piecewisedeterministic

Piecewise Deterministic (PwD) processes are a class of stochastic processes that are widely used in various fields such as finance, telecommunications, and operations research. These processes are characterized by their deterministic behavior between random jump times, and they are often used to model systems that exhibit both continuous and discontinuous changes.

A PwD process can be defined as a stochastic process {X(t), t ≥ 0} that evolves according to a deterministic differential equation between random jump times, and undergoes a random jump at each jump time. The jump times are typically modeled as a Poisson process or a more general renewal process, and the size of the jumps is often modeled as a random variable that is independent of the jump times.

One of the key features of PwD processes is their ability to capture the dynamics of systems that exhibit both continuous and discontinuous changes. For example, in finance, PwD processes are often used to model the price of a stock, which can exhibit both continuous price movements and sudden jumps due to news events or other exogenous shocks.

Another important feature of PwD processes is their ability to capture the dependence between the continuous and discontinuous components of the process. This is achieved by allowing the deterministic differential equation that governs the continuous component to depend on the history of the process, including the times and sizes of the previous jumps.

PwD processes have been extensively studied in the literature, and a number of results are available for their analysis and simulation. For example, it is known that the distribution of the process at a given time can be obtained by solving a system of integral equations, and that the process can be simulated using a combination of deterministic and stochastic algorithms.

In summary, PwD processes are a powerful and flexible class of stochastic processes that are widely used in various fields to model systems that exhibit both continuous and discontinuous changes. Their ability to capture the dynamics of such systems, as well as their dependence between the continuous and discontinuous components, makes them a valuable tool for researchers and practitioners alike.