Dear experts,

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

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

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