# Steven Shreve Bk 2 Pg 467/468 Topic Martingale property

#### kdmfe

Dear experts,

Can anyone please help me clarify the below topic. I have searched the net but could not find any/meaningful explanation

 M(t) Represents compensated Poisson property as martingale property N(t) The Poisson process N(t) counts the number of jumps that occur at or before time t. λt The Poisson process N(t) has intensity λ ‘λt is the Expected value of the Poisson process in interval time [0,t].

Compensated Poisson process is defined as M(t) = N(t) – λt. Then M(t), Compensated Poisson process is a martingale.

Usually, the Poisson process has lines parallel to time (X axis) to show time until the next jump.

I understand the proof but not the above diagram.

Question1: What does straight line with downward slope mean? And what do they characterise?

Question2: Why is it important for me to understand that the compensated Poisson process is a Martingale

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#### PepeQuant

What does straight line with downward slope mean? And what do they characterise?
That is your -\lambda. It is downward sloping.

Why is it important for me to understand that the compensated Poisson process is a Martingale
The "compensation" comes up after you apply the change of measure ( i called it lambda over lambda, to be exact, lambda_tilt / lambda ). Then your discounted*S will become a martingale. This is the master equation you need to remember for change of measure under jump diffusion,

$dS=\mu S dt + \sigma S dW_t + S(t-)dQ_t$$Z_1=\exp {\int_0^t \theta dW_s - \frac{1}{2} \theta^2ds }$$Z_2=\exp{(\lambda - \lambda')t } \prod_s (\frac{\lambda'}{\lambda})^{N(s)}$$Z=Z_1Z_2$
Then you have,
$dS=rSdt+\sigma S dW_t^Q+S(t-)dM_t^Q = ( r-\lambda \mu )Sdt+\sigma S dW_t^Q+S(t-)dQ_t^Q$

#### kdmfe

Thank you PepeQuant for the prompt reply.

My Phd. sir will be coming in the evening. I will show your response to him and if need I will comeback with further doubts/clarifications.

I will be raising few more questions today. If your time permits, kindly take at them.

Thank you once again, a lot. Every small help means a lot to my anxiety to understand this subject.

#### kdmfe

Hello Pepe,

My understanding is lambda is the intensity of the Poisson process. Since Poisson is a counting process, lambda has to be a positive integer.

Here how does lambda take a negative value?

Is the downward slope because of the negative value?

#### kdmfe

You have listed Z1, Z2 and Z formulae. These are from chapter 11, equation 11.6.31/32.33, I think, thank you.

My Tutor is unclear about the derivation of the below differential of the stock process:

dS=rSdt+σSdWtQ+S(t−)dMtQ=(rλμ)Sdt+σSdWtQ+S(t−)dQtQ

Are you able to provide me further information how the above differential was arrived using the previous information

Thank you

Last edited:

#### Qui-Gon

Hello Pepe,

My understanding is lambda is the intensity of the Poisson process. Since Poisson is a counting process, lambda has to be a positive integer.

Here how does lambda take a negative value?

Is the downward slope because of the negative value?
I think you are misunderstanding the construction of the Poisson/compound Poisson process and should consider re-reading the first few sections of Ch. 11 since the rest of the derivations in the chapter will never “click” if you don’t understand the basics. Lambda does not need to be a positive integer.

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