We introduce the class of hybrid marked point processes, which incorporate a state process that interacts with past-dependent events. For example , like in a Hawkes process,  events can exhibit self- or cross-excitation effects  , but these effects can now also depend on the state process. Events of type A will precipitate events of type B only when they move the state process to some critical region, say. In parallel, as each event occurs, the state process transitions to a new value according to transition probabilities that vary with the event type. We prove that such dynamics are equivalent to an intensity process of a specific product form. Our main result addresses the existence of non-explosive marked point processes with given intensities by studying a well-known Poisson-driven SDE (via Poisson embedding). The existing strong existence and uniqueness results rely on a Lipschitz condition that the intensity of a hybrid marked point process may fail to satisfy. This motivates us to propose a natural pathwise construction that instead requires only sublinear behaviour of the intensity. Using a domination argument, we are able to verify that this construction yields indeed a solution. As we restrict ourselves to non-explosive marked point processes, we also manage to prove uniqueness without any specific assumptions on the intensity.
We introduce the class of hybrid marked point processes, which incorporate a state process that interacts with past-dependent events. A particular example of a hybrid marked point process is a state-dependent Hawkes process, where events can exhibit self- or cross-excitation effects , like in classical Hawkes processes , but these effects can now also depend on the state process. Events of type A will precipitate events of type B only when they move the state process to some critical region, say. In parallel, as each event occurs, the state process transitions to a new value according to transition probabilities that vary with the event type. We prove that such dynamics are equivalent to an intensity process of a specific product form. Our main result addresses the existence of non-explosive marked point processes with given intensities by studying a well-known Poisson-driven SDE (via Poisson embedding). The existing strong existence and uniqueness results rely on a Lipschitz condition that the intensity of a hybrid marked point process may fail to satisfy. This motivates us to propose a natural pathwise construction that instead requires only sublinear behaviour of the intensity. Using a domination argument, we are able to verify that this construction yields indeed a solution. As we restrict ourselves to non-explosive marked point processes, we also manage to prove uniqueness without any specific assumptions on the intensity.
R
For example , like in a
A particular example of a hybrid marked point process is a state-dependent
A
where
A
, like in classical Hawkes processes