#### Research Findings

On this page, you can find friendly summaries of my main research findings. First, I introduce a new class of models, called hybrid marked point processes, which allows one to model a state process that interacts with past-dependent events. Second, I find a new way of proving the existence of marked point processes that are specified in terms of their intensity. Note that you can first read about marked point processes, the intensity and Hawkes processes here.

##### HYBRID MARKED POINT PROCESSES

###### OBJECTIVE

Inspired by limit order books, my goal was to find a type of marked point process that is well-suited to the joint modelling of events and the time evolution of the state of a system. The outcome of my research was a new class of models, called hybrid marked point processes, which I introduce in the paragraphs below.

###### SPECIAL MARKS AND THE STATE PROCESS

We begin by specifying the kind of marks that we will consider and the information that these marks will carry. Remember that in a marked point process, one assigns to each event time a mark that describes the event. In a hybrid marked point process, each mark M   will be a couple (E  , X  ) that we will interpret in the following manner:

• the first component E   will correspond to the type of the event (e.g. a buy limit order);

• the second component X   will indicate the new state of the system right after the event (e.g. the volume sitting at the ask and bid prices).

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###### With such marks, we can model both a sequence of random events and the state process of a system. At each time t, we can define the current state of the system, denoted by X(t),  simply as the second component of the most recent mark.​ state of the system at time t

2    component of the most recent mark

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###### DEFINITION: HYBRID DYNAMICS

To define hybrid marked point processes, we need several ingredients. First, let us denote by H(t) the history of the marked point process at time t. In other words, H(t) is the collection of all the event times and corresponding marks before time t. Let us also use the notation (e,x) to refer to any possible mark: e is one of the possible event types and x is one of the possible states of the system. Second, we need a positive function                      of e and the history H(t). Third, we need a function

of x, e and the current state X(t). Besides, as a function of only the first variable x (i.e. when e and X(t) are fixed), the function     should be a probability distribution over all the possible states x. A hybrid marked point process with event function    and transition function      is then defined as a marked point process such that its intensity process satisfies      the intensity at time of events with the mark (e, x)

probability of transitioning from X(t) to x if an event of type e occurs

the intensity of events of type e given the current history H(t)

###### Next, we discuss the dynamics that such an intensity generates. We can prove that, given the current state X(t) and given that the next event is of type e, the first term     is exactly the probability that the system will transition to the state x at the next event. Moreover, we can prove that the second term     is exactly the intensity of events of type e, that is, the intensity of the counting process that counts all the events such that the first component of their marks is equal to e. For example,  the function     can be such that events behave like in a Hawkes process, but with the additional feature that the kernels can now depend on the state process, giving rise to more subtle dynamics. For instance, events of type A will precipitate events of type B only when they move the state process to some critical region, say. Finally, it is interesting to note that this class of marked point processes encompasses both Hawkes processes and continuous-time Markov chains as special cases, whence the qualifier hybrid.   ###### COMPARISON WITH HAWKES PROCESSES

Instead of working with a hybrid marked point process, one might want to consider a Hawkes process where the considered set of possible marks consists of all the couples (e,x) of possible event types e and states x. However, the dynamics of this process will be very different. Indeed, in such a Hawkes process, if an event of type e occurs, the probability distribution of the new state x does not depend on the current state X(t) but on the entire history H(t). Yet, at least in the case of limit order books, knowing the current state and the next event type is enough to (approximately) determine the next state. The above discussion tells us that hybrid marked point processes do replicate this event-state structure, which would not be the case of such a Hawkes process. In this sense, hybrid marked point processes are well-suited to the joint modelling of events and the time evolution of the state of a system.

For rigorous mathematical definitions and statements, please consult my paper.​

##### PATHWISE CONSTRUCTION VIA POISSON EMBEDDING

###### EXISTENCE PROBLEM

Just like Hawkes processes, hybrid marked point processes are defined in terms of their intensity: a given marked point process is hybrid only when its intensity satisfies the above equation. However, this equation depends on the marked point process itself through its history. Due to the self-referential nature of this definition, it is not clear a priori that such marked point processes exist. More generally, can one find a marked point process such that its intensity satisfies a given equation? For simplicity, we restrict here ourselves to the case where there are no marks attached to the events. In other words, we only consider the sequence of random times T  , T  , ... and we forget about the sequence of random marks. The existence problem becomes: how can one generate the random times T  , T  , ... such that their intensity satisfies a given equation (e.g. the one defining a Hawkes process)? I introduce below an intuitive technique, known as Poisson embedding, which can be used to address this problem.

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###### POISSON EMBEDDING TECHNIQUE

The main ingredient used in the Poisson embedding technique is a Poisson random measure, that is something that generates points randomly in the plane using two simple rules:

• if A and B are two disjoint regions of space, then the number of points inside A has to be independent from the number of points inside B;

• the number of points inside any region of space A has to follow a certain distribution, called Poisson distribution, that depends on the surface of A; in particular, the average number of points inside A should be equal to the surface of A.

​Now, let's first consider the case where we want to obtain an intensity         that is equal to a deterministic positive function of time f(t) (i.e. the intensity does not depend on the history of the marked point process). We generate the sequence of random times T  , T  , ... as follows. Use the above Poisson random measure to generate points in the plane. Draw the given deterministic function of time f(t).  Retain only the points below the curve f(t) and above the horizontal axis. Finally, take the time coordinates of these points as the random times T  , T  , ... Then, one can prove that the intensity          of this sequence of random times satisfies               f(t)  for all times t. In other words, we have proved that a point process with an intensity equal to f(t) exists. This result is quite intuitive: the bigger is f(t), the more likely we will have a point below the curve f(t) at time t and, thus, the more likely we will have an event at time t. Going back to the case where we want the intensity to depend on the history of the marked point process, one needs to draw a curve f(t) that changes with each scenario. For each different realisation of the Poisson random measure, a different curve f(t) will be required. Moreover, for each time t, the shape of the curve after time t will have to depend on the points that are below the curve before time t. One outcome of my research was to find a procedure to construct this adaptive curve f(t) and to prove that, under some reasonable conditions, this procedure always works. In particular, one needs to ensure that, in any finite time interval, the number of points below the curve remains finite (we do not want an infinite number of events in a finite amount of time). Besides, this procedure can be used to prove the existence of hybrid marked point processes, which is not the case of the classical procedure developed by Massoulié (1998). 1

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