Question

Suppose we have observed a Poisson process up to time $s$, have seen that $N(s)=k$, and are interested in the value of $N(s+t)$ for small positive $t$. Show that $$ \begin{aligned} & \mathbb{P}\{N(s+t)=k \mid N(s)=k\}=1-\lambda t+O\left(t^2\right), \\ & \mathbb{P}\{N(s+t)=k+1 \mid N(s)=k\}=\lambda t+O\left(t^2\right), \\ & \mathbb{P}\{N(s+t) \geq k+2 \mid N(s)=k\}=O\left(t^2\right), \end{aligned} $$ where $O\left(t^2\right)$ is used to denote terms involving $t^2$ and higher powers of $t$. 526 11 Introduction to Jump Processes

     Suppose we have observed a Poisson process up to time $s$, have seen that $N(s)=k$, and are interested in the value of $N(s+t)$ for small positive $t$. Show that
$$
\begin{aligned}
& \mathbb{P}\{N(s+t)=k \mid N(s)=k\}=1-\lambda t+O\left(t^2\right), \\
& \mathbb{P}\{N(s+t)=k+1 \mid N(s)=k\}=\lambda t+O\left(t^2\right), \\
& \mathbb{P}\{N(s+t) \geq k+2 \mid N(s)=k\}=O\left(t^2\right),
\end{aligned}
$$
where $O\left(t^2\right)$ is used to denote terms involving $t^2$ and higher powers of $t$.
526
11 Introduction to Jump Processes

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 11, Problem 2

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Step 1

This means that the number of events in any interval of time depends only on the length of the interval, not on when the interval occurs. Therefore, $ N(s+t) - N(s) $ is independent of $ N(s) $ and follows a Poisson distribution with parameter $ \lambda t $. Show more…

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Suppose we have observed a Poisson process up to time $s$, have seen that $N(s)=k$, and are interested in the value of $N(s+t)$ for small positive $t$. Show that $$ \begin{aligned} & \mathbb{P}\{N(s+t)=k \mid N(s)=k\}=1-\lambda t+O\left(t^2\right), \\ & \mathbb{P}\{N(s+t)=k+1 \mid N(s)=k\}=\lambda t+O\left(t^2\right), \\ & \mathbb{P}\{N(s+t) \geq k+2 \mid N(s)=k\}=O\left(t^2\right), \end{aligned} $$ where $O\left(t^2\right)$ is used to denote terms involving $t^2$ and higher powers of $t$. 526 11 Introduction to Jump Processes

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