This set of lectures is an introduction to the stochastic analysis of Markov jump processes with applications in biology. The goal is of using stochastic calculus in this context, the analogue of Itô’s calculus for Brownian motion, as an efficient (and simple) tool to study various examples. The Poisson process is the central mathematical object of this domain.

1. Definition and main properties of Poisson processes in a general state space.
2. Applications to stochastic models of gene expression.
3. Stochastic Calculus for marked Poisson processes: martingales and stochastic integrals. Stochastic differential equations driven by Poisson processes. Examples of Ehrenfest models, polymerization processes and neural cells.
4. Functional limit theorems. Law of large numbers and central limit theorems for polymerization processes with the volume as a scaling parameter.

A basic knowledge of martingale theory is assumed.

This is part of the lecture series funded by the NSF RTG grant DMS 2134107.

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