEM 5670 – Advanced Models in Financial Engineering Professor: Nan Chen Nov 21, 2014 Lecture 8: Jump Diﬀusion Models 1 Part I: Review of Poisson Processes 1.1 Poisson Process is an exponential random variable if it is with density ) = λe λt , t 0; , t< To construct a Poisson process, we begin with a sequence , ···} of independent expo- nential random variables, all with the same mean 1 / . The arrival times are deﬁned as =1 The Poisson process is then = max 0 : Theorem 1 The Poisson process with intensity has the distribution ) = λt λt for =

0 ··· Theorem 2 Let be a Poisson process with intensity , and let 0 = ··· be given. Then the increments , N ··· ,N are stationary and independent, and ) = )) for = 0 ··· Theorem 3 Suppose that is a Poisson process with intensity ] = and Var ] = for s Theorem 4 Let be a Poisson process with intensity . Then, the compensated Poisson process λt is a martingale.

Page 2

SEEM 5670 – Advanced Models in Financial Engineering Professor: Nan Chen Nov 21, 2014 1.2 Compound Poisson Process Let be a Poisson process with intensity , and let ,Y ··· be a sequence of i.i.d. random variables with

mean EY . We deﬁne the compound Poisson process =1 , t Theorem 5 Let be the compound Poisson process deﬁned above. Then, the compensated compound Poisson process βλt is a martingale. Let 0 = ··· be given. The increments , Q ··· ,Q are stationary and independent. In particular, the distribution of is the same as the distribution of Theorem 6 Let be a compound Poisson process. Denote the moment-generating function of by ) = uY Then, the moment-generating function of is ) = λt 1)] Theorem 7 Let ,y ··· ,y be a ﬁnite set of nonzero numbers, and let ,p ··· ,p be

positive numbers that sums to 1. Let λ> and ,N ··· ,N be independent Poisson processes, each having intensity λp . Deﬁne =1 Then it is a compound Poisson process. Theorem 8 Let ,y ··· ,y be a ﬁnite set of nonzero numbers, and let ,p ··· ,p be positive numbers that sums to 1. Let ,Y ··· be a sequence of independent identically distributed random variables with ) = = 1 ··· ,M . Let be a Poisson process and deﬁne the compound Poisson process =1 For = 1 ··· ,M , let denote the number of jumps in of size up to time . Then =1 and =1 The processes ··· ,N are independent

Poisson processes, and each has intensity λp

Page 3

SEEM 5670 – Advanced Models in Financial Engineering Professor: Nan Chen Nov 21, 2014 2 Part II: Stochastic Calculus with Jumps 2.1 Jump Process and its Integrals Let ( ,P ) be a probability space on which is given a ﬁltration {F ,t . A process is called a pure jump process if is right-continuous, has left limits, jumps only ﬁnite times on each ﬁnite time interval (0 ,T ], and is constant between jumps. Let dW ds Call dW ds the continuous part of . Deﬁne the stochastic integral of Φ with respect

to to be dX dW ds for any adapted process Φ. Example 1 Let λt , where is a Poisson process with intensity . Let = (i.e., is 1 if has a jump at time , otherwise 0). dX To make the integral be a martingale, we need an additional condition that the integrands must be left-continuous. Theorem 9 Assume that the jump process is a martingale, the integrand is left- continuous and adapted, and ds for all . Then, the stochastic integral dX is also a martingale. Example 2 Let λt , where is a Poisson process with intensity . Let [0 ,S be 1 up to and including the time of the ﬁrst

jump and zero thereafter. dX [0 ,S The quadratic variation of is deﬁned to be X,X = lim k =0 +1 where the partition Π = < t ··· < t . We also need the concept of cross variation Let and be two jump processes. The cross variation of these two is ,X = lim k =0 +1 )( +1

Page 4

SEEM 5670 – Advanced Models in Financial Engineering Professor: Nan Chen Nov 21, 2014 Theorem 10 X,X = [ ,X ( )) and ,X = [ ,c ,X ,c ( ))( )) 2.2 Ito Formula for Jump Processes Theorem 11 Let be a jump process and a function for which and 00 are deﬁned and continuous. Then, ) = ) + dX 00 ,X )]

Theorem 12 Let and be jump processes, and let t,x ,x be a function whose ﬁrst and second partial derivatives appearing in the following formula are deﬁned and continuous. Then, t,X ,X (0 ,X ,X ) + u,X ,X du u,X ,X dX ,c u,X ,X dX ,c u,X ,X ,c ,X ,c u,X ,X ,c ,X ,c u,X ,X ,c ,X ,c s,X ,X s,X ,X )] Example 3 (Geometric Poisson process) Let log( + 1) λσt, where is a Poisson process with intensity . Let with ) = . We have dS σS dM λσS dt σS dN In general, Theorem 13 Let be a jump process. The Doleans-Dade exponential of is deﬁned to be the process

= exp ,X (1 + This process is the solution to the stochastic diﬀerential equation dZ dX with initial condition = 1

Page 5

SEEM 5670 – Advanced Models in Financial Engineering Professor: Nan Chen Nov 21, 2014 Homework Set 8 (Due on Dec 1) 1. Exercise 11.1 in Page 525 of Shreve’s book. 2. Exercise 11.2 in Page 525 of Shreve’s book. 3. Exercise 11.3 in Page 526 of Shreve’s book. 4. Exercise 11.4 in Page 526 of Shreve’s book.

Â© 2020 docslides.com Inc.

All rights reserved.