 ## SEEM Advanced Models in Financial Engineering Professor Nan Chen Nov Lecture Jump Diusion Models Part I Review of Poisson Processes - Description

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

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# SEEM Advanced Models in Financial Engineering Professor Nan Chen Nov Lecture Jump Diusion Models Part I Review of Poisson Processes

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

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## SEEM Advanced Models in Financial Engineering Professor Nan Chen Nov Lecture Jump Diusion Models Part I Review of Poisson Processes

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## Presentation on theme: "SEEM Advanced Models in Financial Engineering Professor Nan Chen Nov Lecture Jump Diusion Models Part I Review of Poisson Processes"— Presentation transcript:

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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.
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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
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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
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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
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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.