Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics

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Abstract

The objective of this paper is to study the filtering problem for a system of partially observable processes (X, Y), where X is a non-Markovian pure jump process representing the signal and Y is a general jump diffusion which provides observations. Our model covers the case where both processes are not necessarily quasi left-continuous, allowing them to jump at predictable stopping times. By introducing the Markovian version of the signal, we are able to compute an explicit equation for the filter via the innovations approach.

Introduction

In this paper we study a stochastic filtering problem for a partially observable system where the unobservable signal process is a pure-jump process, possibly non-Markovian, and takes values in a complete and separable metric space. The observation process is a jump–diffusion with local characteristics that depend on the signal. Our goal is to derive the conditional distribution of the signal process given the available information flow generated by the observation process, i.e., to compute the filter. In other terms we aim to write an explicit evolution equation satisfied by the filter, called filtering equation. Our contribution is methodological in nature: while some examples are shown and some possible applications are briefly outlined, our main intent is to provide a general partially observed stochastic model and to rigorously compute the corresponding filtering equation. The partially observed system considered in this paper has the following two novel features:

  • A non-Markovian pair signal-observed process, which is realized by allowing the data of our model (i.e., local characteristics) to be path-dependent with respect to the signal process.

  • Predictable jump times for both the signal and the observation, to account for models where such jump times arise naturally (for instance, Piecewise Deterministic Processes or reflecting diffusions).

Stochastic filtering in continuous-time is a classic branch of research in applied probability and has been studied under several choices of the pair signal-observation. Starting from the pioneering works of N. Wiener and R. E. Kalman (with its celebrated Kalman filter), research on this subject has vastly expanded in numerous directions and is still ongoing. A fairly detailed account of these developments can be found, for instance in Bain and Crisan [2, Chapter 1]. In this section we summarize the contributions that are more related to our paper.

One of the driving motivations to study stochastic filtering is that partially observed systems naturally arise in fields like engineering, operations research, economics, and finance. These stochastic models are characterized by a lack of information (so called partial information or partial observation), as some of the quantities involved in the model itself are not directly measurable and need to be estimated from the observable quantities.

From a technical perspective, various situations have been considered so far in the literature: linear and nonlinear, finite and infinite dimensional filters (even on manifolds), numerical schemes, and so on. One of the most ubiquitous assumptions in this literature is that the pair signal-observation solves a martingale problem for some infinitesimal generator, implying that it is a Markov process.

This fundamental assumption allows to obtain an explicit filtering equation. Such characterization of the filter has important implications. First, it is key to estimate statistics of the unobserved process. Second, although in general filtering equations do not admit solutions in closed form, one can resort to numerical schemes to approximate such solutions, which require at least explicit filtering equations, see e.g. Bain and Crisan [2] and Damian et al. [18]. Third, to solve optimal control problems with partial observation, one usually needs to have an explicit filtering equation and to apply the so-called separation principle, which enables to switch from the original optimization problem under partial information to an equivalent optimal control problem with complete information, where the hidden state is replaced by the filter, see, e.g., Calvia [8], Altay et al. [1], Colaneri et al. [17] and Calvia and Ferrari [10]. Under the Markovianity hypothesis, several filtering problems have been addressed. Partially observable systems with jump–diffusion observation are studied, for instance, in Grigelionis and Mikulevicius [23], Ceci and Colaneri [13] and the cases of pure-jump signal are discussed, for instance, by Calvia [9], Ceci and Gerardi [14] and Ceci and Gerardi [15].

In contrast to the classical setting, in a non-Markovian context results available in the literature appear to be more abstract in nature. Although general equations for this case are available, see, e.g., Szpirglas and Mazziotto [34], Kallianpur [29] and Liptser and Shiryaev [31], in these works the partially observable system is modeled via an abstract semimartingale decomposition, and consequently, the underlying partially observed models, and hence the filtering equation, are not explicit.

General non-Markovian partially observed models have potential valuable applications. In fact, there are interesting and well-known problems where state variables depend on (part of) the history of some non-directly observable stochastic process, and not exclusively on its current value. They arise, for instance, in models with delay or when dealing with exotic options in finance and in various optimal control problems with partial observation (see, e.g., Bandini et al. [3], [4] and Tang [35]).

In light of this, our first contribution is to set a partially observed model where the pair signal-observation can be non-Markovian and to compute an explicit filtering equation in this context. The lack of the Markov property is due to dependence of our modeling data (i.e., the local characteristics) on the path of the signal process. The signal is a pure-jump process which is allowed to take values in a fairly general state space (metric, complete, and separable). This is in contrast with the usual setting where the state space of the signal is either discrete and finite or Euclidean.

In our model the pair signal-observation has common jumps times, which may also be predictable. To best of our knowledge, this feature has not been yet considered in the literature, where the common assumption is to take only totally inaccessible jump times (cf. the definitions in Jacod and Shiryaev [27, Chapter I], see also Section 2.1). Nevertheless, predictable jumps naturally appear in various stochastic models and even in the definition of some classes of processes, as in the well-known case of Piecewise Deterministic Processes, introduced by M. Davis [19]. In the financial literature, for instance, recent works discussed modelization of real financial markets presenting critical announced random dates, typically related to possible situations of default (see, e.g., Fontana and Schmidt [22], Jiao and Li [28] and discussions therein) or to information delivered at a priori established dates (e.g., central banks updates of interest rates).

Let us now briefly outline some of the details of our analysis.

We consider a pair of stochastic processes (X,Y)=(Xt,Yt)t0, respectively called the signal process and the observed process, on some probability space endowed with a global filtration F=(Ft)t0. Processes X and Y are F-adapted; however, the available information is only given by the subfiltration Y=(Yt)t0 which is the completed and right-continuous natural filtration of Y. This means that the signal is not observable, and the information on the signal can only be retrieved through the process Y whose dynamics depends, more or less directly, on X. We assume that X is a pure-jump process, possibly non Markovian, described in terms of a random counting measure (see, e.g., Brémaud [7], Jacod [25], Jacod and Shiryaev [27]). In contrast to most of the existing literature, we do not assume that the dual predictable projection, or compensator, of X is quasi-left continuous and we allow for predictable jumps. Our objective is to characterize the filter π=(πt)t0, which permits to describe the conditional distribution of the path of X up to time t, given the available information Yt. Therefore, to compute it we need to consider functionals depending on the history of the signal process, which is another novelty of our paper.

Due to the lack of Markovianity of the signal, to solve the filtering problem we construct the history process X. This is an auxiliary process which keeps track of all past values and jump times of X and has the advantage of being Markovian with respect to its natural filtration. Moreover, such process shares the same pure jump nature as the original signal, and it is fully determined by its local characteristics, which can be computed in terms of those of the signal (see Proposition 3.3). The existence of a bijective function that maps X into X and vice versa (see Proposition 3.1), permits to characterize the filter π by addressing an equivalent filtering problem where the conditional distribution of the history process X is derived (see Lemma 4.1).

In this paper, we consider a quite general observation process Y which follows a jump–diffusion, whose local characteristics depend on the trajectory of the signal X, and may also have predictable jumps. We allow for the signal and the observation to have common (both predictable and totally inaccessible) jump times. Having such structure for the observation process permits us to model several information flows. For instance, if X and Y have only common jump times, then the filter would be very informative and able to detect all jump times of the signal, although its positions would still be unknown. Considering the case where Y has also disjoint jump times and a diffusion component, brings additional noise, and the filter is not necessarily able to identify all the jumps of the signal.

To solve the filtering problem we resort to the innovations approach, see, e.g. [2], [7], [29]. This is a classical technique in filtering theory which is particularly convenient when signal and observation have common jump times. Although the idea of the innovations approach is well known, it is anything but easy to apply it in our setting, due to the generality of our partially observable system.

The paper is organized as follows. This introduction concludes with a brief paragraph on the notations and conventions adopted here. In Section 2 we describe the partially observable system that we intend to analyze and we give all the assumptions. In Section 3 we introduce the history process, explain why it is useful to do so, and write the model previously described in this new setting. The filter is introduced in Section 4 and in Section 5 we provide the martingale representation theorem with respect to the filtration generated by the observed process, that is fundamental to derive the filtering equation. The latter is computed in Section 6. Finally, three illustrative examples are collected in Section 7. The proofs of some results stated in the paper are gathered in Appendix.

In this section we collect the main notation used in the paper.

Throughout the paper the set N denotes the set of natural integers N={1,2,} and N0=N{0}.

We indicate by N the collection of null sets in some specified probability space.

The symbol 1C denotes the indicator function of a set C, while 1 is the constant function equal to 1. The symbol ab denotes (a,b] for any <ab<+.

For a fixed metric space E, we denote by dE its metric and by Bb(E) the set of real-valued bounded measurable functions on E. The symbol B(E) indicates the Borel σ-algebra on E and we denote by P(E) the set of probability measures on E. The set of E-valued càdlàg functions on [0,+) is denoted by DE. We always endow it with the Skorokhod topology and the Borel σ-algebra. The set D˜EDE contains all trajectories in DE that are piecewise constant and do not exhibit explosion in finite time (i.e., if (tn)nN(0,+] is the collection of discontinuity points of some trajectory, then limntn=+).

For any given E-valued càdlàg stochastic process η=(ηt)t0 defined on a probability space (Ω,F,P), we denote by (ηt)t0 the left-continuous version of η (i.e., ηt=limstηs,P-a.s., for any t0). The notation ηt indicates the path of process η stopped at time t0, i.e., (ηt)t0 is the DE-valued stochastic process such that, for any t0, ηt={sηts}s0. If η is real-valued, that is E=R, we denote by Δηtηtηt the jump size of η at time t0.

Finally, with the word measurable we refer to Borel-measurable, unless otherwise specified.

Section snippets

The model

The aim of this section is to introduce the model for the partially observed system that we aim to study, which is composed of a pair of processes, respectively called the signal and the observed process. Recall that our objective is to derive the conditional distribution of the signal process given the information provided by the observed process. Put in other words, we aim to characterize the dynamics of the filter, which is rigorously defined in (4.1), equivalently to provide the filtering

The Markovianization procedure

In this section we construct the history process X=(Xt)t0 through a Markovianization procedure. The process X is tightly linked with the signal X and, most importantly, to its stopped trajectory (Xt)t0.

To characterize the dynamics of the filter one typically follows two steps: write, first, the F-semimartingale representation of the process (ϕ(Xt))t0, with ϕ:DER being a bounded a measurable function; then, derive its optional projection with respect to the observation filtration Y.

The filter

We introduce the filter π=(πt)t0, as πt(ϕ)=E[ϕ(Xt)Yt],t0,for any bounded and measurable function ϕ:DER. Recall that DE denotes the set of E-valued càdlàg functions on [0,+) and the notation Xt indicates the path of process X stopped at time t0, i.e., (Xt)t0 is the DE-valued stochastic process such that, for any t0, Xt={sXts}s0.

Since DE is a complete and separable metric space, the process π is well-defined, P(DE)-valued and Y-adapted. Moreover, π admits a càdlàg modification,

Martingale representation theorem under the observed filtration

In this section we consider the Markovianized model introduced in Section 3.1. To characterize the dynamics of the filter Π, see Eq. (4.2), it is necessary to provide a representation theorem for martingales with respect to the observed filtration Y. The form of the observed process Y suggests that Y-martingales can be represented as the sum of two stochastic integrals, respectively driven by a Y-Brownian motion and the Y-compensated jump measure of Y. Therefore, as a first step, we compute the

The filtering equation for Π

In this section we derive the SDE that characterizes the filter Π, see Theorem 6.1. We introduce the following random measures: ηfi((0,t]×B)0t{HΠs(f()1di,K(B,s,,Ys)(h)λm(s,)Qm(s,;dh))+ZΠs(f()1di,G(B,s,,Ys)(z)λn(s,,Ys)Qn(s,,Ys;dz))}ds,ηfp,m((0,t]×B)0tHΠs(f()1dp,K(B,s,,Ys)(h)Rm(s,;dh))dpsm,ηfp,n((0,t]×B)0tZΠs(f()1dp,G(B,s,,Ys)(z)Rn(s,,Ys;dz))dpsn,ρfi((0,t]×B)0tHΠs([f(h)f()]1di,K(B,s,,Ys)(h)λm(s,)Qm(s,;dh))ds,ρfp,m((0,t]×B)0tHΠs([f(h)f()]1dp,K(B,s,

Examples

In this section we present three examples which are covered by our general setting. For these examples we verify, first, that Assumptions 2.1, 2.2, 5.2, and 5.4 are satisfied and then we write the filtering equations.

Example 7.1 Deterministic Jump Times of the Signal

On a filtered probability space (Ω,F,F,P), we consider a pure-jump signal process X taking values in a discrete and finite space E, whose jump times are deterministic, i.e. Xt=nN0ζn1[tn,tn+1),t0,where t0=0, ζ0=eE, (tn)nN(0,+] is a deterministic sequence of time points, and (ζ

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This research was partially supported by the 2018 GNAMPA-INdAM project, Italy Controllo ottimo stocastico con osservazione parziale: metodo di randomizzazione ed equazioni di Hamilton–Jacobi–Bellman sullo spazio di Wasserstein.

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