Compound Poisson Process

Compound Poisson Process

A compound Poisson process is a continuous time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is random. We can write the following expression for the ``value" of the process at time $t$. $$ Y(t) = \sum_{i=1}^{N(t)} D_i $$ Notice that in this expression $N(t)$ and all the $D_i$ values are random variables so $Y(t)$ is thus random. $N(t)$ is a Poisson process and the $D_i$ values are independent and identially distributed random variables. Furthermore, the probability distribution function for the $D_i$ random variables is independent of $N(t)$. It is possible to show that the mean and variance for $Y(t)$ are given by: $$ \mathbb{E}[Y(t)] = \mu \lambda t \qquad \qquad \qquad \textrm{var}[Y(t)] = \lambda t( \sigma^2 + \mu^2 ) $$ where $\mu$ and $\sigma$ are the expectation and variance for the random variable $D_i$. $\lambda$ is the parameter for the Poisson process $N(t)$.

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