
In this work we define a kinetic model for understanding the impact of heterogeneous opinion formation dynamics on epidemics. The considered many-agent system is characterized by nonsymmetric interactions which define a coupled system of kinetic equations for the evolution of the opinion density in each compartment. In the quasi-invariant limit we may show positivity and uniqueness of the solution of the problem together with its convergence towards an equilibrium distribution exhibiting bimodal shape. The tendency of the system towards opinion clusters is further analyzed by means of numerical methods, which confirm the consistency of the kinetic model with its moment system whose evolution is approximated in several regimes of parameters.
Citation: Sabrina Bonandin, Mattia Zanella. Effects of heterogeneous opinion interactions in many-agent systems for epidemic dynamics[J]. Networks and Heterogeneous Media, 2024, 19(1): 235-261. doi: 10.3934/nhm.2024011
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In this work we define a kinetic model for understanding the impact of heterogeneous opinion formation dynamics on epidemics. The considered many-agent system is characterized by nonsymmetric interactions which define a coupled system of kinetic equations for the evolution of the opinion density in each compartment. In the quasi-invariant limit we may show positivity and uniqueness of the solution of the problem together with its convergence towards an equilibrium distribution exhibiting bimodal shape. The tendency of the system towards opinion clusters is further analyzed by means of numerical methods, which confirm the consistency of the kinetic model with its moment system whose evolution is approximated in several regimes of parameters.
Markov processes [1,2], and in particular Markov jump processes (MJPs), are widely used both as models of real-world phenomena and as tools for solving a broad range of analysis, estimation/identification, and control/optimization problems [3]. Their popularity stems from the simplicity of their probabilistic description, the efficiency of their mathematical framework for system analysis, and the potential for stochastic analysis under additional but non-restrictive assumptions. However, the need for more precise modeling of various jump-like phenomena leads to increasingly complex mathematical models, resulting in the emergence of multivariate point processes (MPPs) [4]. Markov renewal processes [5,6] offer a middle ground between MJPs and MPPs. Any jump-like process can be expressed as a sequence of the pair "inter-arrival time – process value". For MRPs, this sequence forms a Markov chain. This allows for the relaxation of the Markov property in continuous time while still describing the process distribution via the transition kernel. However, this simplification comes at the cost of significantly complicating optimal estimation and control of MRPs. Specifically, the dynamics of a general MRP are not governed by a single finite-dimensional stochastic system, but rather by a sequence of such systems glued at random MRP transition times.
The goal of this paper is to introduce a subclass of MRPs of practical interest. This subclass is broader than finite-state CTMCs and shows promise as a model for various jump-like processes arising in fields such as tracking and navigation, telecommunications, financial mathematics, etc. The paper does not aim to describe the broadest class of MRPs for subsequent theoretical inferences. Instead, it focuses on MRPs that allow a straightforward "kinematic" description via the solution to a linear SDS with a martingale on the RHS. The advantage of this approach lies in the relative simplicity of filtering estimates, which, in turn, facilitates the design of efficient numerical filtering algorithms.
The paper is structured as follows: Section 2 introduces the subclass of MRPs under investigation, detailing their key probabilistic properties and showing how any arbitrary function of the MRP can be represented as the solution to a closed linear SDS with a martingale on the RHS.
Section 3 is dedicated to formulating the optimal filtering problem. The hidden system state is modeled by the studied MRP, while the available observations consist of both continuous diffusion and counting components. The drift coefficients of all observations are functions of the system state. The optimality criterion is the traditional mean squared error, meaning the filtering problem involves finding the conditional expectation (CE) of a signal process based on the available observations.
Section 4 forms the core of the paper and contains the solution to the filtering problem. The optimal filtering estimate of the signal process is expressed through the solution of a potentially non-closed SDS with innovation processes on the RHS. To address this non-closedness, we derive an analog of the Kushner-Stratonovich equation, which describes the evolution of the conditional PDF and aids in computing the CE for arbitrary functions of the MRP state.
Section 5 presents the results of an extensive numerical analysis of the proposed filter. A monitoring problem is considered, in which the qualitative state and numerical characteristics of communication channels are inferred from noisy observations of RTT and packet loss flows. The robustness of the filter against uncertainty in the MRP distribution is also demonstrated.
Section 6 concludes the paper with final remarks.
First, we present an informal description of a subclass of Markov renewal processes (MRPs), which we later use as a hidden state in stochastic observation systems.
We introduce the process Zt≜col(θt,Yt)∈RN+M, t∈[0,T]. The first N-dimensional component θt represents a CTMC with the state set SN≜{e1,…,eN} (here, SN stands for the set of the unit coordinate vectors in RN), initial distribution p0≜col(p10,…,pN0), and the transition rate matrix (TRM) Λ(t). The second compound component Yt∈RM also has piece-wise constant paths changing synchronously with θt. If {τn}n∈N is a sequence of θt transition instants and {θτn}n∈N is known, then {Yτn}n∈N is a sequence of mutually independent random vectors with the conditional distribution Π(B)≜col(Π1(B),…,ΠN(B)):
P{Yτn∈B|θ[0,T]}=∫Bθ⊤τnΠ(dy)P−a.s.∀B∈B(RM). |
The distribution of the initial value Y0 is defined via Φ(A)≜col(Φ1(A),…,ΦN(A)) quite analogously:
P{Y0∈B|θ[0,T]}=∫Bθ⊤0Φ(dy)P−a.s.∀B∈B(RM). |
One can interpret the sub-vector Yt as a CTMC with the state set constituted by the random vectors with the distribution Π(⋅). There exists a Wiener–Poisson canonical space with filtration (Ω,F,P,{F}t∈[0,T]) [7], such that the introduced process Zt is properly defined on it and Ft-adapted.
Further in the presentation we use the following notations:
– 1 is a row vector of an appropriate dimensionality formed by units.
– 0 is a zero row vector of an appropriate dimensionality.
– I is a unit matrix of an appropriate dimensionality.
– IB(x) is an indicator function of the set B.
– Nt is the number of process θt transitions occurring on the interval [0,t].
– Tn(s,t)≜P{Nt−Ns=0|θs=en}=exp(∫tsΛnn(u)du) (0⩽s<t) is a survival function of θt inter-arrival time under the condition θs=en; T(s,t)≜row(T1(s,t),…,TN(s,t)).
– λ(t)≜row(Λ11(t),…,ΛNN(t)) is a row-vector collecting the diagonal elements of TRM Λ(t); ˜Λ(t)≜Λ(t)−diag λ(t).
– P(s,t)=‖Pij(s,t)‖i,j=¯1,N is a transition matrix of the CTMC θt on the interval [s,t] (s⩽t): Pij(s,t)≜P{θt=ej|θs=ei}; P(s,t) is a solution to the Kolmogorov system: P′t(s,t)=P(s,t)Λ(t),0⩽s<t,P(s,s)≡I.
– Any function f(e,y):SN×RM→R can be rewritten in the form f(e,y)=¯f(y)e, where ¯f(y)≜row(f(e1,y),…,f(eN,y)).
– Enf≜∫RMf(y,en)Πn(dy), Ef=col(E1f,…,ENf)=∫RMdiag ¯f(y)Π(dy).
– Any probability measure Q(⋅) on (SN×RM,2SN×B(RM)) can be expressed via the distribution mQ(B)=col(mQ1(B),…,mQN(B)), where mn(B)≜Q{θ=en,Y∈B}.
By definition, Zt is Markovian on (Ω,F,P,{Ft}t∈[0,T]) and has the following properties [2]:
1) The matrix-valued transition probability function P(y,s,B,t)=‖Pij(y,s,B,t)‖i,j=¯1,N: Pij(y,s,B,t)≜P{θt=ej,Yt∈B|θs=ei,Ys=y},0⩽s<t⩽T has the form
P(y,s,B,t)=IB(y)diag T(s,t)+diag Π(B)(P(s,t)−diag T(s,t)). | (2.1) |
The function P(y,s,B,t) is a solution to the system
{P′t(y,s,B,t)=P(y,s,B,t)diag λ(t)+diag (Π(B))P(s,t)˜Λ(t),P′t(s,t)=P(s,t)Λ(t),0⩽s<t⩽T,P(y,s,B,s)=IB(y)I,P(s,s)=I. | (2.2) |
2) For an arbitrary bounded Borel function f(e,y):SN×RM→R, the infinitesimal generator At of the process Zt has the form
Atf(e,y)≜limh↓0E{f(θt+h,Yt+h)|θt=e,Yt=y}−f(e,y)h=[¯f(y)diag λ(t)+E⊤f˜Λ⊤(t)]e. | (2.3) |
3) For an arbitrary probability distribution m(B)=col(m1(B),…,mN(B)) on (SN×RM,2SN×B(RM)) the operator A∗t, adjoint to the generator At, has the form
A∗tm(B)=diag (λ(t))m(B)+diag (Π(B))˜Λ⊤(t)m(RM). | (2.4) |
4) The distribution P(t,B)=col(P1(t,B),…,PN(t,B)) of the process Zt (Pn(t,B)≜P{θt=en,Yt∈B}) is a solution to the system
P′t(t,B)=diag (λ(t))P(t,B)+diag (Π(B))˜Λ⊤(t)P(t,RM),P(0,B)=diag (ϕ(B))p0. | (2.5) |
Formally, the compound component Yt of the process Zt represents an MRP. The MJP θt serves as a subordinator: it defines the transition times {τn} for Yt and selects the distributions Πn(⋅) for the values of Yτn. In general, the MRP Yt is non-Markovian. It exhibits the Markov property only when the support sets of the distributions Πn(⋅) are disjointed [8]. Below in the presentation, the components θt and Yt act as the hidden states in stochastic observation systems aimed at estimating these states. Therefore, we combine them into Zt and refer to it as the MRP.
The proposed class of MRPs is a promising tool for modeling real-world phenomena characterized by a combination of "qualitative state" and "numerical characteristics". Let us illustrate this concept.
First, consider the motion of a maneuvering target [9,10]. As is well-known, maneuvers are performed with various accelerations, which we assume to follow a piecewise constant process [11]. Typically, there are only a few types of maneuvers, such as:
– Nearly uniform rectilinear motion.
– Uniformly accelerated or decelerated rectilinear motion.
– Turning.
Thus, the process θt of maneuver changes has three possible states and can be modeled as an MJP [12]. Note, that the numerical parameters Yt of a particular maneuver can vary in each instance. These parameters are described by a three-dimensional vector, including tangential acceleration and two components of normal acceleration.
Second, consider the price evolution of a risky asset. It is assumed that the financial market is governed by a hidden regime-switching process θt [13,14]. There are relatively few possible market regimes [15,16], making it natural to describe them using a finite-state MJP [17]. The local numerical characteristics of the market Yt, including the instantaneous interest rate and volatility, are random and follow specific probability distributions for each regime.
The third example of applying the proposed class of MRPs to mathematical modeling relates to telecommunications and is detailed in Section 5.
Let us introduce a process f(Zt) that is a scalar function of Zt: f=f(e,y):SN×RM→R. It is easy to verify that f(Zt) can be expressed as a linear function of the process Ft≜col(θt,f(Zt)θt)∈R2N: f(Zt)=row(0,1)Ft. The process Ft is called the process associated with f(Zt). Below in the presentation we reserve the notation ft for the second N-dimensional sub-vector of Ft: ft≜f(Zt)θt.
From the Dynkin formula [2], it follows the martingale representation of an arbitrary function of Zt. If f=f(e,y):SN×RM→R is the function, such that
∫RM‖¯f(y)‖2(Π(dy)+Φ(dy))<∞, | (2.6) |
then the process Ft has a finite second moment and represents the unique strong solution to the linear SDS
Ft=F0+∫t0Df(s)Fs−ds+μft, | (2.7) |
where Df(t) is 2N×2N-dimensional matrix-valued non-random function
Df(t)≜[Λ⊤(t)0diag Ef˜Λ⊤(t)diag λ(t)] |
and μft∈R2N is an Ft-adapted square integrable martingale.
If the MRP Zt satisfies the condition ∫RM‖y‖2(Π(dy)+Φ(dy))<∞, then the process Zt≜col(θt,Y1tθt,…,YMtθt)∈R(M+1)N, associated with Zt, is a solution to the linear SDS
Zt=Z0+∫t0DZ(s)Zsds+μZt, | (2.8) |
where DZ(t) is (M+1)N×(M+1)N-dimensional matrix-valued non-random function
DZ(t)≜[Λ⊤(t)00…0diag EY1˜Λ⊤(t)diag λ(t)0…0diag EY2˜Λ⊤(t)0diag λ(t)…0……………diag EYM˜Λ⊤(t)00…diag λ(t)] |
and μZt∈R(M+1)N is an Ft-adapted square integrable martingale.
The generalized Itô rule allows us to derive a formula for the mutual quadratic characteristic of the functions of the MPRs. If g=g(e,y):SN×RM→R is another function, satisfying (2.6) and the process Gt≜col(θt,g(Zt)θt)∈R2N, associated with g(Zt), admits the martingale representation similar to (2.7) with a martingale μgt, then the mutual quadratic characteristic of ft and gt components has the form
⟨f,g⟩t=∫t0[diag Efgdiag (˜Λ⊤(s)θs)−diag (fs)diag λ(s)diag (gs)−diag (fs)˜Λ(s)diag Eg−diag Ef˜Λ⊤(s)diag (gs)]ds. | (2.9) |
The class of MRPs under study is, of course, not as general as the one proposed in [18]. Nevertheless, the fact that an arbitrary function of the studied MRP can be expressed through the solution of a linear SDS has the potential to simplify the form of the optimal filtering equations. Moreover, it could lead to a successful solution of conditionally-optimal filtering problems, such as optimal polynomial filtering. Additionally, the adjoint operator A∗ is sufficiently simple, allowing the temporal evolution of the process distribution to be described by the functions T(s,t) and P(s,t). This provides a relatively simple analog to the Kushner-Stratonovich equation, which describes the evolution of the conditional distribution of the state process. In the case of a diffusion observation system, the Kushner-Stratonovich equation falls within the class of stochastic partial integro-differential equations. However, in the present case, it is expected that the Kushner-Stratonovich equation will take the form of a stochastic integral equation. This simplified form suggests the potential for various efficient algorithms to solve it numerically.
On the Wiener–Poisson basis (Ω,F,P,{F}t∈[0,T]), we consider the observation system
Zt=Z0+∫t0DZ(s)Zsds+μZt,qt=q(Zt)=¯q(Yt)θt,ξt=∫t0g(Zs)ds+∫t0R1/2dws,ηt=∫t0h(Zs)ds+μηt. | (3.1) |
Here,
– Zt∈R(M+1)N is a system state, which is the process associated with the MRP Zt∈RN+M; it is described by the martingale representation (2.8).
– qt∈R is an estimated scalar process, which is a function of the MRP Zt.
– ξt∈RK is a continuous observation process.
– ηt∈RL is an observable process with counting components.
In the observation system (3.1)
– μZt∈R(M+1)N and μηt∈RL are Ft-adapted martingales from the representations of the MRP Zt and counting observations ηt.
– g(Zt)=g(e,y):SN×RM→RK and h(Zt)=h(e,y):SN×RM→RL constitute the "useful signals", governed by the estimated state, in the observations.
– wt∈RK is an Ft-adapted standard Wiener process, characterizing the noise in the continuous observations ξt; the matrix-valued function R denotes the noise intensity in the observations ξt (here, R1/2 stands for the symmetric square root of the non-negative square matrix R).
For the proper formulation of the optimal filtering problem, we should make the following assumptions concerning the observation system (3.1):
A1. In the Wiener-Poisson basis Ft≡σ{Zs,ws,ηs:0⩽s⩽t}.
A2. All trajectories of Zt and ηt are càdlàg functions.
A3. The TRM Λ(t) consists of the piecewise continuous components.
A4. For any n=¯1,N the functions g(en,y), h(en,y), and q(en,y) are continuous in y. There exists a constant C1 such that ∑Nn=1∫RM(‖g(en,y)‖2+‖h(en,y)‖2+q2(en,y))(Π(dy)+Φ(dy))⩽C1<∞.
A5. There exists a constant C2>0 such that minen∈SNh(en,y)>C2 and R⩾C2I.
A6. The components of the martingale μηt are orthogonal to each other, i.e.,
⟨μη,μη⟩t≡∫t0diag h(s,Zs)ds. | (3.2) |
A7. The purely discontinuous martingales μZ and μη are strongly orthogonal, i.e., ⟨μZ,μη⟩t≡0.
Let Ot≜{ξs,ηs:0⩽s⩽t} be a natural filtration generated by the observations available up to time t. The optimal filtering problem for the signal process qt is to find ˆqt≜E{qt|Ot}.
Conditions A1 and A2 ensure the correct application of the stochastic analysis framework [19], and Eq (3.1) represents a properly defined SDS. Condition A1 also implies that all randomness in the canonical space is generated solely by the random processes in (3.1).
Condition A3 guarantees that the transition matrix P(s,t) of the CTMC θt is the solution to the Kolmogorov differential system.
Condition A4 assures that the state Zt, observations ξt, ηt, and the estimated process qt have finite moments up to the second order; hence, the CE ˆqt is optimal in the mean square sense.
Condition A5 is standard for the filtering problem. It means the uniform non-degeneracy of both the observation noise in the continuous observations ξt and intensity of the counting observations ηt. The condition also guarantees the legitimacy of the Girsanov change of measure.
Any observable counting process can be factorized into the process with the orthogonal components, satisfying Condition A6; the mutual jumps in various components of the "genuine" counting observations could be separated into the new counting observable processes.
Finally, Conditions A1–A6 look mainly technical but are non-restrictive in practice.
Below in the presentation we use the notation ˆct≜E{c(Zt)|Ot} for any function c(e,y), for which ∑Nn=1∫Rm‖c(en,y)‖2(Π(dy)+Φ(dy))<+∞, and ˆct−≜lims↑tˆcs.
Theorem 4.1. The optimal filtering estimate ˆqt of the signal process q(Zt) has the form ˆqt=row(0,1)ˆQt, where ˆQt=E{col(θt,q(Zt)θt)|Ot} is a filtering estimate of the process Qt associated with q(Zt); ˆQt is a solution to the SDS
ˆQt=E{Q0}+∫t0DQ(s)ˆQs−ds+∫t0(^Qg⊤s−−ˆQs−^g⊤s−)R−1/2dνs+∫t0(^Qh⊤s−−ˆQs−^h⊤s−)diag −1(ˆhs−)dζs, | (4.1) |
where
νt=R−1/2∫t0(dξs−ˆgs−ds) | (4.2) |
and
ζt=∫t0(dηs−ˆhs−ds) | (4.3) |
are the innovation processes.
Proof of Theorem 4.1 is given in Appendix A.
Remark 4.1. In the case of pure counting observations, the possibility of describing the optimal filtering estimate using Eq (4.1) has been proven in a recent and novel paper [18] for the general class of MRPs.
One can see that Eq (4.1), which describes the evolution of the estimate ˆQt, is not a closed-form equation. Besides the estimate itself, the RHS includes the CEs of the observation drifts and their products with the estimated process. The question of whether or not an optimal filtering estimate is a solution to a finite-dimensional system has its history [20,21,22]. The general answer sounds rather pessimistic: the optimal filters have a finite-dimensional form only for some narrow class of the observation systems [23,24,25].
An alternative approach to solving the optimal filtering problem involves finding the conditional distribution of the state Zt. Let ˆψ(t,y)≜col(ˆψ1(t,y),…,ˆψN(t,y)) denote the conditional PDF, defined such that the identity P{θt=en,Yt∈B|Ot}≡∫Bˆψn(t,y))dy holds P-a.s. for all t∈[0;T], n=¯1,N and B∈B(RM). The optimal filtering estimate ˆqt=E{q(Zt)|Ot} takes the form
ˆqt=N∑n=1∫RMq(en,y)ˆψn(t,y)dy=∫RM¯q(y)ˆψ(t,y)dy. |
If the observation system is described by the diffusion processes, then under certain additional assumptions, the conditional PDF exists and satisfies the Kushner-Stratonovich equation [26]. We impose an additional assumption to derive an analogous equation for the observation systems in (3.1).
A8. The distributions Π(⋅) and Φ(⋅) are absolutely continuous with respect to the Lebesgue measure. The corresponding vector-valued PDFs are π(y)=col(π1(y),…,πN(y)) and ϕ(y)=col(ϕ1(y),…,ϕN(y)).
Condition A8 ensures that the distribution P(t,B)=col(P1(t,B),…,PN(t,B)) of the process Zt has a PDF ψ(t,y)=col(ψ1(t,y),…,ψN(t,y)). This PDF satisfies the following Kolmogorov system
ψ′t(t,y)=diag (λ(t))ψ(t,y)+diag (π(y))˜Λ⊤(t)∫RMψ(t,u)du,ψ(0,y)=diag (p0)ϕ(y). |
Theorem 4.2. Under conditions A1–A8, the evolution of ˆψ(t,y) is described by the system
ˆψ(t,y)=diag (p0)ϕ(y)+∫t0[diag λ(s)ˆψ(s−,y)+diag π(y)˜Λ⊤(s)ˆθs−]ds+∫t0diag (ˆψ(s−,y))(¯g(y)−ˆgs−1)⊤R−1/2dνs+∫t0diag (ˆψ(s−,y))(¯h(y)−ˆhs−1)⊤diag −1(ˆhs−)dζs, | (4.4) |
where ˆθs, ˆgs and ˆhs− are expressed via ˆψ(s,y):
ˆθs=∫RMˆψ(s,y)dy,ˆgs=∫RM¯g(y)ˆψ(s,y)dy,ˆhs−=∫RM¯h(y)ˆψ(s−,y)dy, | (4.5) |
and ¯g(y) and ¯h(y) are the matrix-valued functions
¯g(y)≜[g1(e1,y)…g1(eN,y)………gK(e1,y)…gK(eN,y)],¯h(y)≜[h1(e1,y)…h1(eN,y)………hL(e1,y)…hL(eN,y)]. |
Proof of Theorem 4.2 is given in Appendix B.
To demonstrate the performance of the proposed filtering estimate, we present an applied problem in the field of communications. The problem involves monitoring the qualitative state and numerical characteristics of heterogeneous (wired/wireless) communication channels, which are hidden from direct observation, with only indirect noisy observations available.
The "channel health" is fully described by the MRP Zt≜col(θt,Yt). The first component, θt, which represents the qualitative state, is a CTMC with values in S4:
– θt=e1: moderate load, where the "bottleneck" buffer of the channel is empty.
– θt=e2: pre-congestion state, indicating that the "bottleneck" buffer is non-empty.
– θt=e3: congestion phase, where the "bottleneck" buffer is full.
– θt=e4: signal loss in the wireless channel hop.
The TRM Λ and the initial distribution p0 of the CTMC are
Λ=[−3.825×10−23.75×10−207.5×10−41.5×10−1−2.01×10−15.025×10−27.5×10−402.4975×10−1−2.505×10−17.5×10−42.4975×10−100−2.4975×10−1],p0=[0.7480.1860.0370.029]. |
The second component, Yt∈R2, represents the current numerical characteristics of the channel:
– Y1t is a current RTT value.
– Y2t indicates the fraction of lost packets in the packet flow.
Given the trajectory of the CTMC {θt}, the components Y1t and Y2t are mutually independent and have the uniform conditional PDFs π1(y1|θt) and π2(y2|θt):
π1(y1|θt=e1)=100×I[0.01;0.02](y1),π1(y1|θt=e2)=142.857×I[0.018;0.025](y1), |
π1(y1|θt=e3)=200×I[0.022;0.027](y1),π1(y1|θt=e4)=250×I[0.026;0.03](y1), |
π2(y2|θt=e1)=1000×I[0.0005;0.0015](y2),π2(y2|θt=e2)=1000×I[0.0005;0.0015](y2), |
π2(y2|θt=e3)=11.111×I[0.01;0.1](y2),π2(y2|θt=e4)=4×I[0.05;0.3](y2). |
The observations include the noisy RTT measurements
ξt=∫t0Y1sds+0.0002wt, | (5.1) |
and the counting process of the packet losses
ηt=∫t0Y2sY1sds+μηt. | (5.2) |
The channel state filtering problem involves the online estimation of the process Zt=col(θt,Yt) given the available observations ξt and ηt.
The nonlinearity in the drift term of (5.2) arises because the instantaneous packet loss intensity is proportional to the packet loss fraction, while the total packet sending rate is inversely proportional to the RTT.
To avoid errors introduced by the numerical discretization of continuous-time processes, we use different time scales for simulating the state, observations, and the state filtering procedure: δt=10−4 for the simulation and Δt=10−2 for the estimation procedure.
Since the estimation problem requires solving the Kushner-Stratonovich equation, we apply a grid method with space step sizes δy1=10−3 and δy2=6×10−3. For the filtering process, we utilize a hybrid algorithm combining the MJP filtering method for time-discretized observations [27,28] and the Euler–Maruyama algorithm adapted for jumps [29]. The idea of the algorithm is to approximate the continuous component Yt of the MRP on a finite grid with Nm nodes. The proposed algorithm is stable, ensuring that the resulting numerical solution satisfies the natural conditions of non-negativity and normalization, which are expected for the actual solution ˆψn(t,y). Analyzing the algorithm [28], we can conclude that the computation of each time layer ˆψn(ti,⋅) at time instant ti requires O(N3×N3m) operations.
Although the example is artificial, it exhibits several noteworthy features with practical significance. First, the chosen numerical characteristics of the channel, representing various qualitative states, closely approximate real-world values. Second, the noise intensity in the RTT observations is relatively high. Third, the average duration of states e3 and e4 is rather short, making these states difficult to identify. Fourth, the support sets Dn of the component Y distributions overlap across different channel states θ, complicating the recovery of θ from observations of Y. Lastly, the drift in the counting observations is the nonlinear function of the state. Together, these factors make the example particularly challenging.
Figure 1 presents the results of the observation system simulation, with the following elements:
– Color filling represents the current system state θt: from e1 until e4.
– The true value of the current RTT Y11.
– The true value of the current loss fraction Y2t (displayed on the auxiliary ordinate axis).
– The ratio ΔξtΔt, representing the continuous-time observation.
– The observable process ηt, reflecting the packet losses.
The observations exhibit minor fluctuations, indicating transitions in the state Zt. However, visually identifying the exact current state θt and especially estimating its numerical characteristics Yt, remains challenging. These subtle differences in the observations do not provide sufficient clarity for direct visual interpretation of the channel behavior.
Figure 2 shows the filtering performance for the CTMC θt:
– The true state θt.
– The filtering estimate ˆθct calculated by the continuous observations ξt.
– The filtering estimate ˆθt calculated by the continuous and counting observations ξt and ηt.
The figure highlights that incorporating counting observations significantly enhances the accuracy of the state estimate θt, particularly for the more dynamic "fast" states e3 and e4. These states likely involve more abrupt transitions, where counting observations provide valuable additional information.
Figure 3 presents the filtering results for the components of Yt:
– Color filling indicates the current state θt: from e1 until e4.
– The true values of Y1t and Y2t.
– The filtering estimate ˆYct calculated by the continuous observations ξt.
– The filtering estimate ˆYt calculated by the continuous and counting observations ξt and ηt.
The observation process ηt plays a crucial role in estimating Y2t, the lost packet fraction, as this component directly reflects the packet loss events. In the filtering process, incorporating ηt notably improves the accuracy of the filtering estimate for Y2t. This enhancement is especially evident in the time interval [0;12.0], where the system state remains θt≡e4 (signal loss state). During this interval, the continuous observations alone allow the estimate ˆθct to track the true state θt quite well. However, without access to ηt, the filtering estimate ˆY2,ct for Y2t merely aligns with the expectation of Y2t under the condition θt=e4, yielding an expected value of approximately 0.175, given the uniform distribution R[0.05;0.3]. This causes the estimate to oscillate near this mean value, regardless of the actual variations in Y2t.
By contrast, when ηt is utilized in the filtering process, the estimate ˆY2t becomes much more accurate, as ηt provides direct information about packet losses. This highlights that the lost packet fraction Y2t can only be accurately estimated by observing the flow of packet losses through the counting process ηt. Hence, combining both continuous and counting observations significantly enhances the filtering performance for Y2t.
The filtering errors in both ˆθt and ˆYt often exhibit peaks due to mismatches between the local random behavior of the observations and the true state of the system. This phenomenon is clearly seen in Figures 2 and 3, particularly in the time interval [7;10], where the filtering estimates temporarily deviate from the true values. During this period, both ˆY1t and ˆY2t tend to underestimate their actual values. Moreover, a noticeable peak appears in the conditional probability ˆθ1t, suggesting that the channel is moderately loaded (i.e., θt=e1) even though the true state is θt=e4 (signal loss phase). This discrepancy is explained by examining the observations in Figure 1, which show that during this interval, the continuous observation process δξtht is significantly lower than the actual RTT value. Additionally, no packet losses are recorded, making the local observations resemble those of a channel in a moderately loaded state with high throughput and minimal packet loss fraction. Despite this mismatch, the filtering error is short-lived. The estimation is rapidly corrected as new, more relevant observations become available, which help the filter realign with the true state of the system. This highlights the adaptive nature of the filtering algorithm, which compensates for temporary observation anomalies over time.
To illustrate the joint evolution of the conditional PDF ˆψ(t,y) and the filtering estimates ˆYt, we provide Figures 4 and 5. These figures focus on a short time interval [55.0;75.0], during which the process Zt undergoes several jumps.
Figure 4 and 5 contain similar plots:
– The true values of Y1t and Y2t.
– The filtering estimates ˆY1t and ˆY2t.
– Evolution of marginal conditional PDFs ˆψ1(t,y1) and ˆψ2(t,y2).
One can observe that after each transition of the state Yt, the corresponding filtering estimate is also recalibrated through modifications in the conditional PDF.
In practice, the variance of the filtering error serves not only as a performance index for the estimation but also as a basis for synthesizing control with incomplete information. In only a limited number of fortunate scenarios within stochastic observation systems can one derive the error variance analytically. The observation systems investigated in this article do not fall into this category, leading us to employ the Monte Carlo method (MCM) to compute the sample variance of the filtering error.
For our analysis, we utilize a sample size of Q=10000. We compare the variance of the filtering error to that of the estimated process itself. This comparison is significant because the variance of the estimated process can be viewed as a performance metric for the unconditional mathematical expectation, which serves as a trivial estimator. By examining these variances, we gain valuable insights into the effectiveness of our filtering approach.
Figure 6 presents the performance characteristics of the CTMC θt estimates:
– The value Dθt≜E{‖θt−E{θt}‖2}.
– The second moment Dˆθct≜E{‖ˆθct−θt‖2} of the estimate ˆθct error (calculated by the MCM).
– The second moment Dˆθt≜E{‖ˆθt−θt‖2} of the estimate ˆθt error (calculated by the MCM).
Figure 7 is analogous to Figure 6 and presents the performance characteristics of Yt estimates:
– The variances of the components Yit, i=1,2: DYit≜E{(Yit−E{Yit})2}, i=1,2.
– The second moment DˆYi,ct≜E{(ˆYi,ct−Yit)2} of ˆYi,ct error, i=1,2 (calculated by the MCM).
– The second moment DˆYit≜E{(ˆYit−Yit)2} of ˆYit error, i=1,2 (calculated by the MCM).
In addition to Figures 2 and 3, which illustrate specific trajectories of the system state and its estimates, Figures 6 and 7 support formal conclusions. Moreover, for each filtering estimate of the system state Zt or its sub-vectors, it is possible to calculate the residual variance ratio using the formula
ρˆZ≜∫T0E{‖Zt−ˆZqt‖2}dt∫T0E{‖Zs−E{Zs}‖2}ds. | (5.3) |
From the physical point of view, this integral characteristic represents the ratio of the error "power" to the "power" of the estimated signal itself. A value close to 0 indicates a highly accurate estimate. If the value is slightly lower than 1, the estimate is only marginally better than a trivial one. When the ratio exceeds 1, the proposed estimate is ineffective, as it performs worse than the trivial estimate.
The residual variance ratios of the estimates ˆθc, ˆY1,c, and ˆY2,c calculated using only the continuous observations ξt are as follows: ρˆθc=0.468, ρˆY1,c=0.122, and ρˆY2,c=0.322. In contrast, the corresponding values for the estimates using the entire observation set are ρˆθ=0.384, ρˆY1=0.117, and ρˆY2=0.064. Thus, the use of counting observations of packet losses significantly improves the estimation quality of the channel state θ, has a slight impact on the performance of the current RTT Y1 estimate, and greatly enhances the estimate of the lost packet fraction Y2.
The performance of ˆZt, derived from the entire set of available observations, may be regarded as unimpressive. However, the proposed estimate is optimal in the mean square sense, meaning its precision cannot be improved with the current observations. To enhance estimation precision, additional observers should be employed, and their observations incorporated into the filtering process.
In applied problems, the conditional distribution of the sub-vector Yt in the MRP Zt is partially or completely unknown. To design filtering algorithms that are robust to potential deviations of π(⋅), various approaches can be employed, particularly the minimax approach [30]. This method involves searching for the least favorable distribution π∗(⋅) and subsequently constructing the filter in the form of either (4.1) or (4.4). However, this approach has two significant drawbacks. First, the minimax filtering algorithm is computationally intensive. Second, the resulting estimates are typically overly conservative because they are designed to accommodate the least favorable choice of π(⋅), which often results in inappropriately low precision. The distribution π∗(⋅) can become impractical and unlikely to occur unless there is intentional counteraction.
To develop a more robust version of the filtering algorithm, we propose selecting a specific variant of the distribution π(⋅) and using it in either (4.1) or (4.4). In the case of the bounded support sets Dℓ of the distributions πℓ(⋅), we propose the uniform ones over Dℓ.
Let us illustrate the performance of the proposed robust filtering estimate using the example discussed above. We will consider three variants of distributions with the common support sets Dℓ:
– The continuous uniform distribution over the sets Dℓ (see previous subsection).
– The continuous symmetric triangular distributions over the sets Dℓ.
– Three-point uniform distributions concentrated at the ends of the intervals and their midpoints.
We will compare the filtering estimates using the MCM with a sample size of Q=5000.
Figure 8 presents the results of the numerical study of the CTMC θt estimates for the triangular distribution π(⋅):
– The unconditional mean square Dθt=Etr{‖θt−Etr{θt}‖2}.
– The mean square error (MSE) of the optimal estimate ˆθtr,trt in the case of the triangular distribution: Dˆθtr,trt=Etr{‖ˆθtr,trt−θt‖2} (calculated by the MCM).
– The MSE of the robust estimate ˆθtr,ut in the case of the triangular distribution Dˆθtr,ut=Etr{‖ˆθtr,ut−θt‖2}: the real π(⋅) is triangular, and the filtering algorithm uses the uniform one (calculated by the MCM).
Figure 9 presents similar estimation results for Yt under the triangular distribution π(⋅):
– The unconditional variance DYℓt=Etr{(Yℓt−Etr{Yℓt})2}, ℓ=1,2.
– The MSE of the optimal estimate ˆYℓ,tr,trt in the case of the triangular distribution: DˆYℓ,tr,trt=Etr{(ˆYℓ,tr,trt−Yℓt)2} (calculated by the MCM).
– The MSE of the robust estimate ˆYℓ,tr,ut in the case of the triangular distribution DˆYℓ,tr,ur=Etr{(ˆYℓ,tr,ut−Yℓt)2}: the true distribution π(⋅) is triangular, and the filtering algorithm uses the uniform one (calculated by the MCM).
Figure 10 presents the results of the comparative numerical study of the CTMC θt estimates for the three-point discrete uniform distribution Π(⋅):
– The unconditional mean square Dθt=E3p{‖θt−E3p{θt}‖2}.
– The MSE of the optimal estimate ˆθ3p,3pt in the case of the three-point distribution Dˆθ3p,3pt=E3p{‖ˆθ3p,3pt−θt‖2} (calculated by the MCM).
– The MSE of the robust estimate ˆθ3p,ut in the case of the three-point discrete uniform distribution Dˆθ3p,ut=E3p{‖ˆθ3p,ut−θt‖2}: the real Π(⋅) is three-point discrete uniform, and the filtering algorithm uses the continuous uniform one (calculated by the MCM).
Figure 11 presents similar estimation results for Yt under the three-point discrete uniform distribution Π(⋅):
– The unconditional variance DYℓt=E3p{(Yℓt−E3p{Yℓt})2}, ℓ=1,2.
– The MSE of the optimal estimate ˆYℓ,3p,3pt error in the case of the three-point discrete uniform distribution: DˆYℓ,3p,3pt=E3p{(ˆYℓ,3p,3pr−Yℓt)2} (calculated by the MCM).
– The MSE of the robust estimate ˆYℓ,tr,ut error in the case of the three-point discrete uniform distribution: DˆYℓ,3p,ur=E3p{(ˆYℓ,3p,u,qr−Yℓ,qt)2}: the real Π(⋅) is three-point discrete uniform, and the filtering algorithm uses the continuous uniform one (calculated by the MCM).
To evaluate the performance loss of the robust filtering estimate ˆZi,u (where i= "tr" for triangular distribution or "3p" for three-point distribution) in comparison with the optimal estimate, we use the following integral index. Let ρˆZi,u denote the performance index (5.3) of the robust filtering estimate when the real distribution corresponds to i. Additionally, let ρˆZi,i represent the performance index (5.3) calculated for the optimal filtering estimate ˆZi,i. We propose to consider the value ϰiZ≜(ρˆZi,uρˆZi,i−1)×100%, which is the percentage increase in the index ρ when replacing the optimal estimator with the robust one.
Table 1 contains the characteristics ϰ calculated for the robust filtering estimates of CTMC θ and for the components Y1 and Y2 separately. Note that in the considered case, the performance losses do not exceed 19%. In the adjacent estimation problem involving a priori uncertainty, one can select the least favorable distribution from the discrete options [30]. Therefore, it is not surprising that the performance loss is greater for the three-point uniform distribution Π: the actual least favorable distribution is concentrated at distant points within the support sets Dℓ, and three-point distribution is close to it. By analyzing the presented figures and the table, we can conclude that the proposed robust filtering algorithm demonstrates acceptable accuracy and could be beneficial in situations where there is uncertainty in the probability distribution π(⋅) of the sub-vector Yt.
Estimated component | Triangular distribution | Three-point distribution |
θ | 0.4 % | 18.7 % |
Y1 | 3.6 % | 8.9 % |
Y2 | 1.6 % | 11.6 % |
In summary, we can present the results of the paper as follows:
1) The paper introduces a subclass of MRPs that has practical value for the mathematical modeling of real-world objects and phenomena.
2) The optimal filtering problem for the MRP, considering both continuous and counting observations, is properly formulated and solved. The optimal filtering estimate of a scalar signal process is defined through the solution of a potentially non-closed SDS. Additionally, a variant of the Kushner-Stratonovich equation, which describes the evolution of the conditional PDF of the system state, is derived.
3) The high quality of the derived estimate is illustrated through an applied example related to telecommunications, where the filter enables monitoring of the network channel qualitative state and numerical characteristics based on observations of the RTT and packet loss flow.
4) The presented filtering algorithm demonstrates robustness to a priori uncertainty in the probability distribution of the estimated MRP.
These results can serve as a foundation for future studies.
First, the class of observation systems can be expanded to include the considered MRPs as states for subsequent solutions to the optimal filtering problem. A promising direction involves utilizing continuous-time observations with multiplicative noise [31] and estimating the CTMC θt using the noiseless observations of the component Yt. The filtering problems in this context are challenging because they do not allow for a Girsanov change of measure, which would reduce the original filtering problem to one involving Wiener and Poisson processes as observations [32].
Second, the Kushner-Stratonovich equation (4.4) is the non-linear stochastic partial integro-differential equation. It is simpler than the original Kushner-Stratonovich equation derived for diffusion observation systems because it does not include partial derivatives with respect to the state variable y. However, to address various applied estimation problems, efficient numerical algorithms for solving (4.4) and an analysis of their accuracy are required.
Third, the proposed filtering algorithm can be adapted to account for a priori uncertainty in the parameters of the MRP distribution. The current version already demonstrates some robustness to imprecise knowledge of the Yt distribution. However, the algorithm can be further developed in several directions, including uncertain parameter identification [33], the design of a corresponding guaranteed filter [30], and fuzzy logic adaptations [34,35].
Fourth, the paper highlights telecommunications as an application area for the mathematical modeling of real phenomena using the studied MRPs and the subsequent solutions to estimation problems. The applicability of MRPs and the corresponding estimation framework could be broader, encompassing areas such as navigation and maneuvering target tracking [9,10], financial mathematics [36], biology [37], medicine [38], etc.
The author is deeply grateful to the anonymous Referee for valuable remarks and comments on the initial version of the manuscript.
The research was supported by the Ministry of Science and Higher Education of the Russian Federation, project No. 075-15-2024-544.
The research was carried out using the infrastructure of the Shared Research Facilities "High Performance Computing and Big Data" (CKP "Informatics") of the Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences.
The author declares no conflict of interest.
We anticipate the proof of the theorem itself by
Lemma A.1. The innovation process νt is an Ot-adapted Wiener process. The innovation process ζt is an Ot-adapted purely discontinuous martingale with the quadratic characteristic
⟨ζ,ζ⟩t=∫t0diag ˆhs−ds. |
Proof. First, we verify the martingale property of νt. To do this, we consider two time instants 0⩽u<t⩽T:
E{νt−νu|Ou}=E{R−1/2∫tu(g(Zs)−ˆgs−)ds+wt−wu|Ou}=R−1/2∫tuE{g(Zs)−ˆgs−|Ou}ds+E{E{wt−wu|Fu}|Ou}=0, |
because ˆgs≠ˆgs− at a finite subset of [0,T] P−a.s. So, νt is a martingale.
Further, to define ⟨ν,ν⟩t, we apply the Itô rule to the product νtν⊤t, keeping in mind that ⟨w,w⟩t=tI:
νtν⊤t=∫t0νs−dν⊤s+∫t0dνsν⊤s−+⟨ν,ν⟩t=∫t0νs(R−1/2(g(Zs)−ˆgs−)ds+dws)⊤+∫t0(R−1/2(g(Zs)−ˆgs−)ds+dws)ν⊤s+⟨w,w⟩t. |
We have the equality ⟨ν,ν⟩t=⟨w,w⟩t=tI, and since the quadratic characteristic of wt is the non-random function with respect to the filtration {Ft}, it also is with respect to {Ot}. The martingale νt has the quadratic characteristic tI and P-a.s. continuous trajectories, hence this is a K-dimensional Wiener process [39, Thm 14.4.1].
Now, we verify the martingale property of ζt. We consider two time instants 0⩽u<t⩽T and use the fact that ˆhs−≠ˆhs at most at a finite set of instants on [0,T] with probability 1:
E{ζt−ζu|Ou}=E{∫tu(h(Zs−)−ˆhs−)ds+μηt−μηu|Ou}=∫tuE{h(Zs)−ˆhs|Ou}ds+E{E{μηt−μηu|Fu}|Ou}=0.P−a.s. |
So, ζt is a martingale.
Further, to define ⟨ζ,ζ⟩t, we apply the Itô rule to the product ζtζ⊤t keeping in mind the equality ⟨η,η⟩t=∫t0diag h(Zs−)ds:
ζtζ⊤t=∫t0ζs−dζ⊤s+∫t0dζsζ⊤s−+∑0⩽τ⩽tΔζτΔζ⊤τ=∫t0ζs−dζ⊤s+∫t0dζsζ⊤s−+∑0⩽τ⩽tΔητΔη⊤τ=∫t0ζs−dζ⊤s+∫t0dζsζ⊤s−+[η,η]t=∫t0diag ˆhs−ds+μ4t+∫t0ζs−dζ⊤s+∫t0dζsζ⊤s−⏟Ot−adapted martingale. |
Hence, ⟨ζ,ζ⟩t=∫t0diag ˆhs−ds.
Now, we go back to the proof of Theorem 4.1. First, if ˆQt is an optimal estimate of the process Qt associated with q(Zt), and q(Zt)=row(0,1)Qt is the linear transformation of Qt, then ˆqt=row(0,1)ˆQt. To derive an equation describing the evolution of ˆQt, we use an approach suggested in [32, Chap. 7] based on the uniqueness of the special semimartingale decomposition [19].
The associated process Qt can be described by an SDS, which is an appropriate version of (2.7)
Qt=Q0+∫t0DQ(s)Qs−ds+dμQt, | (A.1) |
where μQt∈R2N is an Ft-adapted square integrable martingale. Conditioning both sides of (A.1) by Ot, one can obtain, that
ˆQt=E{Q0}+∫t0DQ(s)ˆQs−ds+∫t0γsdνs+∫t0Γsdζs, | (A.2) |
where γs=γ(s,ωs)∈R2N×K and Γs=Γ(s,ωs)∈R2N×L are Ot-predictable random matrix-valued processes that should be determined [32, Lemmas 7.4.1 and 7.4.2]. However, decomposition (A.2) is true when there exists a suitable Girsanov measure transform for which ξt is the Wiener process, and ηt is the Poisson one. To meet this condition, the equality
E{ΛT}=1 | (A.3) |
should be true [39] for
Λt=exp(−∫t0g⊤(Zs)R−1/2dws−12∫t0g⊤(Zs)R−1g(Zs)ds+L∑ℓ=1∫t0((hℓ(Zs)−1)ds−lnhℓ(Zs−)dηℓs)). |
The processes g(⋅) and h(⋅) in (3.1) can be expressed via the solution to some linear SDS, and this is sufficient condition for the fulfillment of (A.3) [40, Theorem 4.1].
Let us consider the product ˆQtξ⊤t. Due to the Itô rule and the fact ⟨Q,ξ⟩t≡0, we have
Qtξ⊤t=∫t0Qsdξ⊤s+∫t0dQsξ⊤s=∫t0(Qsg(Zs)⊤R−1/2+DQ(s)Qsξ⊤s)ds+∫t0(Qsdw⊤s+dμQsξ⊤s)⏟Ft−measurable martingale. |
Conditioning both sides of the last expression with respect to Ot, we obtain the first variant of decomposition of ˆQtξ⊤t:
E{Qtξ⊤t|Ot}=ˆQtξ⊤t=∫t0(^Qg⊤s−R−1/2+DQ(s)ˆQs−ξ⊤s)ds+μ5t, | (A.4) |
where μ5t is an Ot-adapted martingale.
Now, we use (4.1) and the expression ξt=∫t0ˆgs−ds+R1/2νt:
ˆQtξ⊤t=∫t0ˆQs−dξ⊤s+∫t0dˆQsξ⊤s+⟨ˆQ,ξ⟩t=∫t0(ˆQs−ˆg⊤s−R−1/2+DQ(s)ˆQs−ξ⊤s+γs)ds+∫t0(ˆQs−dν⊤sR1/2+(γsdνs+Γsdζs)ξ⊤s)⏟≜μ6t, | (A.5) |
where μ6t is an Ot-adapted martingale.
Formulae (A.4) and (A.5) represent the same semimartingale, and this presumes P-a.s. fulfillment of the equality ^Qg⊤t−R−1/2+DQ(t)ˆQt−ξ⊤t=ˆQt−ˆg⊤t−R−1/2+DQ(t)ˆQt−ξ⊤t+γt for almost any t∈[0,T]. It is easy to verify that the Ot-predictable process
γt=(^Qg⊤t−−ˆQt−ˆg⊤t−)R−1/2 | (A.6) |
satisfies the last equality.
We define the integrand Γt analogously. First, we obtain the decomposition of the product ˆQtη⊤t:
Qtη⊤t=∫t0Qs−dη⊤s+∫t0dQsη⊤s−+∑0⩽τ⩽tΔQτΔη⊤τ=∫t0(Qsh(Zs)⊤+DQ(s)Qsη⊤s)ds+∫t0(Qs−d(μηs)⊤+dμQsη⊤s−)⏟Ft−measurable martingale. |
Conditioning both sides of the last expression with respect to Ot, we obtain the first variant of decomposition of ˆQtη⊤t:
E{Qtη⊤t|Ot}=ˆQtη⊤t=∫t0(^Qh⊤s−+DQ(s)ˆQs−η⊤s−)ds+μ7t, | (A.7) |
where μ7t is an Ot-adapted martingale.
Now, we apply the Itô rule to ˆQtη⊤t and use the expression ηt=∫t0ˆhs−ds+ζt:
ˆQtη⊤t=∫t0ˆQs−dη⊤s+∫t0dˆQsη⊤s−+∑0⩽τ⩽tΔˆQτΔη⊤τ=∫t0(ˆQs−ˆh⊤s−+DQ(s)ˆQsη⊤s+Γsdiag ˆhs−)ds+μ8t, | (A.8) |
where μ8t is an Ot-adapted martingale. Formulae (A.7) and (A.8) represent the same semimartingale, and this presumes P-a.s. fulfillment of the equality ^Qh⊤t+DQ(t)ˆQtη⊤t=ˆQtˆh⊤t−+DQ(t)ˆQtη⊤t+Γtdiag ˆht− for almost any t∈[0,T]. It is easy to verify that the Ot-predictable process
Γt=(^Qh⊤t−−ˆQt−ˆh⊤t−)diag −1ˆht− | (A.9) |
satisfies the last equality. The substitution of (A.6) and (A.9) into (A.2) completes the proof.
The existence of the conditional PDF ˆψ(t,y) follows from an abstract variant of the Bayes rule given both the continuous and counting observations by analogy with [41, Thm 7.23]. Conditions A1–A8 guarantee the legitimacy of this existence. If ˆψ(t,y) does exist, then expressions (4.5) for ˆθs, ˆgs, and ˆhs− are obvious.
Using (4.1), one can obtain the following representation for the estimate ˆqt=E{q(Zt)|Ot}:
ˆqt=E{q(Z0)}+∫t0[λ(s)ˆqs−+E⊤q˜Λ⊤(s)ˆθs−]ds+∫t0(^qg⊤s−−ˆqs−^g⊤s−)R−1/2dνs+∫t0(^qh⊤s−−ˆqs−^h⊤s−)diag −1(ˆhs−)dζs. | (B.1) |
We choose the function q(⋅), defining the estimated process, in the form q(z)=q(e,y)=e⊤neIB(y) for some n∈{1,…,N} and B∈B(RM), hence q(Zt)=e⊤nθtIB(Yt). The terms in both sides of (B.1) can be written as follows:
ˆqt=∫Bˆψn(t,y)dy=e⊤n∫Bˆψ(t,y)dy, | (B.2) |
E{q(Z0)}=pn0∫Bϕn(y)dy=e⊤n∫Bdiag (p0)ϕ(y)dy, | (B.3) |
∫t0λ(s)ˆqs−ds=∫t0λ(s)E{θs−e⊤nθs−IB(Ys−)|Os−}ds=∫t0λ(s)diag (en)∫Bˆψ(s−,y)dyds=e⊤n∫B[∫t0diag (λ(s))ˆψ(s−,y)ds]dy, | (B.4) |
∫t0E⊤q˜Λ⊤(s)ˆθs−ds=∫t0e⊤ndiag (∫Bπ(y)dy)˜Λ⊤(s)ˆθs−ds=e⊤ndiag (∫Bπ(y)dy)∫t0˜Λ⊤(s)ˆθs−ds, | (B.5) |
∫t0^qg⊤s−R−1/2dνs=∫t0E{e⊤nθs−IB(Ys−)g⊤(θs−,Ys−)|Os−}R−1/2dνs=e⊤n∫t0[∫Bdiag (ˆψ(s−,y))¯g⊤(y)dy]R−1/2dνs=e⊤n∫B[∫t0diag (ˆψ(s−,y))¯g⊤(y)R−1/2dνs]dy, | (B.6) |
∫t0ˆqs−ˆg⊤s−R−1/2dνs=∫t0e⊤n[∫Bˆψ(s−,y)dy]ˆg⊤s−R−1/2dνs=e⊤n∫B[∫t0diag (ˆψ(s−,y))(ˆgs−1)⊤R−1/2dνs]dy, | (B.7) |
∫t0^qh⊤s−diag −1(ˆhs−)dζs=e⊤n∫B[∫t0diag (ˆψ(s−,y))¯h⊤(y)diag −1(ˆhs−)dζs]dy, | (B.8) |
∫t0ˆqs−ˆh⊤s−diag −1(ˆhs−)dζs=e⊤n∫B[∫t0diag (ˆψ(s−,y))(ˆhs−1)⊤diag −1(ˆhs−)dζs]dy. | (B.9) |
The change of the integration order in (B.4)–(B.9) is proper due to the Fubini theorem. From the expressions above and (B.1), it follows that
e⊤n∫B[ˆψ(t,y)−diag (p0)ϕ(y)−∫t0(diag (λ(s))ˆψ(s−,y)+diag (∫Bπ(y)dy)˜Λ⊤(s)ˆθs−)ds−∫t0diag (ˆψ(s−,y))(¯g⊤(y)−ˆgs−1)⊤R−1/2dνs−∫t0diag (ˆψ(s−,y))(¯h⊤(y)−ˆhs−1)⊤diag −1(ˆhs−)dζs]dy=0. | (B.10) |
From the arbitrariness of n∈{1,…,N} and B∈B(RM), it follows that (B.10) holds when the equality
ˆψ(t,y)=diag (p0)ϕ(y)+∫t0[diag λ(s)ˆψ(s−,y)+diag π(y)˜Λ⊤(s)ˆθs−]ds+∫t0diag (ˆψ(s−,y))(¯g(y)−ˆgs−1)⊤R−1/2dνs+∫t0diag (ˆψ(s−,y))(¯h(y)−ˆhs−1)⊤diag −1(ˆhs−)dζs, |
is true P-a.s. and almost everywhere with respect to the Lebesgue measure. The theorem is proved.
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Estimated component | Triangular distribution | Three-point distribution |
θ | 0.4 % | 18.7 % |
Y1 | 3.6 % | 8.9 % |
Y2 | 1.6 % | 11.6 % |