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 processof 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 assumptionson the intensity .
We introduce a class of hybrid marked point processes that encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class provides a unified treatment of such existing processes, it also contains new processes with complex dynamics. These marked point processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. In fact, this property completely characterises the class of hybrid marked point processes. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process that interacts with its state process . Our main theoretical results establish the existence and uniqueness of hybrid marked point processes under very general assumptions, extending the results of Massouli\'e (1998) on interacting point processes .
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that encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class provides a unified treatment of such existing processes, it also contains new processes with complex dynamics. These marked point processes are defined implicitly via their intensity and are endowed with
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In fact, this property completely characterises the class of
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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 processof a specific product form
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result addresses the existence
theoretical results establish the existence and uniqueness
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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
R
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 assumptionson the intensity
processes under very general assumptions, extending the results of Massouli\'e (1998) on interacting point processes