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

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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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.