Nonlinear filtering of partially observed systems arising in singular stochastic optimal control
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Calvia, Alessandro; Ferrari, Giorgio Working Paper Nonlinear filtering of partially observed systems arising in singular stochastic optimal control Center for Mathematical Economics Working Papers, No. 651 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Calvia, Alessandro; Ferrari, Giorgio (2021) : Nonlinear filtering of partially observed systems arising in singular stochastic optimal control, Center for Mathematical Economics Working Papers, No. 651, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29554928 This Version is available at: https://hdl.handle.net/10419/238129 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
651 June 2021 Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control Alessandro Calvia and Giorgio Ferrari Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en
NONLINEAR FILTERING OF PARTIALLY OBSERVED SYSTEMS ARISING IN SINGULAR STOCHASTIC OPTIMAL CONTROL ALESSANDRO CALVIA AND GIORGIO FERRARI Abstract. This paper deals with a nonlinear ltering problem in which a multi-dimensional signal process is additively aected by a process ν whose components have paths of bounded variation. The presence of the process ν prevents from directly applying classical results and novel estimates need to be derived. By making use of the so-called reference probability measure approach, we derive the Zakai equation satised by the unnormalized ltering process, and then we deduce the corresponding Kushner-Stratonovich equation. Under the condition that the jump times of the process ν do not accumulate over the considered time horizon, we show that the unnormalized ltering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. Our analysis paves the way to the study of stochastic control problems where a decision maker can exert singular controls in order to adjust the dynamics of an unobservable Itô-process. Keywords: Stochastic ltering; singularly controlled systems; reference probability measure; Zakai equation; Kushner-Stratonovich equation. AMS 2020: 93E11, 60G35, 60H15, 60J25, 60J76. 1. Introduction This paper studies a stochastic ltering problem on a nite time horizon [0, T] , T > 0 , in which the dynamics of a multi-dimensional process X= (Xt)t∈[0,T ] , called signal or unobserved process , are additively aected by a process having components of bounded variation. The aim is to estimate the hidden state Xt , at each time t∈[0, T] , using the information provided by a further stochastic process Y= (Yt)t∈[0,T ] , called observed process ; said otherwise, we look for the conditional distribution of Xt given the available observation up to time t . This leads to derive an evolution equation for the ltering process, which is a probability measure-valued process satisfying, for any given bounded and measurable function ϕ:Rm→R , πt(ϕ):=ZRm ϕ(x)πt(dx) = Eϕ(Xt)Yt, t ∈[0, T], where (Yt)t∈[0,T ] is the natural ltration generated by Y and augmented by P -null sets. The process π provides the best estimate (in the usual L2 sense) of the signal process X , given the available information obtained through the process Y . Stochastic ltering is nowadays a well-established research topic. The literature on the subject is vast and many dierent applications have been studied: the reader may nd a fairly detailed historical account in the book by Bain and Crisan [2]. Classic references are the books by Bensoussan [5], Kallianpur [26], Liptser and Shiryaev [32] (cf. also Brémaud [6, Chapter 4] for stochastic ltering with point process observation); more recent monographs are, e.g., the aforementioned book by Bain and Crisan [2], Crisan and Rozovski [15], and Xiong [37] (see also Cohen and Elliott [13, Chapter 22]). Recently, dierent cases where the signal and/or the observation processes can have discontinuous trajectories (as in the present work) have been studied and explicit ltering equations have been derived: see, for instance, Bandini et al. [4], Calvia [8], Ceci and Gerardi [11, 12], Ceci and Colaneri [9, 10], Confortola and Fuhrman [14], Grigelionis and Mikulevicius [23]. The main motivation of our analysis stems from the study of singular stochastic control problems under partial observation. Consider a continuous-time stochastic system whose position or level Xt at time t∈[0, T] is subject to random disturbances and can be adjusted instantaneously through (cumulative) actions that, as A. Calvia LUISS University, Department of Economics and Finance, Viale Romania 32, 00197 Rome (Italy). E-mail: [email protected] . G. Ferrari Bielefeld University, Center for Mathematical Economics (IMW), Universitätstrasse 25, 33615, Bielefeld (Germany). E-mail: giorgio.fer[email protected] . This research was supported by the 2019 INdAM-GNAMPA project Problemi di controllo ottimo stocastico con osservazione parziale in dimensione innita , of which the rst author was Principal Investigator. Financial support by the German Research Foundation (DFG) through the Collaborative Research Centre 1283 is also gratefully acknowledged by the second author. 1
2 A. CALVIA AND G. FERRARI functions of time, do not have to be absolutely continuous with respect to Lebesgue measure. In particular, they may present a Cantor-like component and/or a jump component. The use of such singular control policies is nowadays common in applications in Economics, Finance, Operations Research, as well as in Mathematical Biology. Typical examples are, amongst others, (ir)reversible investment choices (e.g., Riedel and Su [36]), dividends' payout (e.g., Reppen et al. [35]), inventory management problems (e.g., Harrison and Taksar [24]), as well as harvesting issues (e.g., Alvarez and Shepp [1]). Suppose also that the decision maker acting on the system is not able to observe the dynamics of the controlled process X , but she/he can only follow the evolution of a noisy process Y , whose drift is a function of the signal process. Mathematically, we assume that the pair (X, Y ) is dened on a ltered complete probability space (Ω,F,F:= (Ft)t∈[0,T ],P) and that its dynamics are given, for any t∈[0, T] , by the following system of SDEs: (dXt=b(t, Xt) dt+σ(t, Xt) dWt+ dνt, X0−∼ξ∈ P(Rm), dYt=h(t, Xt) dt+γ(t) dBt, Y0=y∈Rn. (1.1) Here: ξ is a given probability distribution on Rm ; W and B are two independent F -standard Brownian motions; coecients b, σ, h, γ are suitable measurable functions; ν is a càdlàg, Rm -valued process with (components of) bounded variation, that is adapted to the previously introduced observation ltration (Yt)t∈[0,T ] . Clearly, the decision maker might want to adjust the dynamics of X in order to optimize a given performance criterion. Since X is unobservable, this leads to a stochastic optimal control problem under partial observation, which can be tackled by deriving and studying the so-called separated problem , an equivalent problem under full information (see, e.g., Bensoussan [5]), where the signal X is formally replaced by its estimate provided by the ltering process π . However, to eectively solve the original optimization problem by means of the separated one, a rst necessary step concerns the detailed study of the associated ltering problem. To the best of our knowledge, the derivation of explicit ltering equations in the setting described above has not yet received attention in the literature. In this paper we provide a rst contribution in this direction. Indeed, the recent literature treating singular stochastic control problems under partial observation assumes that the observed process, rather than the signal one, is additively controlled (cf. Callegaro et al. [7], De Angelis [16], Décamps and Villeneuve [17], and Federico et al. [22]). Clearly, such a modeling feature leads to a ltering analysis that is completely dierent from ours. By making use of the so-called reference probability measure approach, we derive the Zakai stochastic partial dierential equation (SPDE) satised by the so-called unnormalized ltering process , which is a measure-valued process, associated with the ltering process via a suitable change of probability measure. Then, we deduce the corresponding evolution equation for π , namely, the so-called Kushner-Stratonovich equation or Fujisaki-Kallianpur-Kunita equation . Furthermore, we show that the unnormalized ltering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. The latter result is proved under the technical requirement that the jump times of the process ν aecting X in (1.1) do not accumulate over the considered time-horizon. Although such a condition clearly poses a restriction on the generality of the model, we also acknowledge that it is typically satised by optimal control processes arising in singular stochastic control problems. It is important to notice that establishing conditions under which the unnormalized ltering process possesses a density paves the way to recast the separated problem as a stochastic control problem in a Hilbert space, as we will briey explain in the next section. The rest of the introduction is now devoted to a discussion of our approach and results at a more technical level. 1.1. Methodology and main results. In this paper we are going to study the ltering problem described above through the so-called reference probability approach , that we briey summarize here. To start, let us notice that the model introduced in (1.1) is somewhat ill-posed. In fact, the dynamics of the signal process X depend on the (Yt)t∈[0,T ] -adapted process ν while, simultaneously, the dynamics of the observed process Y depend on X . Otherwise said, it is not clear how to dene ν , which has to be given a priori , and circularity arises if one attempts to introduce the partially observed system (X, Y ) as in (1.1). A possible way out of this impasse is to dene Y as a given Gaussian process independent of X (see (2.2)). In this way, it makes sense to x a (Yt)t∈[0,T ] -adapted process ν and to dene the dynamics of the signal process X as in the rst SDE of (1.1) (see also (2.8)). Finally, under suitable assumptions, there exists a probability measure change (cf. (2.12)) that allows us to recover the dynamics of Y as in the second SDE of (1.1) (see also (2.13)). It is important to notice that the resulting probability depends on the initial law ξ of X0− and on ν .
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 3 To derive the associated Kushner-Stratonovich equation there are two main approaches in the literature: The Innovations approach and the aforementioned reference probability approach. Although it might be possible to derive the ltering dynamics in our context by using the former approach, we follow the latter method. Our rst main results is Theorem 3.4, where we deduce the Zakai equation veried by the unnormalized ltering process (see (3.3) for its denition). From this result, as a byproduct, we deduce in Theorem 3.6 the Kushner-Stratonovich equation satised by the ltering process. It is worth noticing that, given the presence of the bounded-variation process ν in the dynamics of X , Theorem 3.4 cannot be obtained by invoking classical results, but novel estimates need to be derived (cf. Lemma A.1 and Proposition A.2). In particular, we employ a change of variable formula for Lebesgue-Stieltjes integrals. It is clear that in applications, for instance to optimal control problems, establishing uniqueness of the solution to the Zakai equation or to the Kushner-Stratonovich equation is essential. In the literature there are several approaches to tackle this problem, most notably the following four: The ltered martingale problem approach, originally proposed by Kurtz and Ocone [31], and later extended to singular martingale problems in [29] (see also [28]); the PDE approach, as in the book by Bensoussan [5] (see also [2, Section 4.1]); the functional analytic approach, introduced by Lucic and Heunis [33] (see also [2, Section 4.2]); the density approach, studied in Kurtz and Xiong [30] (see also [2, Section 7] and [37]). The rst three methods allow to prove uniqueness of the solution to the Zakai equation in a suitable class of measure-valued processes. However, they do not guarantee that the unique measure-valued process solution to the Zakai equation admits a density process, a fact that has an impact on the study of the separated problem. Indeed, without requiring or establishing conditions guaranteeing existence of such a density process, the separated problem must be formulated in an appropriate Banach space of measures and, as a consequence, the Hamilton-Jacobi-Bellman (HJB) equation associated to the separated problem must be formulated in such a general setting as well. As a matter of fact, only recently some techniques have been developed to treat this case, predominantly in the theory of mean-eld games (an application to optimal control problems with partial observation is given in [3]). A more common approach in the literature considers, instead, the density process as the state variable for the separated problem. If it is possible to show that such a density process is the unique solution of a suitable SPDE in L2(Rm) , the so-called Duncan-Mortensen-Zakai equation, then this L2(Rm) -valued process can be equivalently used as state variable in the separated problem. This is particularly convenient, since for optimal control problems in Hilbert spaces a well-developed theory is available, at least in the regular case (see, e.g., the monograph by Fabbri et al. [21]). Therefore, in view of possible future applications to singular optimal control problems under partial observation, we adopted the density approach to prove that, under suitable assumptions, the unnormalized ltering process is the unique solution to the Zakai equation in the class of measure-valued processes admitting a density with respect to Lebesgue measure. We show this result, rst, in the case where ν is a continuous process (cf. Theorem 4.6) and, then, in the case where the jump times of ν do not accumulate in the time interval [0, T] (see Theorem 4.7). As we already observed, although this assumption prevents to achieve full generality, it has a clear interpretation and it is usually satised by the examples considered in the literature. From a technical side, it seems that a direct approach using the method proposed by [30] is not feasible to treat the case of accumulating jumps, due to diculties in estimating crucial quantities in the arguments used, that are related to the jump component of ltering process. A possible workaround might consists in approximating the process ν by cutting away jumps of size smaller than some δ > 0 and then, provided that a suitable tightness property holds, pass to the limit, as δ→0 , in the relevant equations. However, this is a delicate and lengthy reasoning, which is left for future research. The rest of this paper is organized as follows. Section 1.2 provides notation used throughout this work. Section 2 introduces the ltering problem. The Zakai and Kushner-Stratonovich equations are then derived in Section 3, while the uniqueness of the solution to the Zakai equation is proved in Section 4. Finally, Appendix A collects the proof of technical results. 1.2. Notation. In this section we collect the main notation used in this work. Throughout the paper the set N denotes the set of natural integers N={1,2, . . . } , N0={0,1, . . . } , and R is the set of real numbers. For any m×n matrix A= (aij) , the symbol A∗ denotes its transpose and kAk is its Frobenius norm; i.e., kAk= (Pm i=1 Pn j=1 a2 ij)1/2 . For any x, y ∈Rd , kxk denotes the Euclidean norm of x and x·y=x∗y indicates the inner product of x and y . For a xed Hilbert space H , we denote its inner product by h·,·i and by k·kH its norm. The symbol 1C denotes the indicator function of a set C , while 1 is the constant function equal to 1 . The symbol Rb a denotes R[a,b] for any −∞ < a ≤b < +∞ .
4 A. CALVIA AND G. FERRARI For any d∈N and T > 0 , we denote by C1,2 b([0, T]×Rd) the set of real-valued bounded measurable functions on [0, T]×Rd , that are continuously dierentiable once with respect to the rst variable and twice with respect to the second, with bounded derivatives. For any such function, the symbol ∂t denotes the derivative with respect to the rst variable, while Dx= (∂1, . . . , ∂d) and D2 x= (∂2 ij)d i,j=1 denote, respectively, the gradient and the Hessian matrix with respect to the second variable. Furthermore, we simply write C2 b(Rd) , when we are considering a real-valued bounded function on Rd that is twice continuously dierentiable with bounded derivatives. For any d∈N we indicate by L2(Rd) the set of all square-integrable functions with respect to Lebesgue measure and for all k∈N we denote by W2 k(Rd) the Sobolev space of all functions f∈L2(Rd) such that the partial derivatives ∂α exist in the weak sense and are in L2(Rd) , whenever the multi-index α= (α1, . . . , αd) is such that α1+···+αd≤k . For a xed metric space E , endowed with the Borel σ -algebra, we denote by P(E) , M+(E) , and M(E) the sets of probability, nite positive, and nite signed measures on E , respectively. If µ∈ M(E) , then |µ|∈M+(E) is the total variation of µ . For any given càdlàg stochastic process Z= (Zt)t≥0 dened on a probability space (Ω,F,P) , we denote by (Zt−)t≥0 the left-continuous version of Z (i.e., Zt−= lims→t−Zs,P -a.s., for any t≥0 ), and by ∆Zt:= Zt−Zt− the jump of Z at time t≥0 . If Z has nite variation over [0, t] , for all t≥0 , |Z| (resp. Z+ , Z− ) is the variation process (resp. the positive part process, the negative part process) of Z , i.e., the process such that, for each t∈[0, T] and ω∈Ω , |Z|t(ω) (resp. Z+ t(ω) , Z− t(ω) ) is the total variation (resp. the positive part, the negative part) of the function s7→ Zs(ω) on [0, t] . It is useful to remember that Z=Z+−Z− , |Z|=Z++Z− , and that Z+ , Z− are non-decreasing processes. Finally, with the word measurable we refer to Borel-measurable , unless otherwise specied. 2. Model formulation Let T > 0 be a given xed time horizon and (Ω,F,F:= (Ft)t∈[0,T ],P) be a complete ltered probability space, with F satisfying the usual assumptions. Dene on (Ω,F,F,P) two independent F -adapted standard Brownian motions W and B , taking values in Rd and Rn , respectively, with d, n ∈N . Let then γ: [0, T]→Rn×n be a measurable function such that, for each t∈[0, T] , γ(t) is symmetric, with γij(t)∈L2([0, T ]) , for all i, j = 1, . . . , n , and uniformly positive denite; that is, there exists δ > 0 such that for all t∈[0, T] and all x∈Rm γ(t)x·x≥δkxk2. (2.1) These requirements guarantee in particular that the observed process Y= (Yt)t∈[0,T ] , dened as Yt=y+Zt 0 γ(t) dBt, t ∈[0, T], y ∈Rn, (2.2) is an Rn -valued F -adapted martingale, of which we take a continuous version. Clearly, it holds dYt=γ(t) dBt, t ∈[0, T], Y0=y∈Rn. (2.3) Remark 2.1 . It is not restrictive to require that γ is symmetric (and uniformly positive denite). Indeed, suppose that B is an Rk -valued F -adapted standard Brownian motion and that γ: [0, T]→Rn×k is such that γγ∗(t) := γ(t)γ∗(t) is uniformly positive denite. Then, we can obtain an equivalent model dening the Rn -valued F -adapted standard Brownian motion e B= ( e Bt)t∈[0,T ] through: de Bt:=γγ∗(t)−1/2γ(t) dBt, t ∈[0, T ]. In fact, in this case (2.3) becomes: dYt=γγ∗(t)1/2de Bt, t ∈[0, T], Y0=y∈Rn, and clearly γγ∗(t)1/2 is symmetric (and uniformly positive denite). We indicate with the symbol Y the completed natural ltration generated by Y , i.e., Y:= (Yt)t∈[0,T ] , with Yt:={Ys: 0 ≤s≤t}∨N , where N is the collection of all P -null sets. Remark 2.2 . Notice that since γ is invertible, Y coincides with the completed natural ltration generated by B and is, therefore, right-continuous. These facts will be useful in the sequel. Next, we consider a probability distribution ξ on Rm ; measurable functions b: [0, T]×Rm→Rm and σ: [0, T]×Rm→Rm×d , with m∈N ; a Y -adapted, càdlàg, Rm -valued process ν whose components have paths of nite variation. We introduce the following requirements, that will be in force throughout the paper.
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 5 Assumption 2.1. (i) There exist constants Cb and Lb such that for all t∈[0, T] kb(t, x)−b(t, x0)k ≤ Lbkx−x0k and kb(t, 0)k ≤ Cb,∀x, x0∈Rm. (2.4) (ii) There exist constants Cσ and Lσ such that for all t∈[0, T] kσ(t, x)−σ(t, x0)k ≤ Lσkx−x0k and kσ(t, 0)k ≤ Cσ,∀x, x0∈Rm. (2.5) (iii) The probability law ξ∈ P(Rm) satises ZRmkxk2ξ(dx)<+∞. (2.6) (iv) The Rm -valued process ν is Y -adapted, càdlàg, with ν0−= 0 . Its components have paths of nite variation, which in particular satisfy |νi|T≤K, ∀i= 1, . . . , m, (2.7) for some constant K > 0 . Under Assumption 2.1, for any such ν , the following SDE for the signal process X= (Xt)t∈[0,T ] admits a unique strong solution: dXt=b(t, Xt) dt+σ(t, Xt) dWt+ dνt, t ∈[0, T], X0−∼ξ∈ P(Rm). (2.8) It is important to bear in mind, especially in applications to optimal control problems, that the solution to (2.8) and all the quantities that are related to it depend on the the probability distribution ξ and on ν . However, for the ease of exposition, we will not stress this dependence in the sequel. Remark 2.3 . Conditions (2.4) and (2.5) ensure that SDE (2.8) admits a unique strong solution for any ν . If we assume, in addition, that (2.6) and (2.7) hold, then we have that, for some constant κ depending on T , b , σ , and ν , E[ sup t∈[0,T ]kXtk2]≤κ(1 + E[kX0−k2]) <+∞, (2.9) since E[kX0−k2] = RRmkxk2ξ(dx) . Proofs of these statements are standard and can be found, for instance, in [13, 34]. We nally arrive to the model we intend to analyze via a change of measure. Let h: [0, T]×Rm→Rn be a measurable function satisfying the following condition, that will stand from now on. Assumption 2.2. There exists a constant Ch such that for all t∈[0, T] kh(t, x)k ≤ Ch(1 + kxk),∀x∈Rm. (2.10) For all t∈[0, T] dene then: ηt:= exp Zt 0 γ−1(s)h(s, Xs) dBs−1 2Zt 0kγ−1(s)h(s, Xs)k2ds. (2.11) By Proposition A.2, η is a (P,F) -martingale, under Assumptions 2.1 and 2.2. Therefore, we can introduce the probability measure e P on (Ω,FT) satisfying de P dPFT =ηT. (2.12) By Girsanov's Theorem, the process B= (Bt)t∈[0,T ] given by Bt:=Bt−Rt 0γ−1(s)h(s, Xs) ds , t∈[0, T] , is a (e P,F) -Brownian motion, and under e P the dynamics of the observed process are provided by the SDE: dYt=h(t, Xt) dt+γ(t) dBt, t ∈[0, T], Y0=y∈Rn. (2.13) We see that equations (2.8) and (2.13) are formally equivalent to model (1.1). Observe, however, that the Brownian motion driving (2.13) is not a source of noise given a priori , but it is obtained through a probability measure change; moreover, our construction implies that it depends on the initial law ξ and on process ν . This formulation is typical in optimal control problems under partial observation (see, e.g., [5, Chapter 8]) and has the advantage of avoiding the circularity problem discussed in the Introduction. Remark 2.4 . If the partially observed system dened by (2.8) and (2.13) describes the state variables of a singular optimal control problem, where ν is the control process, then condition (2.7) implies that the singular control is of nite fuel type (see El Karoui and Karatzas [19], Karatzas et al. [27] for early contributions).
6 A. CALVIA AND G. FERRARI Remark 2.5 . It is worth noticing that all the results in this paper remain valid if we allow b to depend also on ω , as long as the map (ω, t)7→ b(ω, t, x) is Y -adapted and càdlàg, for each x∈Rm , and condition (2.4) holds uniformly with respect to ω (i.e., Lb and Cb do not depend on ω ). To extend our subsequent results to this case, it suces to apply the so-called freezing lemma whenever necessary. This modeling exibility is important when it comes to treating controlled dynamics where b is a deterministic function, depending on an additional parameter representing the action of a regular control α= (αt)t∈[0,T ] . Clearly, this control must be càdlàg and Y -adapted, i.e., based on the available information. The measurability requirement above ensures that the map (ω, t)7→ b(t, x, αt(ω)) is Y -adapted. 3. The Zakai and Kushner-Stratonovich equations In this section we will deduce the Zakai equation satised by the unnormalized ltering process, dened in (3.3). As a byproduct, we will deduce the Kushner-Stratonovich equation satised by the ltering process (see (3.1) for its denition). As anticipated in the Introduction, we will use the reference probability approach to achieve these results. The reference probability will be precisely P , under which the observed process is Gaussian and satises (2.2). However, the probability measure that matters from a modelization point of view is e P , which dened in (2.12). Indeed, we will dene the ltering process under this measure. It is important to bear in mind that e P and P are equivalent probability measures. Hence, any result holding P -a.s., holds also e P -a.s., and we will write only the rst of these two wordings. The following technical lemma is needed. Its proof is a consequence of the facts highlighted in Remark 2.2 and it is omitted (the reader may refer, for instance, to [2, Prop. 3.15]). In what follows we will denote Y:=YT . Lemma 3.1. Let Z be an Ft -measurable, P -integrable random variable, t∈[0, T] . Then E[Z| Yt] = E[Z| Y]. As previously anticipated, the ltering process π= (πt)t∈[0,T ] is a P(Rm) -valued process providing the conditional law of the signal X at each time t∈[0, T] , given the available observation up to time t . It is dened for any bounded and measurable ϕ: [0, T]×Rm→R as: πt(ϕt):=e Eϕ(t, Xt)Yt, t ∈[0, T], (3.1) where ϕt(x):=ϕ(t, x) , for any (t, x)∈[0, T]×Rm . Since Rm is a complete and separable metric space, π is a well-dened, P(Rm) -valued and Y -adapted process. 1 Moreover, π admits a càdlàg modication, since X is càdlàg (see, e.g. [2, Cor. 2.26]). Hence, in the sequel we shall consider π as a Y -progressively measurable process. We recall the useful Kallianpur-Striebel formula, which holds thanks to Proposition A.2 for any bounded and measurable ϕ: [0, T]×Rm→R and for any xed t∈[0, T] (for a proof see, e.g., [2, Prop. 3.16]) πt(ϕt) = Eηtϕ(t, Xt)Y EηtY,P -a.s. (3.2) This formula allows us to dene the measure-valued process ρ= (ρt)t∈[0,T ] , called unnormalized conditional distribution of X , or unnormalized ltering process , dened, for any bounded and measurable ϕ: [0, T]×Rm→R , as: ρt(ϕt):=Eηtϕ(t, Xt)Yt, t ∈[0, T]. (3.3) Given the properties of π and of η it is possible to show (see, e.g., [2, Lemma 3.18]) that ρ is càdlàg and Y -adapted, hence Y -progressively measurable. Moreover, the Kallianpur-Striebel formula implies that for any bounded and measurable ϕ: [0, T]×Rm→R and for any xed t∈[0, T] : πt(ϕt) = ρt(ϕt) ρt(1),P -a.s. , (3.4) where 1:Rm→R is the constant function equal to 1 . To describe the local dynamics of the signal process X , let us introduce the operator A , dened for any ϕ∈C1,2 b([0, T]×Rm) as: Aϕ(t, x):= Dxϕ(t, x)·b(t, x) + 1 2trD2 xϕ(t, x)σσ∗(t, x),(t, x)∈[0, T ]×Rm. (3.5) We can also dene the family of operators At , t∈[0, T] , given by: Atϕ(x)=Dxϕ(x)·b(t, x) + 1 2trD2 xϕ(x)σσ∗(t, x), x ∈Rm, ϕ ∈C2 b(Rm). 1 Without any particular assumptions on Y , the ltering process is adapted with respect to the right-continuous enlargement of Y . However, as previously observed, in our model Y is already right-continuous.
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 7 To obtain the Zakai equation we need, rst, to write the semimartingale decomposition of the process ϕ(t, Xt)t∈[0,T ] . For any ϕ∈C1,2 b([0, T]×Rm) we have, applying Itô's formula: ϕ(t, Xt) = ϕ(0, X0−) + Zt 0∂s+Aϕ(s, Xs) ds+Zt 0 Dxϕ(s, Xs−) dνs +X 0≤s≤thϕ(s, Xs)−ϕ(s, Xs−)−Dxϕ(s, Xs−)·∆νsi+Mϕ t, t ∈[0, T]. (3.6) Here, Mϕ t:=Rt 0Dxϕ(t, Xt)σ(t, Xt) dWt , t∈[0, T] , is a square-integrable (P,F) -martingale, thanks to conditions (2.4) and (2.5) (see also Remark 2.3). We need the following two technical Lemmata. Up to minor modications, their proofs follow that of [2, Lemma 3.21]. Lemma 3.2. Let Ψ = (Ψt)t∈[0,T ] be a real-valued (P,F) -progressively measurable process such that EZT 0 Ψ2 sds<+∞. Then, for any j= 1, . . . , k we have EZt 0 ΨsdBj sY=Zt 0 E[Ψs| Y] dBj s, t ∈[0, T]. Lemma 3.3. Let Ψ = (Ψt)t∈[0,T ] be a real-valued (P,F) -progressively measurable process satisfying 2 EZT 0 Ψ2 sdhMϕis<+∞. Then, EZt 0 ΨsdMϕ sY= 0, t ∈[0, T]. We are now ready to state the main result of this section, namely, to provide the Zakai equation. Theorem 3.4. Suppose that Assumptions 2.1 and 2.2 are satised and, moreover, that ZRmkxk3ξ(dx)<+∞. (3.7) Then, for any ϕ∈C1,2 b([0, T]×Rm) , the unnormalized conditional distribution ρ satises the Zakai equation: ρt(ϕt) = ξ(ϕ0) + Zt 0 ρs∂s+Asϕsds+Zt 0 ρs−Dxϕsdνs+Zt 0 γ−1(s)ρs(ϕshs) dBs +X 0≤s≤thρs−ϕs(·+ ∆νs)−ϕs−Dxϕs·∆νsi,P -a.s. , t ∈[0, T], (3.8) where ξ(ϕ0):=RRmϕ(0, x)ξ(dx) and, for all t∈[0, T] , ht(·):=h(t, ·) , Zt 0 ρs−Dxϕsdνs:= m X i=1 Zt 0 ρs−∂iϕsdνi s, Zt 0 γ−1(s)ρs(ϕshs) dBs:= n X i=1 n X j=1 Zt 0 γ−1 ij (s)ρs(ϕshj s) dBi s. Proof. Fix t∈[0, T] and ϕ∈C1,2 b([0, T]×Rm) . Let us introduce the constants Cϕ:= sup t,x |ϕ(t, x)|, C0 ϕ:= sup t,x kDxϕ(t, x)k, C00 ϕ:= sup t,x kD2 xϕ(t, x)k, where the suprema are taken over [0, T]×Rm . The proof is organized in several steps. Step 1. (Approximation) For any xed ε > 0 , dene the bounded process ηε= (ηε t)t∈[0,T ] : ηε t:=ηt 1 + εηt , t ∈[0, T], (3.9) where η is dened in (2.11). Both η and ηε have continuous trajectories and this fact will be used in what follows without further mention. 2 If M is any (P,F) -square integrable martingale, hMi denotes its (P,F) -predictable quadratic variation.
14 A. CALVIA AND G. FERRARI Proof. To ease notations, for any ε > 0 denote by Zε the process Zε t:=Tεζt , t≥0 . Fix ε > 0 and consider an orthonormal basis {ϕk}k∈N of H such that ϕk∈C2 b(Rm) , for any k∈N . Writing the Zakai equation for the function Tεϕk (recall that ν is continuous by assumption) we get: ζt(Tεϕk) = ξ(Tεϕk) + Zt 0 ζsAsTεϕkds+Zt 0 ζs−DxTεϕkdνs+Zt 0 γ−1(s)ζs(Tεϕkhs) dBs, (4.4) for all t∈[0, T] . Notice that, for any ϕ∈C2 b(Rm) and any t∈[0, T] , we can write: Atϕ(x) = m X i=1 bi(t, x)∂iϕ(x) + m X i,j=1 aij(t, x)∂ijϕ(x), x ∈Rm, where a is the function dened in (4.1). For any i, j = 1, . . . , m , `= 1, . . . , n , and t∈[0, T] , we dene the random measures on Rm : bi tζt(dx):=bi(t, x)ζt(dx), aij tζt(dx):=aij(t, x)ζt(dx), γh` tζt(dx):= n X p=1 γ−1 `p (t)hp(t, x)ζt(dx). These measures are P -almost surely nite, for any t∈[0, T] , thanks to Assumption 4.1 and to (A.3) (see also (A.19) for the last measure). Applying Lemma 4.2 and the integration by parts formula we get: ζtAtTεϕk= m X i=1 ZRm bi(t, x)∂iTεϕk(x)ζt(dx) + m X i,j=1 ZRm aij(t, x)∂ijTεϕk(x)ζt(dx) = m X i=1 ZRm bi(t, x)Tε∂iϕk(x)ζt(dx) + m X i,j=1 ZRm aij(t, x)Tε∂ijϕk(x)ζt(dx) = m X i=1 bi tζt(Tε∂iϕk) + m X i,j=1 aij tζt(Tε∂ijϕk) = m X i=1hTε(bi tζt), ∂iϕki+ m X i,j=1hTε(aij tζt), ∂ijϕki= m X i,j=1hϕk, ∂ijTε(aij tζt)i− m X i=1hϕk, ∂iTε(bi tζt)i. In a similar way, we obtain ζt∂iTεϕk=−hϕk, ∂iTεζti , and n X j=1 γ−1 ij (t)ζtTεϕkhj t=hϕk, Tε(γhi tζt)i, i = 1, . . . , n. Putting together all these facts, we can rewrite (4.4) as hϕk, Zε ti=hϕk, Zε 0−i+ m X i,j=1 Zt 0hϕk, ∂ijTε(aij sζs)ids− m X i=1 Zt 0hϕk, ∂iTε(bi sζs)ids − m X i=1 Zt 0hϕk, ∂iTεζs−idνi s+ n X i=1 Zt 0hϕk, Tε(γhi sζs)idBi s,P -a.s. , t ∈[0, T]. Applying Itô's formula we get that, for all t∈[0, T] , P -a.s., hϕk, Zε ti2=hϕk, Zε 0−i2+ m X i,j=1 Zt 0 2hϕk, Zε sihϕk, ∂ijTε(aij sζs)ids − m X i=1 Zt 0 2hϕk, Zε sihϕk, ∂iTε(bi sζs)ids+ n X i=1 Zt 0hϕk, Tε(γhi sζs)i2ds − m X i=1 Zt 0 2hϕk, Zε s−ihϕk, ∂iTεζs−idνi s+ n X i=1 Zt 0 2hϕk, Zε sihϕk, Tε(γhi sζs)idBi s. Using Assumption 4.1 and (A.3), it is possible to show that the stochastic integral with respect to Brownian motion B is a P -martingale. By the optional sampling theorem, this stochastic integral has zero expectation even when evaluated at any bounded stopping time. Therefore, picking an F -stopping time τ≤t , for arbitrary t∈[0, T] , summing over k up to N∈N , and taking the expectation, by Fatou's lemma we have that EkZε τ−k2 H=Elim N→∞ N X k=1hϕk, Zε τ−i2≤lim inf N→∞ EN X k=1hϕk, Zε τ−i2≤ kZε 0−k2 H
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 15 + lim inf N→∞ (m X i,j=1 EZτ− 0 N X k=1 2hϕk, Zε sihϕk, ∂ijTε(aij sζs)ids − m X i=1 EZτ− 0 N X k=1 2hϕk, Zε sihϕk, ∂iTε(bi sζs)ids+ n X i=1 EZτ− 0 N X k=1hϕk, Tε(γhi sζs)i2ds − m X i=1 EZτ− 0 N X k=1 2hϕk, Zε s−ihϕk, ∂iTεζs−idνi s), (4.5) where we used the fact that, since Zε 0−∈H , lim N→∞ N P k=1hϕk, Zε 0−i2=kZε 0−k2 H . More generally, since Zε t∈H , for all t∈[0, T] , P -a.s. (cf. Remark 4.1), we have that N X k=1hϕk, Zε ti2≤ ∞ X k=1hϕk, Zε ti2=kZε tk2 H, t ∈[0, T]. (4.6) We want now to estimate the quantities appearing inside the limit inferior, in order to exchange the limit and the integrals in (4.5). First of all, let us notice that, thanks to Assumption 4.1, the following estimates hold P -a.s., for all i, j = 1, . . . , m , all `= 1, . . . , n , and all t∈[0, T] : k∂ijTε(aij tζt)k2 H≤K1kTε|ζ|tk2 H,k∂iTε(bi tζt)k2 H≤K2kTε|ζ|tk2 H, kTε(γh` tζt)k2 H≤K3kTε|ζ|tk2 H,k∂iTεζtk2 H≤K4kTε|ζ|tk2 H, where K1=K1(ε, m, σ) , K2=K2(ε, m, b) , K3=K3(n, h, γ) , K4=K4(ε, m) . They can be proved following a reasoning analogous to that of [2, Lemma 7.5] (see also [37, Chapter 6]). Recalling that 2|ab| ≤ a2+b2 , for all a, b ∈R , using the estimates provided above, Lemma 4.2, and (4.6), we get that, for all N∈N , all i, j = 1, . . . , m , and all s∈[0, T] , 1s<τ N X k=1 2hϕk, Zε sihϕk, ∂ijTε(aij sζs)i ≤ N X k=1hϕk, Zε si2+ N X k=1hϕk, ∂ijTε(aij sζs)i2 ≤ kZε sk2 H+k∂ijTε(aij sζs)k2 H≤(1 + K1)kTε/2|ζ|sk2 H. With analogous computations, we get, for all i= 1, . . . , m , all N∈N , and all s∈[0, T] , 1s<τ N X k=1 2hϕk, Zε sihϕk, ∂iTε(bi sζs)i ≤ (1 + K2)kTε/2|ζ|sk2 H, 1s<τ N X k=1 2hϕk, Zε s−ihϕk, ∂iTεζs−i ≤ (1 + K4)kTε/2|ζ|sk2 H, and, for all N∈N and all s∈[0, T] , n X i=1 1s<τ N X k=1hϕk, Tε(γhi sζs)i2≤nK3kTε|ζ|sk2 H. The terms appearing on the r.h.s. of these estimates are dt⊗dP - and d|νi|t⊗dP -integrable on [0, T]×Ω , for all i= 1, . . . , m , since, for any ε > 0 , E"ZT 0kTε|ζ|sk2 Hds#≤TE[ sup s∈[0,T ]kTε|ζ|sk2 H]<+∞, E"ZT 0kTε|ζ|sk2 Hd|νi|s#≤KE[ sup s∈[0,T ]kTε|ζ|sk2 H]<+∞. Therefore, by the dominated convergence theorem, we can pass to the limit in (4.5), as N→ ∞ , EkZε τ−k2 H≤ kZε 0−k2 H+ m X i,j=1 EZτ− 0 2hZε s, ∂ijTε(aij sζs)ids− m X i=1 EZτ− 0 2hZε s, ∂iTε(bi sζs)ids + n X i=1 EZτ− 0kTε(γhi sζs)k2 Hds− m X i=1 EZτ− 0hZε s−, ∂iTεζs−idνi s, (4.7)
16 A. CALVIA AND G. FERRARI We nally get the claim, bounding the terms on the r.h.s. of (4.7) by using the following results: for the second one, apply [37, Lemma 6.11]; for the third and the last one, apply [37, Lemma 6.10]; for the fourth one, use the fact that the constant K3 above does not depend on ε . Proposition 4.4 allows to deduce that any M+(Rm) -valued solution of the Zakai equation (3.8) admits a density with respect to Lebesgue measure. Proposition 4.5. Suppose that Assumption 4.1 holds. Let ζ= (ζt)t∈[0,T ] be a Y -adapted, càdlàg, M+(Rm) - valued solution of (3.8) , with ζ0−=ξ∈ P(Rm) . If ν is continuous and if ξ admits a square-integrable density with respect to Lebesgue measure on Rm , then there exists an H -valued process Z= (Zt)t∈[0,T ] such that, for all t∈[0, T] , ζt(dx) = Zt(x)dx, P -a.s. Moreover, Z is Y -adapted, continuous, and satises E[kZtk2 H]<+∞ , for all t∈[0, T] . Proof. As a consequence of Lemma 4.3, the assumptions of Proposition 4.4 hold and we have that for each ε > 0 and all F -stopping times τ≤t , t∈[0, T] , E[kTεζτ−k2 H]≤ kTεζ0−k2 H+MZτ− 0 E[kTεζs−k2 H] dAs. Therefore, we can apply Lemma A.1 and get that, for all t∈[0, T] , E[kTεζt−k2 H] = E[kTεζtk2 H]≤ kTεζ0−k2 HeM(T+mK), (4.8) where we used the fact that ζ is continuous, since ν is, and that At≤AT≤T+mK , for all t∈[0, T] . Notice that, denoting by Z0− the density of ξ with respect to Lebesgue measure on Rm , Tεζ0−(y) = ZRm ψε(x−y)ξ(dx) = ZRm ψε(x−y)Z0−(x) dx=TεZ0−(y), y ∈Rm. By point ii. of Lemma 4.2 and since the constants appearing in (4.8) do not depend on ε , we get sup ε>0 E[kTεζtk2 H]≤ kZ0−k2 HeM(T+mK), t ∈[0, T ]. Taking, as in the Proof of Proposition 4.4, an orthonormal basis {ϕk}k∈N of H such that ϕk∈C2 b(Rm) , for any k∈N , the dominated convergence theorem entails that, for all k∈N , lim ε→0hTεζt, ϕki= lim ε→0ZRmZRm ψε(x−y)ϕk(y) dyζt(dx) = ZRm ϕk(x)ζt(dx) = ζt(ϕk). Applying Fatou's Lemma we get that, for all t∈[0, T] , E"∞ X k=1 ζt(ϕk)2#=E"∞ X k=1 lim ε→0hTεζt, ϕki2#≤lim inf ε→0 E"∞ X k=1hTεζt, ϕki2# ≤sup ε>0 E[kTεζtk2 H]≤ kZ0−k2 HeM(T+mK)<+∞, (4.9) and hence, from Lemma 4.1 we deduce that, P -a.s., ζt is absolutely continuous with respect to Lebesgue measure on Rm , for all t∈[0, T] . Moreover, its density process Z= (Zt)t∈[0,T ] takes values in H and, by standard results, is Y -adapted and continuous (because ν is). Finally, since ζt(ϕk) = RRmϕk(x)Zt(x) dx=hϕk, Zti , for all k∈N , and all t∈[0, T] , we get E[kZtk2 H] = E"∞ X k=1hϕk, Zti2#=E"∞ X k=1 ζt(ϕk)2#<+∞, t ∈[0, T]. We are now ready to state our rst uniqueness result for the solution to the Zakai equation, in the case where ν is continuous. Theorem 4.6. Suppose that Assumptions 2.1, 2.2, 4.1, and (3.7) hold. If ν is continuous and if ξ∈ P(Rm) admits a square-integrable density with respect to Lebesgue measure on Rm , then the unnormalized ltering process ρ , dened in (3.3) , is the unique Y -adapted, continuous, M+(Rm) -valued solution to the Zakai equation (3.8) . Moreover, there exists a Y -adapted, continuous, H -valued process p= (pt)t∈[0,T ] satisfying, for all t∈ [0, T] , E[kptk2 H]<+∞ and ρt(dx) = pt(x)dx , P -a.s.
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 17 Proof. Clearly, the unnormalized ltering process ρ , dened in (3.3), is a Y -adapted, continuous (since ν is), M+(Rm) -valued solution to (3.8). Therefore, the second part of the statement follows directly from Proposition 4.5. Uniqueness can be established as follows. Let ζ(1), ζ(2) be two Y -adapted, càdlàg, M+(Rm) -valued solutions to (3.8). Dene ζ:=ζ(1) −ζ(2) ∈ M(Rm) and let Z:=Z(1) −Z(2) ∈H be its density process, where Z(1) and Z(2) are the density processes of ζ(1) and ζ(2) , respectively, which exist thanks to Proposition 4.5. Standard facts from measure theory show that, for all non-negative, bounded, measurable functions ϕ:Rm→R and all t∈[0, T] , |ζ|t(ϕ)≤ζ(1) t(ϕ) + ζ(2) t(ϕ) . From this fact, applying Lemma 4.3 we deduce that E[ sup t∈[0,T ]kTε|ζ|t−k2 H]≤2E[ sup t∈[0,T ]kTεζ(1) t−k2 H]+2E[ sup t∈[0,T ]kTεζ(2) t−k2 H]<+∞. Therefore, from Proposition 4.4 we get that for all ε > 0 and all F -stopping times τ≤t , t∈[0, T] , E[kTεζτ−k2 H]≤MZτ− 0 E[kTε|ζ|s−k2 H] dAs, where A is dened in (4.2). An application of the dominated convergence theorem shows that kTε|ζ|t−k2 H−→ kZtk2 H , as ε→0 , for all t∈[0, T] , and hence, by Fatou's lemma E[kZtk2 H] = E[lim ε→0kTεζτ−k2 H]≤lim inf ε→0 E[kTεζτ−k2 H] ≤lim inf ε→0MZτ− 0 E[kTε|ζ|s−k2 H] dAs=MZτ− 0 E[kZs−k2 H] dAs. Finally, Proposition 4.5 ensures that E[kZt−k2 H]≤2E[kZ(1) t−k2 H]+2E[kZ(2) t−k2 H]<+∞, for all t∈[0, T], This allows us to use Lemma A.1 to get that, for all t∈[0, T] , E[kZt−k2 H] = E[kZtk2 H] = 0 , whence we obtain kZtk2 H= 0 , P -a.s., and therefore uniqueness of the solution to the Zakai equation. 4.2. The case in which the jump times of ν do not accumulate. Exploiting the recursive structure of (3.19), we can prove uniqueness of the solution to the Zakai equation (3.8), also in the case where the jump times of ν do not accumulate. Theorem 4.7. Suppose that Assumptions 2.1, 2.2, 4.1, and (3.7) hold. If the jump times of ν do not accumulate over [0, T] and if ξ∈ P(Rm) admits a square-integrable density with respect to Lebesgue measure on Rm , then the unnormalized ltering process ρ , dened in (3.3) , is the unique Y -adapted, càdlàg, M+(Rm) - valued solution to the Zakai equation (3.8) . Moreover, there exists a Y -adapted, càdlàg, H -valued process p= (pt)t∈[0,T ] satisfying, for all t∈[0, T] , E[kptk2 H]<+∞ and ρt(dx) = pt(x)dx , P -a.s. Proof. Let us denote by ρ the unnormalized ltering process associated with the initial law ξ and process ν , and by p0− the density of ξ with respect to Lebesgue measure on Rm . Let T0= 0 and dene the sequence of jump times of ν Tn:= inf{t>Tn−1: ∆νt6= 0}, n ∈N, with the usual convention inf ∅= +∞ . Recall that also T0 can be a jump time of ν . Moreover, since the jump times of ν do not accumulate over [0, T] , we have that Tn≤Tn+1 , P -a.s., and Tn<+∞=⇒Tn< Tn+1 , for all n∈N0 . We start noticing that the formula ρTn(ϕ) = ρTn−ϕTn(·+ ∆νTn) , n∈N0 , appearing in (3.19) holds for all ϕ∈Cb(Rm) . Indeed, continuity of the observation ltration Y implies that e E[ϕ(XT− n)| YTn] = e E[ϕ(XT− n)| YT− n] = πT− n(ϕ). Using continuity of process η , Kallianpur-Striebel formula (3.2), and the freezing lemma, we get ρTn(ϕ) = e E[ϕ(XTn)| YTn]EηTnY=e E[ϕ(XT− n+ ∆νTn)| YTn]EηT− nY =πT− n(ϕ(·+ ∆νTn)) EηT− nY=ρT− n(ϕ(·+ ∆νTn)), for all ϕ∈Cb(Rm) and all n∈N0 . This, in turn, entails that if ρT− n admits a density pT− n with respect to Lebesgue measure, then ZRm ϕ(x)ρTn(dx) = ρT− n(ϕ(·+ ∆νTn)) = ZRm ϕ(x+ ∆νTn)pT− n(x) dx=ZRm ϕ(x)pT− n(x−∆νTn) dx.
18 A. CALVIA AND G. FERRARI Therefore, since Cb(Rm) is a separating set (see, e.g., [20, Chapter 3, Section 4]), we have the equivalence of measures ρTn(dx) and pT− n(x−∆νTn) dx , implying that ρTn admits density with respect to Lebesgue measure on Rm , given by pT− n(·−∆νTn) . We can now use the recursive structure of (3.19) to get the claim. Dene the process ν(1) t:=νt1t<T1+νT11t≥T1, t ∈[0, T], and the random measure ξ(1)(dx):=p0−(x−∆ν0) dx , on Rm . Consider, for all ϕ∈C2 b(Rm) , the Zakai equation ρ(1) t(ϕ) = ξ(1)(ϕ) + Zt 0 ρ(1) s∂s+Asϕds +Zt 0 ρ(1) s−Dxϕdν(1) s+Zt 0 γ−1(s)ρ(1) s(ϕhs) dBs,P -a.s. , t ∈[0, T]. (4.10) Since ν(1) satises point (iv) of Assumption 2.1, we have that (4.10) is the Zakai equation for the ltering problem of the partially observed system (2.8)(2.13), with initial law ξ(1) and process ν(1) , which is continuous on [0, T] . Therefore, by Theorem 4.6, ρ(1) is its unique solution and admits a density p(1) with respect to Lebesgue measure on Rm , with E[kp(1) tk2 H]<+∞ , for each t∈[0, T] . It is clear that, since νt=ν(1) t on {t<T1} , we have that ρt=ρ(1) t on the same set, and hence ρt admits density p(1) t on {t<T1} . Next, let us dene the process ν(2) t:=νt+T11t<T2−T1+νT21t≥T2−T1, t ∈[0, T], and the random measure ξ(2)(dx) = pT− 1(x−∆νT1) dx , on Rm . Consider, for all ϕ∈C2 b(Rm) , the Zakai equation ρ(2) t(ϕ) = ξ(2)(ϕ) + Zt 0 ρ(2) s∂s+As+T1ϕds +Zt 0 ρ(2) s−Dxϕdν(2) s+Zt 0 γ−1(s+T1)ρ(2) s(ϕhs+T1) dBs+T1,P -a.s. , t ∈[0, T]. (4.11) Since ν(2) satises point (iv) of Assumption 2.1, we have that (4.11) is the Zakai equation for the ltering problem of the partially observed system (2.8)(2.13), with initial law ξ(2) and process ν(2) , which is continuous on [0, T] . Therefore, by Theorem 4.6, ρ(2) is its unique solution and admits a density p(2) with respect to Lebesgue measure on Rm , with E[kp(2) tk2 H]<+∞ , for each t∈[0, T] . It is clear that, since νt=ν(2) t−T1 on {T1≤t < T2} , we have that ρt=ρ(2) t−T1 on the same set, and hence ρt admits density p(2) t−T1 on {T1≤t<T2} . Continuing in this manner, we construct a sequence of solutions (ρ(n))n∈N and corresponding density processes (p(n))n∈N . We deduce that the unnormalized ltering process is represented by ρt= ∞ X n=1 ρ(n) t−Tn 1Tn−1≤t<Tn, t ∈[0, T], and hence is the unique Y -adapted, càdlàg, M+(Rm) -valued solution to the Zakai equation (3.8), admitting a Y -adapted, càdlàg, H -valued density process p , given by pt= ∞ X n=1 p(n) t−Tn 1Tn−1≤t<Tn, t ∈[0, T]. The fact that E[kptk2 H]<+∞ , for all t∈[0, T] , follows from the analogous property for each of the processes p(n) , n∈N . Appendix A. Techincal results Let us recall that if A (dened on a given ltered complete probability space) is a càdlàg, adapted, nonnegative process, with A0−= 0 , and H is an optional process, satisfying Rt 0|Hs|dAs<+∞ , for all t≥0 , P -a.s., then for any stopping time τ we have that Zτ− 0 HsdAs:=Z+∞ 0 Hs1s<τ dAs.
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 19 Lemma A.1. Let (Ω,F,F,P) be a given ltered complete probability space, x T > 0 , and let A and H be two càdlàg, F -adapted real-valued processes. Suppose that A is non-decreasing, with A0−= 0 and AT≤K , P -a.s., for some constant K > 0 , and that H satises one of the following: a. E[supt∈[0,T ]|Ht−|]<+∞ ; b. H is non-negative and such that E[Ht−]<+∞ , for all t∈[0, T] . Assume, moreover, that for any F -stopping time τ≤T we have E[Hτ−]≤M+E"Zτ− 0 Hs−dAs#, (A.1) for some constant M . Then E[HT−]≤MeK . Proof. The following reasoning is inspired by the proof of [25, Lemma IX.6.3]. Let us dene ˜ At:=At1t<T +K1t≥T, t ≥0. ˜ A is still a càdlàg, F -adapted and non-decreasing process, with ˜ A0−= 0 . Moreover, for any stopping time τ≤T , random measures 1s<τ dAs and 1s<τ d˜ As agree, therefore (A.1) implies E[Hτ−]≤M+E"Zτ− 0 Hs−d˜ As#. (A.2) Next, dene Ct:= inf{s≥0: ˜ As≥t} , t≥0 , which (see, e.g., [18, Chapter VI, Def. 56] or [25, Proposition I.1.28]) is an F -stopping time for all t≥0 , satisfying Ct≤T , thanks to the denition of ˜ A . We now x t∈[0, K] . Using (A.2), we get E[H(Ct)−]≤M+EZ+∞ 0 Hs−1s<Ctd˜ As=M+EZ+∞ 0 H(Cu)−1Cu<Ctdu. Since C is a non-decreasing process, we have that {Cu< Ct}⊂{u<t} , and hence 1Cu<Ct≤1u<t . Therefore E[H(Ct)−]≤M+EZt 0 H(Cu)−du. If H satises condition b. we can directly apply Fubini-Tonelli's theorem as below. If, instead, condition a. holds, since Cu≤T and, for each xed ω∈Ω , the image of the map u7→ Cu(ω) is a subset of [0, T] , we have that supu∈[0,K]|H(Cu)−| ≤ sups∈[0,T ]|Hs−| , so EZt 0|H(Cu)−|du≤KE[ sup s∈[0,T ]|Hs−|]<+∞. Therefore, we can apply Fubini-Tonelli's theorem and get E[H(Ct)−]≤M+Zt 0 E[H(Cu)−] du, whence we obtain, from the usual Gronwall's lemma, E[H(Ct)−]≤Met . Thanks to the denition of ˜ A , we have that CK=T and the claim follows letting t=K in the last inequality. Proposition A.2. Under Assumptions 2.1 and 2.2, the process η , dened in (2.11) , is a (P,F) -martingale. Proof. Let us notice, rst, a fact that will be useful in this proof. It can be easily shown that condition (2.1) implies, for some constant Cγ , kγ−1(t)k ≤ Cγ,∀t∈[0, T]. (A.3) Let us dene, for all t∈[0, T] , Zt:=Zt 0 γ−1(s)h(s, Xs) dBs. Thanks to condition (2.10) and using (A.3) and (2.9), we easily get E"ZT 0kγ−1(s)h(s, Xs)k2ds#≤nE"ZT 0kγ−1(s)k2kh(s, Xs)k2ds# ≤nChCγE"ZT 0 (1 + kXsk2) ds#≤nChCγT[1 + κ(1 + E[kX0−k2])] <+∞. (A.4)
20 A. CALVIA AND G. FERRARI Therefore, Z is an (F,P) -martingale, and hence η , which is the Doléans-Dade exponential of Z , is a nonnegative local (F,P) -martingale (see, e.g., [13, Lemma 15.3.2]). Thus, to prove the claim it is enough to show that E[ηt]=1 for all t∈[0, T] . We start proving, rst, that E[ηtkXt−k2]≤C , for all t∈[0, T] , where C is an appropriately chosen constant. For the sake of brevity, let us write bs:=b(s, Xs) , σs:=σ(s, Xs) , and hs:=h(s, Xs) . Applying Itô's formula we get kXtk2=kX0−k2+Zt 02X∗ s−bs+kσsk2ds+ 2 Zt 0 X∗ s−σsdWs + 2 Zt 0 Xs−dνs+X 0≤s≤t{kXsk2−kXs−k2−2Xs−·∆νs}, and using the integration by parts rule we have ηtkXtk2=kX0−k2+Zt 02ηs−X∗ s−bs+ηs−kσsk2ds+ 2 Zt 0 ηs−X∗ s−σsdWs +Zt 0kXs−k2ηsγ−1(s)hsdBs+ 2 Zt 0 ηs−Xs−dνs +X 0≤s≤t ηs−{kXsk2−kXs−k2−2Xs−·∆νs}. Therefore, for any xed ε > 0 , we obtain ηtkXtk2 1 + εηtkXtk2=kX0−k2 1 + εkX0−k2+Zt 0 ηs− [1 + εηs−kXs−k2]22X∗ s−bs+kσsk2ds −Zt 0 εη2 s− [1 + εηs−kXs−k2]34kX∗ s−σsk2+kXs−k4kγ−1(s)hsk2ds +Zt 0 2ηs− [1 + εηs−kXs−k2]2Xs−dνc s+Zt 0 2ηs− [1 + εηs−kXs−k2]2X∗ s−σsdWs +Zt 0 ηs−kXs−k2 [1 + εηs−kXs−k2]2γ−1(s)hsdBs+X 0≤s≤tηskXsk2 1 + εηskXsk2−ηs−kXs−k2 1 + εηs−kXs−k2, (A.5) where νc denotes the continuous part of the process ν . With standard estimates (see, e.g., [2, Solution to Exercise 3.11]) it is possible to show that the stochastic integrals with respect to Brownian motions W and B are (F,P) -martingales. This implies, thanks to the optional sampling theorem, that these stochastic integrals have zero expectation even when evaluated at any bounded stopping time. Fixing a F -stopping time τ≤t , for arbitrary t∈[0, T] , taking the expectation and noticing that the third term in (A.5) is non-negative, we get Eητ−kXτ−k2 1 + εητ−kXτ−k2≤EkX0−k2 1 + εkX0−k2 +EZτ− 0 ηs−2X∗ s−bs+kσsk2 [1 + εηs−kXs−k2]2ds+EZτ− 0 2ηs−Xs− [1 + εηs−kXs−k2]2dνc s +EX 0≤s<τηskXsk2 1 + εηskXsk2−ηs−kXs−k2 1 + εηs−kXs−k2. (A.6) We proceed, now, to nd suitable estimates for the terms appearing in (A.6). Notice that, thanks to conditions (2.4) and (2.5), we have that for some constant C1 2X∗ s−bs+kσsk2≤C1(1 + kXs−k2),P -a.s. , s ∈[0, T], Recalling that η is non-negative and that E[ηt]≤1 , for any t∈[0, T] , we get EZτ− 0 ηs−2X∗ s−bs+kσsk2 [1 + εηs−kXs−k2]2ds≤C1EZτ− 0 ηs−(1 + kXs−k2) [1 + εηs−kXs−k2]2ds ≤C1EZτ− 0 ηs−ds+C1EZτ− 0 ηs−kXs−k2 1 + εηs−kXs−k2ds ≤C1T+C1EZτ− 0 ηs−kXs−k2 1 + εηs−kXs−k2ds. (A.7)
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 21 Next, we see that EZτ− 0 2ηs−Xs− [1 + εηs−kXs−k2]2dνc s= m X i=1 EZτ− 0 2ηs−Xi s− [1 + εηs−kXs−k2]2dνi,c s ≤ m X i=1 EZτ− 0 2ηs−|Xi s−| [1 + εηs−kXs−k2]2d|νi,c|s≤ m X i=1 EZτ− 0 ηs−(1 + |Xi s−|2) [1 + εηs−kXs−k2]2d|νi,c|s ≤ m X i=1 EZτ− 0 ηs−d|νi,c|s+ m X i=1 EZτ− 0 ηs−kXs−k2 1 + εηs−kXs−k2d|νi,c|s. (A.8) Similarly to what we did in the proof of Lemma A.1, let us dene ˜νi t:=|νi|t1t<T +K1t≥T, t ≥0, i = 1, . . . , m. For each i= 1, . . . , m , ˜νi is a Y -adapted, càdlàg, non-decreasing process, with ˜νi 0−= 0 . Moreover, random measures 1s<τ d|νi|s and 1s<τ d˜νi s agree, therefore EZτ− 0 ηs−d|νi|s=EZτ− 0 ηs−d˜νi s, i = 1, . . . , m, and, in particular, EZτ− 0 ηs−d|νi,c|s=EZτ− 0 ηs−d˜νi,c s, i = 1, . . . , m. Let us dene the changes of time Ci t:= inf{s≥0: ˜νi s≥t} , for all t≥0 and all i= 1, . . . , m . Then, noticing that {Ci s≤t}={˜νi t≥s} and recalling that η is non-negative and ˜νi T=K , we get EZτ− 0 ηs−d˜νi,c s≤E"ZT 0 ηs−d˜νi,c s#≤E"ZT 0 ηs−d˜νi s#=EZ+∞ 0 η(Ci s)−1Ci s≤Tds =EZ+∞ 0 η(Ci s)−1s≤˜νi Tds=E"ZK 0 η(Ci s)−ds#=ZK 0 E[η(Ci s)−] ds. Since E[ηt]≤1 , for any t∈[0, T] , and Ci s≤T , for all s∈[0, K] , we get that EZτ− 0 ηs−d|νi,c|s=EZτ− 0 ηs−d˜νi,c s≤K, i = 1, . . . , m. (A.9) Similarly, we obtain also EZτ− 0 ηs−d|νi|s=EZτ− 0 ηs−d˜νi s≤K, i = 1, . . . , m. (A.10) Therefore, putting together (A.8) and (A.9) we obtain EZτ− 0 2ηs− [1 + εηs−kXs−k2]2Xs−dνc s≤mK + m X i=1 EZτ− 0 ηs−kXs−k2 1 + εηs−kXs−k2d|νi,c|s (A.11) We are left with estimating the last term of (A.6). We have: EX 0≤s<τηskXsk2 1 + εηskXsk2−ηs−kXs−k2 1 + εηs−kXs−k2≤EX 0≤s<τηs−(kXsk2−kXs−k2) 1 + εηs−kXs−k2 =EX 0≤s<τηs−(k∆νsk2+ 2Xs−·∆νs) 1 + εηs−kXs−k2≤EX 0≤s<τ m X i=1 ηs−(|∆νi s|+1+|Xi s−|2) 1 + εηs−kXs−k2|∆νi s|, where we used the fact that η is continuous. Since all quantities in the last term are non negative and |∆νi s| ≤ K , for all s∈[0, T] and all i= 1, . . . , m , P -a.s., we get that EX 0≤s<τ m X i=1 ηs−(|∆νi s|+1+|Xi s−|2) 1 + εηs−kXs−k2|∆νi s| ≤(1 + K) m X i=1 EX 0≤s<τ ηs−|∆νi s|+ m X i=1 EX 0≤s<τ ηs−kXs−k2 1 + εηs−kXs−k2|∆νi s| ≤(1 + K) m X i=1 EZτ− 0 ηs−d|νi|s+ m X i=1 EX 0≤s<τ ηs−kXs−k2 1 + εηs−kXs−k2∆|νi|s
22 A. CALVIA AND G. FERRARI ≤mK(1 + K) + m X i=1 EX 0≤s<τ ηs−kXs−k2 1 + εηs−kXs−k2∆|νi|s, (A.12) where we used (A.10) and the fact that |∆νi|= ∆|νi| . Therefore, feeding (A.7), (A.11), and (A.12) back into (A.6), we obtain Eητ−kXτ−k2 1 + εητ−kXτ−k2≤M1 + EZτ− 0 ηs−kXs−k2 1 + εηs−kXs−k2dAs, (A.13) where M is a suitable constant, not depending on ε , and A is the process At:=t+ m X i=1|νi|t, t ∈[0, T ]. Clearly, A is a càdlàg, Y - (and hence F -) adapted, non-negative process, with A0−= 0 and AT≤T+mK . Moreover, ηt−kXt−k2 1+εηt−kXt−k2≤1 ε , for all t∈[0, T] , P -a.s. Therefore, we can apply Lemma A.1 and obtain Eηt−kXt−k2 1 + εηt−kXt−k2≤MeM(T+mK). Recalling that η is continuous we get, applying Fatou's lemma, E[ηtkXt−k2] = Elim ε→0 ηtkXt−k2 1 + εηtkXt−k2≤lim inf ε→0 EηtkXt−k2 1 + εηtkXt−k2≤MeM(T+mK). (A.14) It is important to stress that (A.14) holds for any t∈[0, T] , since t was arbitrarily chosen. Now we can nally obtain that E[ηt] = 1 , for all t∈[0, T] . By Itô's formula, for an arbitrarily xed ε > 0 and all t∈[0, T] , ηt 1 + εηt =1 1 + ε−Zt 0 εη2 s (1 + εηs)3kγ−1(s)h(s, Xs)k2ds+Zt 0 ηs (1 + εηs)2γ−1(s)h(s, Xs) dBs. Thanks to conditions (2.10) and (2.1), standard computations show that the stochastic integral is a (P,F) - martingale. Therefore, taking the expectation we get Eηt 1 + εηt=1 1 + ε−EZt 0 εη2 s (1 + εηs)3kγ−1(s)h(s, Xs)k2ds. Notice that εη2 s (1+εηs)3kγ−1(s)h(s, Xs)k2−→ 0 , as ε→0 , dP⊗dt -a.s. Moreover, εη2 s (1 + εηs)3kγ−1(s)h(s, Xs)k2≤ηskγ−1(s)h(s, Xs)k2, s ∈[0, T], that, using conditions (2.10) and (2.1), satises (see also (A.4)) E"ZT 0 ηskγ−1(s)h(s, Xs)k2ds#≤nChCγE"ZT 0 ηs(1 + kXsk2) ds# =nChCγ(ZT 0 E[ηs] ds+ZT 0 E[ηskXs−k2] ds)≤nChCγT[1 + MeM(T+mK)], where we used the fact that E[ηt]≤1 , for all t∈[0, T] , and (A.14). Similarly, ηt 1+εηt→ηt , as ε→0 , dP⊗dt -a.s., and E[RT 0ηsds]≤T . Therefore, by the dominated convergence theorem E[ηt] = lim ε→0 Eηt 1 + εηt= lim ε→01 1 + ε−EZt 0 εη2 s (1 + εηs)3kγ−1(s)h(s, Xs)k2ds= 1, and this concludes the proof. Proof of Lemma 4.3. Fix ε > 0 . To start, let us notice that continuity of process ν implies that also ζ is continuous and, therefore, ζt=ζt− and Tεζt=Tεζt− , dt⊗dP -almost everywhere. Since ψ2ε is bounded by (4πε)−m 2 , we get that for all t∈[0, T] , kTεζtk2 H=ZRmZRm ψε(x−y)ζt(dx)2 dy (A.15) =ZRmZRmZRm ψε(x−y)ψε(z−y)ζt(dx)ζt(dz) dy (A.16) =ZRmZRm ψ2ε(x−z)ζt(dx)ζt(dz)≤(4πε)−m 2ζt(1)2. (A.17)
FILTERING OF SINGULARLY CONTROLLED SYSTEMS 23 Taking into account (3.8) and the fact that ν is continuous, the process ζ(1) satises ζt(1) = 1 + Zt 0 γ−1(s)ζs(hs) dBs, t ∈[0, T], where ht(·):=h(t, ·) , t∈[0, T] . Thanks to Assumption 4.1, ζt(ht)<+∞ , P -a.s., for all t∈[0, T] . Therefore, since ζt is P -a.s. a nite (non-negative) measure, for any t∈[0, T] , we get that ζ(1) is a non-negative (P,Y) -local martingale, and hence a (P,Y) -supermartingale. The next step is to prove that ζ(1) is a square-integrable 3 (P,Y) -martingale. We follow, rst, a reasoning analogous to that of [5, Lemma 4.3.1] (see also [2, Lemma 3.29]) to provide an explicit representation of ζ(1) . By Itô's formula we obtain, for any δ > 0 and all t∈[0, T] , log pδ+ζt(1)2= log √1 + δ+Zt 0 ζs(1) δ+ζs(1)2γ−1(s)ζs(hs) dBs +1 2Zt 0 δ−ζs(1)2 [δ+ζs(1)2]2 n X i=1 n X j=1 γ−1 ij (s)ζs(hj s) 2 ds. (A.18) Since, thanks to Assumption 4.1 and (A.3), n X i=1n X j=1 γ−1 ij (t)ζt(hj t)2 ≤(nCγKhζt(1))2,P -a.s. ,∀t∈[0, T], (A.19) and δ−ζt(1)2 [δ+ζt(1)2]2≤1 δ+ζt(1)2 , P -a.s., for all t∈[0, T] , we have ζs(1)2 [δ+ζs(1)2]2 n X i=1 n X j=1 γ−1 ij (t)ζt(hj t) 2 ≤nCγKh ζs(1)2 δ+ζs(1)22 ≤(nCγKh)2,∀t∈[0, T], and δ−ζs(1)2 [δ+ζs(1)2]2 n X i=1 n X j=1 γ−1 ij (s)ζs(hj s) 2 ≤ζs(1)2 δ+ζs(1)2(nCγKh)2≤(nCγKh)2,∀t∈[0, T]. Both the r.h.s. of the last two inequalities are integrable on [0, T] , therefore we can pass to the limit, as δ→0 , in (A.18), getting that, for all t∈[0, T] , log(ζt(1)) = 1 + Zt 0 γ−1(s)ζ1 s(hs) dBs−1 2Zt 0 n X i=1n X j=1 γ−1 ij (s)ζ1 s(hj s)2 ds, (A.20) where ζ1 t(dx):=ζt(dx) ζt(1) , t∈[0, T] , is the normalized process associated to ζ . From (A.20) we get the explicit representation for ζ(1) , i.e., for all t∈[0, T] , ζt(1) = expZt 0 γ−1(s)ζ1 s(hs) dBs−1 2Zt 0 n X i=1n X j=1 γ−1 ij (s)ζ1 s(hj s)2 ds. (A.21) This entails that ζ(1) coincides with the Doléans-Dade exponential of the continuous (P,Y) -local martingale Rt 0γ−1(s)ζ1 s(hs) dBs , t∈[0, T] . Using once more (A.19) we have that, for any k > 1 , Eexpk 2ZT 0 n X i=1n X j=1 γ−1 ij (t)ζ1 t(hj t)2 dt≤exp kT(nCγKh)2 2. Applying [13, Theorem 15.4.6], we get that, for any p > 1 , ζ(1) is a p -integrable (in particular, squareintegrable) (P,Y) -martingale. Therefore, from (A.15) we get E[ sup t∈[0,T ]kTεζtk2 H]≤(4πε)−m 2E[ sup t∈[0,T ] ζt(1)2]<+∞, whence, recalling the remark at the beginning of the proof, the claim. 3 If M= (Mt)t∈[0,T ] is any martingale, we say that M is a p -integrable martingale, with p≥1 , if E[supt∈[0,T ]|Mt|p]1/p < +∞ .