On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

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1 On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen Grades DOCTOR RERUM NATURALIUM (Dr.rer.na.) im Fachgebie Mahemaik. vorgeleg von Maser of Science Ling Xu geboren am in Hebei Die Annahme der Disseraion haben empfohlen: 1. Professor Dr. Rüdiger Frey (Universiä Leipzig) 2. Professor Wolfgang Runggaldier(Universiy of Padova) Die Verleihung des akademischen Grades erfolge mi Besehen der Vereidigung am mi dem Gesamprädika magna cum laude.

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3 Absrac We are ineresed in a nonlinear filering problem moivaed by an informaion-based approach for modelling he dynamic evoluion of a porfolio of credi risky securiies. We solve his problem by change of measure mehod and show he exisence of he densiy of he unnormalized condiional disribuion which is a soluion o he Zakai equaion. Zakai equaion is a linear SPDE which, in general, canno be solved analyically. We apply Galerkin mehod o solve i numerically and show he convergence of Galerkin approximaion in mean square. Lasly, we design an adapive Galerkin filer wih a basis of Hermie polynomials and we presen numerical examples o illusrae he effeciveness of he proposed mehod. The work is closely relaed o he paper Frey and Schmid (21).

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5 Acknowledgemens I am very graeful o Prof. Frey and Prof. Schmid. They gave me he suggesions abou my research and discussed wih me he problems ha happened during he accomplishmen of his hesis. Wihou heir seady suppor, compeen help and criicism, i would have been impossible o finish his disseraion. My hanks are also going o Frau Göz and Frau Fricke as well as oher members of he group, for heir enhusiasic help. Furhermore, I hank he Max Planck Insiue for Mahemaics in he Sciences, and DFG (Deuschen Forschungsgemeinschaf) for heir financial suppor o my say in Universiy Leipzig. I wan o say hanks o my husband, Chuangye Wang for his never-ending suppor.

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7 Conens 1 Inroducion Preliminary work Overview of he hesis Filering model and he Zakai equaion Sochasic filering A general inroducion The filering equaions Finie-dimensional filers Filering from poin processes observaions The filering model Unobserved sae process Observaions The objecive The innovaions processes The Zakai equaions A new measure The unnormalized filering equaions Numerical mehods A finie dimensional filer Two special cases A finie dimensional filer The finie-sae Markov chain approximaion Approximaing Markov chain Filer Numerical soluion Paricle mehods i

8 ii CONTENTS 4 Linear sochasic PDEs Semigroup approach Semigroup Linear SPEDs The variaional approach o linear parabolic SPDEs General seing Basic resuls for linear parabolic SPDEs wih Gaussian noise Unnormalized condiional densiy Assumpions Main resul Finding he unnormalized condiional densiy Proof of Theorem Some exisence and uniqueness resuls on sochasic PDEs The backward SPDEs The unnormalized condiional densiy Convergence resuls Inroducion The mild form of he Zakai equaion The sochasic inegral Coninuiy heorem Coninuiy heorem Proofs Galerkin approximaion Galerkin approximaion SDEs Analyical soluion Numerical mehods for SDEs Basis funcions Gaussian series Hermie expansion The adapive Galerkin approximaion Inroducion The adapive Galerkin approximaion The adapive Galerkin approximaion wih Hermie polynomials

9 CONTENTS iii 7.4 Proofs Simulaion sudies Tables Figures Summary A Appendix 159 A.1 Preliminaries A.1.1 Sobolev spaces A.1.2 Some spaces of processes A.2 Some Hilber spaces B Lis of frequenly used noaion and symbols 165

10 iv CONTENTS

11 Chaper 1 Inroducion 1.1 Preliminary work Financial moivaion: Credi risk model under incomplee informaion The credi markes have developed a a remendous speed in recen years and he demand for credi derivaives, such as credi defaul swaps (CDSs) and collaeralized deb obligaions (CDOs), are growing rapidly. Consequenly, recen research has highlighed he sudy of he credi risk model. A good model is one ha capures he dynamic evoluion of credi spreads and he dependen srucure of defaul in a realisic way. Also, from a compuaion poin of view, i should be racable and parsimonious. Exising credi risk models can be divided ino wo classes: srucural models and reducedform models. In srucural models, defaul occurs when an asses value falls below a hreshold, generally represening liabiliies. Srucural credi risk models are discussed in, for insance, Black and Scholes (1973), Meron (1974), Black and Cox (1976). In reduced-form models, one models direcly he law of he defaul ime where he precise mechanism leading o defaul is no specified. In pracice, reduced-form models are usually preferred for racabiliy reasons. Reduced-form credi risk models are discussed in, for insance, Jarrow and Turnbull (1995), Lando (1998), Duffie and Singleon (1999), Blanche-Scallie and Jeanblanc (24). In mos of he credi risk models, he disribuion of defaul imes depends on a sae variable process X and in pracice, X can no be fully observed by invesors. This, in urn, leads o a nonlinear filering problem in a naural way. Srucural credi risk model under incomplee informaion are discussed in, for insance, Kusuoka (1999), Duffie and Lando (21), Jarrow and Proer (24), Coculescu, Geman, and Jeanblanc (28), and Frey and Schmid (29). Reduced-form credi risk models under incomplee informaion are discussed in, for insance, Frey and Schmid (21), Frey and Runggaldier (21), Schönbucher (24), Collin-dufresne, Goldsein, and Helwege (23), Giesecke (24) and McNeil, Frey, and Embrechs (25), page The general heories of nonlinear filering are well-developed, see for insance Bain and Crisan (29), and are increasingly being used in financial mahemaics, see for insance Frey (2), Frey and Runggaldier (21), Frey and Schmid (29). Filering echniques come ino play when he facors canno be observed direcly. A shor inroducion for nonlinear filering is presened in Secion

12 2 CHAPTER 1. INTRODUCTION Model and noaion Frey and Schmid (21) sudy reduce-form porfolio credi risk models under incomplee informaion. They consider models where he defaul inensiies of he firms under consideraion depend on an unobservable sochasic processes X. The informaion for invesors, he so-called marke informaion, conains only he defaul hisory of firms and noisy price observaions of raded credi derivaives. Specifically, hey work on some filered probabiliy space (Ω, F, F, P) and on a finie ime inerval [,T. Here P is a risk-neural measure. F = {F } T is he full-informaion filraion and all processes will be F-adaped. In associaion wih a generic process ξ, define for each a sub-σ-field of F, denoed F ξ,by F ξ = σ{ξ s,s [,}. (1.1.1) Consider defaulable securiies issued by a firm where he random ime τ denoes he defaul ime of he firm. Y = 1 {τ } is he corresponding defaul indicaor. The defaul inensiy is assumed o depend on some sae process X, which is modelled as a d-dimensional finie sae Markov chain. ( ) Assume τ is a doubly sochasic random ime wih P, F -defaul inensiy λ = λ(x ), i.e., here is a funcion λ : R (, ), such ha N := Y τ λ(x s )ds is an F-maringale. The informaional advanage of informed marke paricipans is modelled via observaions of a process Z. Formally, he marke filraion is given by F Z,Y := {F Z,Y } T, where process Z, which is l-dimensional, solves he SDE dz = h(x )d + db, [,T. Here B is a l-dimensional sandard Brownian moion independen of X and Y.Andh( ) isa funcion from R d o R l. Consider a liquidly raded credi derivaives wih mauriy T and FT Y -measurable payoff P.In order o simplify he compuaion, we assume he full-informaion value of he securiies given by E(P F )=: P (X,Y ). The marke price of he securiy, which is deermined by informed marke-paricipans wih marke informaion, is defined as, by ieraed condiional expecaions, [ P := E(P F Z,Y )=E E(P F ) F Z,Y = E[ P (X,Y ) F Z,Y. The objecive of financial mahemaics is o derive he dynamics of he marke price. In order o compuer he price P, one need o obain he condiional measure of X given F Z,Y,given by π. This leads o a nonlinear filering problem in a naural way. 1.2 Overview of he hesis Moivaed by he credi risk model sudied by Frey and Schmid (21), we exend he model wih wo viewpoins, realisic and mahemaical. Realisically, he sae process X is modelled as a diffusion process wih generaor L, which is a second order differenial operaor wih domain D(L). Mahemaically, he jump observaion Y is modelled as a doubly sochasic Poisson process wih sochasic inensiy λ(x ), hen τ can be viewed as he firs jumping ime of Y.

13 1.2. OVERVIEW OF THE THESIS 3 Given he pas observaion of Z and Y, he objecive of his hesis is o deermine he condiional measure of X. We solve his problem by change of measure mehod and we show he exisence of he densiy of he unnormalized condiional disribuion which is a soluion of he so-called Zakai equaion. The Zakai equaion is a linear sochasic parial differenial equaion which canno ypically be solved analyically. We will apply Galerkin mehod o solve i numerically, and will show he convergence of Galerkin approximaion in mean square. To conclude, we design an adapive Galerkin filer wih a basis of Hermie polynomials and presen numerical examples o illusrae he effeciveness of he proposed mehod. Following are specifics abou he opics of his hesis. The unnormalized filering equaions We deduce he evoluion equaion for π using he change of measure mehod: A new measure P is consruced under which Z becomes a Brownian moion, Y becomes a Poisson process wih inensiy 1, and he process X, Z, Y are independen. Then, π has a represenaion in erms of an associaed unnormalized version ρ. Thisρ is hen shown o saisfy he so-called Zakai equaion, for any f D(L), ρ (f) =E[f(X ) F Z,Y + ρ s (Lf)ds + ρ s (fh )dz s ) + ρ s (f(λ 1) d(y s s), P a.s., [,T. For deail, see Theorem 2.9. This leads o he evoluion equaion for π by an applicaion of Iô s formula. The unnormalized condiional densiy The following quesion is ineresing o answer: Does here exis a densiy of he condiional disribuion of X given accumulaed observaion? In Chaper 5, we prove, under fairly mild condiions, he unnormalized condiional disribuion ρ has a square inegrable densiy q wih respec o Lebesgue measure, which is a weak soluion of a linear sochasic parial differenial equaion, known as he Zakai Equaion. For he deail, see Theorem 5.3. Numerical approach Our objecive is o seek a numerical mehod ha is implemenable and provides accurae soluions o he Zakai Equaion. One approach is o approximae he diffusion model of X by a finie-sae Markov chain and o wrie down he Zakai equaion for i, which yields a sochasic ODE, see Frey and Runggaldier (21), Frey and Schmid (21). In his work, a differen approach is proposed, namely he Zakai equaion is direcly approximaed by means of he classical Galerkin mehod for solving deerminisic PDEs, see Ahmed and Radaideh (1997). The soluion of he Zakai equaion is firs approximaed by a finie combinaion of orhogonal series. Then, i is approximaed by he soluion of a family of finie dimensional sochasic ordinary differenial equaions, which can be solved numerically or analyically. This work consiss of wo main pars, heoreical and numerical.

14 4 CHAPTER 1. INTRODUCTION Theoreical par In Chaper 6, we prove he convergence of he Galerkin approximaion of he Zakai equaion in mean square sense, wih usual assumpions. This is done by using a general coninuiy resul for he soluion of a mild sochasic linear differenial equaion on a Hilber space wih respec o he semigroup. Wih bounded and square inegrable funcion ϕ, wehave sup E ϕ(x)q (n) (x)dx ϕ(x)q (x)dx 2, as n, [,T R d R where E is he expecaion w.r.. P, n is he number of basis funcions used in he Galerkin filer. Numerical par Concerned wih he Galerkin approximaion convergence rae, in Chaper 7, we design an adapive Galerkin filer wih a basis of Hermie polynomials and presen numerical examples o illusrae he effeciveness of he proposed mehod. In simulaion sudy, we compare he proposed mehod wih paricle mehods and show ha he Galerkin approximaion converges well. In Chaper 7, we presen he Galerkin approximaion sraegy for soluions of he Zakai equaion (5.2.1) by solving a sequence of finie dimensional sochasic differenial equaions. The soluion of he Zakai equaion can be consruced by he Galerkin mehod using any suiable se of basis funcions from Hilber space. I is possible o choose a complee se of basis funcions, like Gaussian series and Hermie funcions. However, in mos nonlinear filering problems, paricularly when he observaion noise is small, he condiional densiy is well-localized in a small region of he sae space and generally canno be prediced in advance. To overcome his difficuly, we design an adapive Galerkin filer wih Hermie polynomials. Finally, we presen examples and he corresponding simulaion resuls. Srucure of he hesis Lised below is a brief summary of he remaining chapers. In Chaper 2: We inroduce he nonlinear filering model which we sudy hroughou his hesis and deduce he corresponding filering equaion by change of measure mehod. In Chaper 3: We presen an overview of he main compuaional mehods currenly available for solving he filering problem. Three classes of numerical mehod are presened: he finie dimensional filer based on approximaing he condiional densiy by a linear combinaion of Gaussian funcions, finie sae Markov chain, and he paricle filer. In Chaper 4: Since he Zakai equaion is a linear SPDE, we survey some exising resuls on linear SPDEs. Two approaches presened are he semigroup and he variaional. In Chaper 5: We show he exisence of he unnormalized condiional densiy which is he soluion of a Zakai equaion. In Chaper 6: We solve he Zakai equaion numerically by Galerkin approximaion, and we show he convergence of Galerkin approximaion in mean square. In Chaper 7: We design an adapive Galerkin filer wih Hermie polynomials and presen examples of he corresponding simulaion resuls o illusrae he effeciveness of he proposed mehod.

15 Chaper 2 Filering model and he Zakai equaion In his chaper, we mainly inroduce he filering model which will be sudied hroughou he hesis. Furhermore we deduce he corresponding filering equaion by a change of measure mehod. Presened in Secion 2.1 is a shor review of a nonlinear filering problem. In Secion 2.2, we inroduce he nonlinear filering model sudied in his hesis and include he sae process which we are ineresed bu, unable o observe direcly. Also included are he observaion processes which are he parial observaions of he sae, and he objecive of he nonlinear filering problem. The objecive is o obain he condiional disribuion of he parially observed processes recursively. In Secion 2.3, we inroduce he innovaion process and discuss heir maringale processes, which are useful for he numerical sudy inroduced in Chaper 7. In Secion 2.4, by change of measure mehod, we derive he Zakai equaion, which describes he evoluion of an unnormalized version of he condiional disribuion. 2.1 Sochasic filering This secion is devoed o a shor inroducion on nonlinear filering. For deails, we refer o he book Bain and Crisan (29). We begin Secion wih a general nonlinear filering problem. In Secion 2.1.2, we pursue he Markov case and presen he filering equaion obained by wo approaches. Secion is devoed o finie dimensional filers. Finally, in Secion 2.1.4, we inroduce he nonlinear filering problem for jump-diffusion case A general inroducion The objecive, in his secion, is o presen a shor inroducion o a general nonlinear filering problem. I concerns he following: Denoe by T R + a se of ime poins (usually T = R + ). We are ineresed in a signal or sae process X = {X } T which can no be observed direcly. Insead, he so called observaion process Z = {Z } T, a noisy nonlinear observaion of X is obained. 5

16 6 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION The objecive of a filering problem is o deermine he condiional disribuion of he sae X given F Z = σ(z s, s ),whichwedenoebyπ. Recall, by Equaion (1.1.1), F Z denoes he pas observaion of Z unil ime. Bain and Crisan (29), Corollary 2.26, page 31, show ha under some assumpions, he condiional disribuion of he signal can be viewed as a sochasic process wih values in he space of probabiliy measures. Addiionally, from a compuaional poin of view, i is desired ha π be obained recursively. This means ha π +s can be buil up from π and he new observaion raher han he whole observaion hisory. This allows for quick updaing of he filer and avoids serious daa sorage issues. Typically, π is infinie dimensional. However, someimes π can be aken as a finie-dimensional objec. This will be inroduced in Secion The filering equaions To derive deailed resuls we inroduce some noaions as follows. Le (Ω, F, P) be a probabiliy space ogeher wih a filraion {F } [,T which saisfies he usual condiions. Assume ha X is a R d -valued Markov process wih sae space S R d. For example, X can be a diffusion process or a finie sae Markov chain, for deails, see Bain and Crisan (29), page 49. Denoe by L be he generaor of he Markov process: for x S, E[f(X ) X = x f(x) Lf(x) = lim, he limi being uniform in x S and D(L) denoes he se of all bounded, measurable, realvalued funcions f : S R such ha his limi exiss. The generaor gives he expeced rae of change of he process {f(x )} [,T. Z, is he noisy nonlinear observaion of X, Z = h(x s )ds + B, T, (2.1.1) where B is a sandard l-dimensional Brownian moion which is independen of X. h : R d R l is a measurable funcion such ha, T P( h(x s ) ds < ) =1, (2.1.2) where he Euclidean norm is defined in he usual fashion for vecors. This ensures he Riemann inegral in Equaion (2.1.1) is well defined. As we inroduced before, he objecive is o recursively derive π (f) = E[f(X ) F Z, which denoes he condiional of X given he pas observaion. The problem can be solved by wo approaches, one is he innovaion approach, he oher is he change of measure mehod. In he innovaion approach, if Equaion (2.1.2) is saisfied, Bain and Crisan (29), Theorem 3.3, page 68, show ha, wih addiional assumpion for h such ha he sochasic inegral in he following equaion is well defined, f D(L), [,T, π (f) =π (f)+ π s (Lf)ds + [ ( ) π s (hf) π s (h)π s (f) dz s π s (h)ds. (2.1.3)

17 2.1. STOCHASTIC FILTERING 7 This is called he Kushner-Sraonovich equaion. The second erm describes he evoluion of he disribuion of X. The hird describes he evoluion of he condiional disribuion of X given accumulaion of observaions. Equaion (2.1.3) is a nonlinear sochasic equaion. I is no only an infinie dimensional, bu i has a complicaed srucure due o he presence of he erm π s (h). In general, i is no useful for compuaion. By he change of measure approach and wih he same assumpions as Equaion (2.1.3), one obains an unnormalized version of π, denoed by ρ which saisfies he following linear equaion. See Theorem 3.24, Bain and Crisan (29), page 62, ρ (f) =ρ (f)+ ρ s (Lf)ds + ρ s (hf)dz s, [,T, which has a much simpler srucure han Equaion (2.1.3). This equaion is called Zakai equaion. And π can be obained from ρ afer normalizing π (f) = ρ (f) ρ (1), [,T Finie-dimensional filers Recall ha π denoes he condiional disribuion of X given he pas observaion. In general, π can no be deermined by finie number of parameers. Bu, in some special cases, π will be deermined by a finie dimensional sochasic differenial equaions driven by observaions. The aim of his secion is o inroduce some special filers for which he corresponding π is finie-dimensional. The Kalman-Bucy filer In his secion, we inroduce he very special filering problem where he signal is Gaussian and he observaion funcion is linear. The corresponding heory is called Kalman-Bucy filer. For his case, he condiional disribuion of X given F Z is a normall disribuion. Hence i is deermined by he condiional mean and he condiional variance. Therefore, in his case, he filer is 2-dimensional. Finally, we give he evoluion equaions of he wo parameers. Here, o simplify, we assume ha he coefficiens are 1-dimensional. We consider he following linear model, for [,T, { dx =( b X + b )d + σ dv, dz =( h X + h )d + db, (2.1.4) where X is normal disribued wih mean μ R and variance r 2 R+, V and B are independen 1-dimensional Brownian moions, { b }, { b }, { h } and { h } are deerminisic R-valued processes, { σ } is a deerminisic R + -valued process, and Z =. For his model, he condiional disribuion of he sae X given he pas observaion of Z is Gaussian, applying Lemma 6.12, Bain and Crisan (29), page 149. A normal disribuion is deermined by is mean and variance. Therefore, he condiional disribuion is uniquely deermined by is mean, defined by ˆX := E[X F Z, and variance, defined by P := E [(X ˆX ) 2 F Z. By Proposiion 6.14, Bain and Crisan (29), page 152, he process { ( ˆX,P ) } is he unique soluion o he following T

18 8 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION equaions, for [,T, { d ˆX =( b ˆX + b )d + h P [ dz ( h ˆX + h )d, d d P = σ 2 +2 b P h 2 P 2, (2.1.5) wih ˆX = E(X )=μ and P = E[(X ˆX ) 2 =r 2. The deails can be found in Bain and Crisan (29), page Finie sae Markov chain Nex we inroduce a filer, for which he corresponding condiional disribuion is a finiedimensional sochasic process, since he sae space is finie dimensional. We will show ha, in his case, (2.1.3) gives rise o a finie-dimensional filer. We consider Model (2.1.1) and, for simpliciy, we assume ha he coefficiens are 1-dimensional. Moreover,wespecifyohecasehaX be a finie-sae Markov-chain wih sae space S = {1,...,n}, wheren N. The generaor associaed o process X is a Marix Q =(q ij ) 1 i,j n, which denoes he ransiion inensiies of X. Le p,i := P(X = i F Z ) and p := (p,1,...,p,n ) (2.1.6) be he vecor process represening he vecor of condiional probabiliies. Le vecor h := (,andin h(1),h(2),...,h(n)) beheideniymarixofsizen. Davis and Marcus (1981), Example 1, page 66, and Bain and Crisan (29), Remark 3.26, page 65, show ha he condiional probabiliies solve he following n-dimensional sochasic differenial equaion, ) p = p + Q p s ds + (B h p s I n )p s (dz s h p s ds, [,T, (2.1.7) ( ) where B := diag h(1),...,h(n) is a diagonal marix. Equaion (2.1.7) is a recursive equaion for he compuaion of p. Moreover Equaion (2.1.7) is an n-dimensional SDE sysem for he vecor p of condiional probabiliies, and hence a finie-dimensional filer Filering from poin processes observaions The nonlinear filering problem for jump-diffusion is of grea ineres. The fundamenal o poin process filering was presened in Brémaud (1972) and Brémaud (1981). Frey and Runggaldier (21) deal wih a general case and provide recursive updaing rules for he filer. Shown below is a simple example wih 1-dimensional coefficiens: Le n N, and we assume ha X can be a finie-sae Markov-chain wih sae space S = {1,...,n}. The generaor associaed o he process X is a Marix Q =(q ij ) 1 i,j n. Here, observaion Z is a doubly sochasic Poisson process wih inensiy λ(x ), where λ : R R +. Le p, defined by Equaion (2.1.6), be he vecor process represening he vecor of condiional probabiliies. Le vecor λ (. := λ(1),λ(2),...,λ(n)) Frey and Runggaldier (21) show ha hese condiional probabiliies solve he following n-dimensional sochasic differenial equaion,

19 2.2. THE FILTERING MODEL 9 for [,T, p = p + Q p s ds + ( ) where B λ := diag λ(1),...,λ(n) 1 λ (B λ p λ ( ) p s I n )p s dz s λ p s ds, (2.1.8) s is a diagonal marix. 2.2 The filering model The objecive of his secion is o inroduce he nonlinear filering problem in which we are ineresed. We have a sae process X which can no be observed. The informaion of X is obained by he observaion processes Z and Y,whereZ is a nonlinear coninuous observaion, wih some noise, and Y is a doubly sochasic Poisson process wih he sochasic inensiy which is a nonlinear funcion of X. We are ineresed in he condiional expecaion of a funcion of X given he pas observaion. All sochasic processes will be defined on a probabiliy space (Ω, F, P) and on a finie ime inerval [,T. F = {F, [,T}, which saisfies he usual condiions, is he full-informaion filraion. All processes considered are by assumpion F-adaped. We consider he following filering problem hroughou his hesis Unobserved sae process This secion is devoed o an inroducion of he sae process which is a diffusion process in our case. To be precise, le X = {X, T } be he unobserved d-dimensional sae. The sae process is a sochasic process which can no be observed direcly. We assume X is he soluion of a d-dimensional sochasic differenial equaion driven by a sandard m-dimensional Brownian moion V = {V, [,T}, X = X + b(x s )ds + σ(x s )dv s, T. (2.2.1) Here, X has finie second momen. Le p L 2 (R d ) be he densiy of he law of X,hen p (x) a.e.and R d p (x)dx = 1. We assume ha b : R d R d and σ : R d R d m saisfy he following condiions: here exis a posiive consan K, such ha for all x, y R d,wehave b(x) b(y) K x y, σ(x) σ(y) K x y. (2.2.2) b(x) K(1 + x ), σ(x) K(1 + σ(x) ). (2.2.3) Here he Euclidean norm is defined in he usual fashion for vecors, and exended o d m d marices by considering hen as d m-dimensional vecors : σ = m i=1 j=1 (σ ij) 2. Under he globally Lipschiz condiion, he SDE has a unique soluion. For he exisence and uniqueness of he soluion for Equaion (2.2.1), see Øksendal (198), Theorem 5.5, page 48, or Bain and Crisan (29), page 7. Le L (R d ) be he space of bounded measurable real-valued funcions on R d. The generaor L : D(L) L (R d ) associaed o he process X is he second order differenial operaor L = d i=1 b i (x) x i d 2 a ij (x), (2.2.4) x i x j i,j=1

20 1 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION where a(x) = [a ij (x) := σ(x)σ (x), and b i and x i denoe he ih componen of b and x, respecively. D(L) consiss of all funcion f L (R d ), such ha Lf L (R d )and M f = f(x ) f(x ) Lf(X s )ds, [,T, (2.2.5) is an {F }-adaped maringale Observaions X is parially observed, ha is, informaion concerning X is obained from he observaion processes Z, which is coninuous, and Y, which is a pure jump process. Z is a l-dimensional noisy nonlinear observaion of he sae X, Z = h(x s )ds + B, T. (2.2.6) Here B = {B, [,T} is a l-dimensional sandard Brownian moion independen of X and Y.Wehavehefollowing: Assumpion 2.1. We assume ha h : R d R l is a measurable funcion such ha ( T ) P h(x s ) ds < =1. (2.2.7) The Equaion (2.2.7) ensures ha he Riemann inegral in Equaion (2.2.6) exiss P a.s. Assumpion 2.2. We assume ha λ : R d [ϖ 1,ϖ 2 is a coninuous funcion, where <ϖ 1 < ϖ 2 are consans. We furher assume ha Y is a doubly sochasic Poisson process wih he sochasic inensiy {λ(x )} [,T and Y =. Then he process Y λ(x s )ds, [,T, is a (P, F)-maringale. Denoe he jumping imes of Y by he sequence {τ n } n 1,henτ n = inf{ Y n} The objecive Recall ha F Z,Y is he σ-algebra generaed by {Z u, u } {Y u, u }. Then he observaion filraion is given by F Z,Y := {F Z,Y } T. We are ineresed in, for all f L (R d ), [ E f(x ) F Z,Y, [,T. (2.2.8) The subjec of he mahemaical heory of filering is finding suiable ways of compuing his condiional expecaion recursively, eiher exacly or approximaely. Equivalenly, he principal aim of solving a filering problem is o deermine he condiional disribuion of he sae X given he observaion hisory.

21 2.3. THE INNOVATIONS PROCESSES 11 To conclude, we sudy he following nonlinear model hroughou he hesis, for [,T, X = X + b(x s)ds + σ(x s)dv s, Z = h(x s)ds + B, Y is a doubly sochasic Poisson process wih inensiy λ(x ), (2.2.9) where X is he sae process, Z and Y are observaions. The objecive is o deermine he condiional disribuion of he sae X given he observaion hisory. 2.3 The innovaions processes In his secion, we inroduce he innovaions processes and discuss heir maringale properies. These heoreical properies will be used in our compuaion sraegy, see Secion In Chaper 7, we will use he Galerkin mehod o approximae he unnormalized condiional densiy. When he numerical resul is inconsisen wih is heoreical properies, i is necessary o increase he number of basis funcions used in he approximaion. For a generic process η, denoe η := E[η F Z,Y. Now we inroduce he innovaions processes as follows: for [,T, M = Y μ = Z In wha follows, we show M and μ are maringales. λ s ds, (2.3.1) ĥ s ds. (2.3.2) Lemma 2.1. Under Assumpions 2.2, {M } [,T is a (P, F Z,Y )-maringale. Proof. I follows from Equaion (2.3.1) ha for [,T, E M =E Y λ(x s )ds + λ(x s )ds E Y λ(x s )ds +2 λ T<. λ s ds The las inequaliy follows as Y λ(x s)ds is a maringale and λ is bounded by Assumpion 2.2. For s< T, [ E[M Fs Z,Y =E Y Y s On he one hand, [ E Y Y s [ =E Y Y s s s λ(x u )du s λ u du λ(x u )du Fs Z,Y Fs Z,Y + M s Fs Z,Y + E [ ( = E E Y Y s [ (λ(x u ) λ u )du s s Fs Z,Y + M s. ) F Z,Y λ(x u )du F s =. s

22 12 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION On he oher hand, [ E (λ(x u ) λ u )du s Fs Z,Y = = s =, s ( E λ(x u ) λ u F Z,Y s where he firs equaliy follows from Fubini s Theorem. So and M is a (P, F Z,Y )-maringale. ) du ( E λ(x u ) E(λ(X u ) F Z,Y E[M F Z,Y s =M s, u ) Fs Z,Y Lemma 2.2. Under Assumpion 2.1, {μ } [,T is a (P, F Z,Y )-Brownian moion. Proof. I follows from Equaion (2.3.2) ha, for s< T, [ E[μ Fs Z,Y =E Z Z s [ = E B B s + = s = μ s, s s [ E h(x u ) E ĥ u du Fs Z,Y + μ s ( ) h(x u ) ĥu ( h(x u ) F Z,Y u du Fs Z,Y + μ s ) F Z,Y s du + μ s where he fourh equaliy follows from he independen incremens of he Brownian moion B, Fubini s Theorem and ieraed condiional expecaion. Therefore μ is a (P, F Z,Y )-maringale. On he oher hand, i follows from Equaion (2.3.2) ha μ = Z =. Since μ is an F Z,Y -maringale saring from zero a ime zero wih coninuous pahs and wih quadraic variaion equal o a each ime, μ is an F Z,Y -Brownian moion, by Shreve (24), Theorem 4.6.4, page 168. ) du 2.4 The Zakai equaions In his secion, we proceed o esimae he sae X based on he available informaion. Generally, he condiional disribuion can be compued by wo mehods. The firs mehod is o solve Kushner-Sraonovich equaion, which is a nonlinear sochasic parial differenial equaion. The second mehod is o solve Zakai equaion, which is a linear sochasic parial differenial equaion describing he dynamics of he unnormalized disribuion. The objecive of his secion is o deduce he Zakai equaion of he nonlinear filering problem. The secion is organized as follows. In Secion 2.4.1, a new measure is consruced under which Z becomes a Brownian moion, Y becomes a sandard Poisson process. In Secion 2.4.2, we show ha he condiional expecaion has a represenaion in erms of an associaed unnormalized version ρ and ρ saisfies a linear evoluion equaion, he so-called Zakai equaion.

23 2.4. THE ZAKAI EQUATIONS A new measure Before deriving he Zakai Equaion, we inroduce a new measure where Z becomes a sandard Brownian moion and Y becomes a Poisson process wih inensiy 1. Furhermore, under he new measure, hey are independen. Prior o ha, we define sochasic process Λ. We show i is a maringale and consequenly, a new measure is consruced based on Λ. Define sochasic process Λ by 1 ( Λ := τ n 1 λ(x τn ) 1 2 ) ( exp h(x s ) 2 ds + [h(x s ) db s ) (λ(x s ) 1)ds, [,T. (2.4.1) In wha follows, we will show ha Λ isa(p, F)-maringale under some assumpions. The necessary and sufficien condiions for absolue coninuiy of measures have been sudies for diffusion ype by Lipser and Shiryaev (1974a), page , for poin processes, by Lipser and Shiryaev (1974b), page , Brémaud (1981), Theorem T4, page 168, Theorem T11, page 242. Now we inroduce condiions under which Λ is a maringale. For general case, he classical condiion is Novikov s condiion, see Proer (25), Theorem 41, page 14. Noice, for his special case, B is independen of X. We have an alernaive condiion provided as follows. Assumpion 2.3. We assume ha h : R d R l is a measurable funcion such ha [ T E h(x s ) 2 ds <. (2.4.2) Lemma 2.3. Suppose ha Assumpions 2.3 and 2.2 are fulfilled, hen E( Λ )=1 for [,T. Proof. For sake of simpliciy, define [ Λ 1, := exp { Λ 2, := τ n 1 λ(x τn ) h(x s ) 2 ds, [h(x s ) db s 1 2 } [ exp (λ(x s ) 1)ds. Wih Assumpion 2.3, apply Lipser and Shiryaev (1974a), Example 4, page 234, or Noe 3, page 278, we ge E( Λ 1, F X )=1. (2.4.3) Wih Assumpion 2.2, Brémaud (1981), Theorem T11, page 242, shows 1 The produc τ n is aken o be 1 if τ1 >. E( Λ 2, F X )=1. (2.4.4)

24 14 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION Combining Equaion (2.4.3) and (2.4.4), we ge [ E( Λ )=E E( Λ F X ) [ =E E( Λ 1, Λ2, F X ) [ =E E( Λ 1, F X )E( Λ 2, F X ) =1. The desired resul is obained. For reader s convenience we presen he proof of Equaion (2.4.3) and (2.4.4) here. Noe ha B is independen of X, by Assumpion (2.4.2), applying Shreve (24), Theorem 4.4.9, given F X, [h(x s) db s is normally disribued wih mean zero and variance h(x s) 2 ds. So we obain Equaion (2.4.3). Furhermore, noice ) F X E ( Λ2, =exp [ ( (λ(x s ) 1)ds E τ n 1 λ(x τn ) Given F X and τ j <τ j+1,hevariablesτ 1 <τ 2... <τ j are disribued like j order saisics from a sample of independen random variable wih densiy λ(x s )/( λ(x s)ds),s [,. So ( E τ n 1 λ(x τn ) F X Equaion (2.4.4) is obained. ) ( =E 1 {τ1 >} F X [ =exp [ =exp I is now ime o show ha Λ is a maringale. ) + j=1 λ(x s )ds j E (1 τj τ j+1 j= λ(x s )ds exp(). n=1 ( λ(x s)ds) j j! F X ). 1 λ(x τn ) j F X ) ( λ(x s)ds) j Proposiion 2.4. Suppose ha Assumpions 2.3 and 2.2 are fulfilled, hen process { Λ } T, defined by Equaion (2.4.1), is a nonnegaive (P, F)-maringale. Proof. Firs we show { Λ } T is a (P, F)-nonnegaive-local maringale and a (P, F)-supermaringale. By Iô s formula, process Λ saisfies he equaion Define Λ =1 Λ s [h(x s ) db s λ(x s ) 1 ( Λ s d Y s λ(x s ) s ) λ(x u )du. { inf T Λ n S n = or h(x s) 2 ds n or } λ(x s) 1 ds n, if {...}, T, oherwise. Applying Brémaud (1981), (II, T8) and Shreve (24), Theorem , we obain Λ is a Sn (P, F)-maringale. Now λ is bounded, herefore, Y has only finiely many jumps in [, T. Noice

25 2.4. THE ZAKAI EQUATIONS 15 ha Λ is a lef-coninuous process, we have sup [,T Λ <, a.s.. Moreover we have T h(x s) 2 ds <, P a.s., λ is bounded, P a.s.. Therefore Λ is a (P, F)-nonnegaive local maringale. I is nonnegaive, by Lemma 2.3 E( Λ )=E( Λ )=1<. I is inegrable and herefore, by Brémaud (1981), (I, E8), page 8, i is a (P, F)-supermaringale. Furhermore, applicaion of Lemma 2.3, Brémaud (1981) I, E6, page 7, shows ha { Λ } T is a(p, F)-maringale. Define he new measure P on he measureable space (Ω, F) by P (A) = Λ T (ω)p(dω), for all A F.DenoebyE he expecaion w.r.. P. Proposiion 2.5. If Assumpion 2.3 and 2.2 hold, hen, 1 P is a probabiliy measure. 2 The law of he process X under P ishesameasislawunderp. 2UnderP, Z is a sandard Brownian moion. 4UnderP, Y is a sandard Poisson process wih inensiy 1. 5UnderP, he processes X, Z and Y are independen. A Proof. Resuls follow from Proposiion 2.4 and Girsanov s heorem for semimaringales, see for insance, Jacod and Shiryaev (23), Theorem 3.24, page 172. Lemma 2.6. Suppose ha Assumpion 2.3 and 2.2 are fulfilled. Define Λ := Λ 1, [,T, (2.4.5) where Λ is defined by Equaion (2.4.1). ThenΛ is a (P, F)-maringale. E (Λ )=1and Λ = E ( dp dp F ) for all. Proof. Firs we have E (Λ )=E( Λ Λ )=1. (2.4.6) By Brémaud (1981), (I, E6), page 7, in order o show Λ is a maringale wih respec o F and P, i is suffices o show i is a (P, F)-supermaringale. This can be obained similarly as he proof of Proposiion 2.4. To be precisely, by he definiion of Λ, { Λ := τ n { = τ n } λ(x τn ) exp } ( λ(x τn ) exp ( [h(x s ) db s [h(x s ) dy s 1 2 h(x s ) 2 ds h(x s ) 2 ds ) (λ(x s ) 1)ds ) (λ(x s ) 1)ds.

26 16 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION Applying Iô s formula Λ =1+ } Λ s {[h(x s ) dz s +(λ(x s ) 1)d(Y s s). Define { { inf T Λ S n = n or h(x s) 2 ds n or } λ(x s) 1 ds n, if {...}, T, oherwise. Applying Brémaud (1981), (II, T8) and Shreve (24), Theorem , we obain Λ Sn is a (P, F)-maringale. Now, Y is a Poisson processes wih inensiy 1, herefore, Y has only finiely many jumps in [,T. Noice ha Λ is a lef-coninuous process, we have sup [,T Λ <, a.s.. Moreover we have T h(x s) 2 ds <, P a.s., λ is bounded, P a.s.. Therefore, Λ is a (P, F)-nonnegaive local maringale. I is nonnegaive, by Equaion (2.4.6), E( Λ )=E( Λ )=1<. I is inegrable and herefore, By Brémaud (1981), (I, E8), page 8, a (P, F)-supermaringale The unnormalized filering equaions Followed he idea from Bain and Crisan (29), Proposiion 3.15, page 56, we have he following resul. Proposiion 2.7. Le U be an inegrable F -measurable random variable. Then Z,Y F E [U F Z,Y =E [U F Z,Y T. Proof. Le us denoe by = σ(z +u Z,Y +u Y ; u T ), hen F Z,Y T = Z,Y, F ). Under P Z,Y, F F Z,Y T is independen of F because Z is an F -adaped Brownian moion and Y is Poisson process wih inensiy 1. Noing ha U is F -adaped, using he properies of he condiional expecaion, we ge E [U F Z,Y =E [ U σ(f Z,Y, F Z,Y ) = E [U F Z,Y T. (2.4.7) σ(f Z,Y In he following proposiion, known as Kallianpur-Sriebel formula, see Kallianpur and Sriebel (1968), we show ha he disribuion of X given F Z,Y under he original measure P can be calculaed in erms of condiional expecaions under he new measure P.Inoherword,iis suffice o compue he numeraor on he righ hand side of Equaion (2.4.8). Proposiion 2.8 (Kallianpur-Sriebel). Suppose ha Assumpions 2.3 and 2.2 are fulfilled. For any f L (R d ), [,T, ( E f(x ) F Z,Y ) = E (f(x )Λ F Z,Y T ) E (Λ F Z,Y =: ρ (f) T ) ρ (1). (2.4.8)

27 2.4. THE ZAKAI EQUATIONS 17 Proof. See he proof of Bain and Crisan (29), Proposiion 6.1. In he following, we furher assume ha P [ T [ρ s ( h ) 2 ds < =1, (2.4.9) P [ T [ρ s (1) 2 ds < =1. (2.4.1) Under condiion (2.4.9), he sochasic inegral ρ s(fh )dz s is a (P, F)-local maringale for any bounded measurable funcion f. Moreover, he sochasic inegral ρ ) s ( f(λ 1) d(ys s) is a (P, F)-maringale for any bounded measurable funcion f in view of bounded of λ, by inegraion heorem. For deails see Brémaud (1981), Theorem T8, page 27. Now, we are ready o presen he main resul of his secion. In he following, we show {ρ } [,T saisfies he following linear SPDE. Theorem 2.9. Suppose ha Assumpions 2.3 and 2.2 are fulfilled. Furher more, if condiions (2.4.9) and (2.4.1) are saisfied hen he processes ρ saisfies for any f D(L). ρ (f) =π (f)+ ρ s (Lf)ds + ρ s (fh )dz s ) + ρ s (f(λ 1) d(y s s), P a.s., [,T, (2.4.11) The approach leading o he dynamics evoluion of for he unnormalized condiional measure has been developed in docoral Duncan (1967), Morensen (1966) and he imporan paper of Zakai (1969). The linear SPDE (2.4.11) is herefore known as he Duncan-Morensen-Zakai equaion, or simply, Zakai equaion. Proof of Theorem 2.9. We firs approximae Λ wih Λ ε given by Λ ε = Λ 1+εΛ, ε >. The definiion of Λ implies ha dλ =Λ [h(x ) dz +(λ(x ) 1)d(Y ). (2.4.12) The Iô formula for jump process, see for insance Shreve (24) Theorem , shows Λ ε =Λε + Λ s (1 + ελ s ) 2 h(x s) dz s + Λ s (1 + ελ s ) 2 (λ(x s) 1)ds + τ n ελ 2 s (1 + ελ s ) 3 h(x s) 2 ds Λ τ n. (2.4.13) 1+εΛ τn

28 18 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION The produc rule for semimaringales, ogeher wih Equaion (2.4.13) and (2.2.5), implies ha Λ ε f(x )=Λ ε f(x )+ E F Z,Y T =Λ ε f(x ) Λ ε s df (X s)+ Λ ε s (Lf)(X s)ds + Λ s f(x s )dλ ε s +[Λε,f(X) f(x s ) (1 + ελ s ) 2 h(x s) dz s ελ 2 s f(x s )( (1 + ελ s ) 3 h(x s) 2 )ds Λ s f(x s ) (1 + ελ s ) 2 (λ(x s) 1)ds Λ ε s dm f s f(x s )Λ s (λ(x s ) 1) (1 + ελ s λ(x s ))(1 + ελ s ) dy s. Taking condiional expecaions on boh sides, we ge [ [ Λ ε f(x ) Λ ε f(x ) =E F Z,Y T [ + E [ Λ ε F s(lf)(x Z,Y s )ds + E Λ ε T F sdm Z,Y s f T [ + E Λ s f(x F Z,Y s ) T (1 + ελ s ) 2 h(x s) dz s [ + E ελ 2 s f(x F Z,Y s )( T (1 + ελ s ) 3 h(x s) 2 )ds [ E Λ s f(x F Z,Y s ) T (1 + ελ s ) 2 (λ(x s) 1)ds [ + E f(x s )Λ s (λ(x s ) 1) F Z,Y T (1 + ελ s λ(x s ))(1 + ελ s ) dy s. :=E 1 + E 2 + E 3 + E 4 + E 5 + E 6 + E 7, (2.4.14) correspondingly. Compare o he Equaion (2.4.11), i remains o show ha, P a.s., asε, E F Z,Y T E 4 [ Λ ε f(x ) ρ (f), E 1 π (f), E 2 ρ s (fh )dz s, E 5 =, E 6 ρ s (Lf)ds, E 3 =, ) ρ s (λ 1)ds, E 7 ρ s (f(λ 1) dy s. In he following, we show his sep by sep. Firs, by he poinwise convergence of Λ ε Λ, We have ha lim ε Λε f(x )=Λ f(x ). E Λ ε f(x ) f E (Λ )= f E( Λ Λ )= f <, as Λ Λ = 1. Then, he dominaed convergence heorem gives ha lim ε E F Z,Y T [Λ ε f(x ) = E [Λ F Z,Y f(x ) = ρ (f), P a.s. (2.4.15) T

29 2.4. THE ZAKAI EQUATIONS 19 In a similar way, lim E 1 = E [Λ ε F Z,Y f(x ) = π (f), P a.s. (2.4.16) T Now we consider E 2.Noeha, E F Z,Y Λ ε s (Lf)(X s)ds = E ελ s 1 T F Z,Y T 1+εΛ s ε (Lf)(X s)ds 1 ε Lf T<. By Fubini s heorem, we rewrie E 2 as Moreover, We have ha E ( E 2 = E Λ ε F Z,Y s (Lf)(X s) ds T E F Z,Y T [ Λ ε s(lf)(x s ) ds. (2.4.17) lim ε Λε s(lf)(x s )=Λ s (Lf)(X s ). ) E ( ) E (Λ F Z,Y s Lf )ds T = Lf E (Λ s )ds = Lf E( Λ s Λ s )ds = Lf <, such ha he dominaed convergence heorem implies he desired resul for E 2 : lim E 2 = ε E F Z,Y T [ Λ s (Lf)(X s ) ds = ρ s (Lf)ds, P a.s. (2.4.18) Since Λ ε is bounded, apply Proer (25), Corollary 3, page 73 and Bain and Crisan (29), Lemma 3.21, page 59, E 3 =. (2.4.19) Before considering E 4, we firs have he following square inegrabiliy, E [ ( f 2 Λ ) s 2 h(xs (X s ) (1 + ελ s ) 2 ) 2 ds = 1 ε E[ f 2 ελ s (X s ) (1 + ελ s ) 4 Λ s h(x s ) 2 ds f 2 E [ Λ s h(x s ) 2 ds ε = f 2 ε = f 2 ε E[ Λs Λ s h(x s ) 2 ds [ E h(x s ) 2 ds <. The las inequaliy follows from Equaion (2.4.2). According o Bain and Crisan (29), Lemma 6.6, we change he order of condiional expecaion and sochasic inegral, and we rewrie E 4 equivalenly as E 4 = E F Z,Y T [ Λ s f(x s ) (1 + ελ s ) 2 h(x s) dz s.

30 2 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION Now consider he following process E F Z,Y T [ Λ s f(x s ) (1 + ελ s ) 2 h(x s) dz s. (2.4.2) We now show his is a maringale by using Jensen s inequaliy, Fubini s Theorem and Equaion (2.4.2), E { ( [ E Λ ) s 2ds } f(x F Z,Y s ) T (1 + ελ s ) 2 h(x s) E { [( E Λ ) s 2 } f(x F Z,Y s ) T (1 + ελ s ) 2 h(x s) ds E { [ E f 2 (X F Z,Y s ) 1 ελ s T ε (1 + ελ s ) 4 Λ s h(x s ) 2 } ds f 2 E [ E [Λ ε F Z,Y s h(x s ) 2 ds T = f 2 ε = f 2 ε E [ Λ s h(x s ) 2 ds [ E h(x s ) 2 ds. As Z is a sandard Brownian moion under measure P, he sochasic inegral defined in Equaion (2.4.2) is a (P, F)-maringale by Shreve (24), Theorem 4.3.1, page 134. Moreover, from he condiion (2.4.9), he posulaed limi process as ε, ρ s (fh )dz s (2.4.21) is a local maringale. Thus, he difference of (2.4.2) and (2.4.21) is a local maringale: Define E F Z,Y T [ ελ 2 s (2 + ελ s ) (1 + ελ s ) 2 f(x s)h(x s ) dz s. (2.4.22) ξ s := ελ2 s(2 + ελ s ) (1 + ελ s ) 2 f(x s)[h(x s ). Since he poinwise limi lim ξ s = ε and, aking ino accoun Equaion (2.4.9), we ge he following dominaing, E F Z,Y T ξ s E (2 f F Z,Y Λ s h(x s ) ) =2 f ρ s ( h ) <, P a.s., T for almos every s [,T, dominaed convergence heorem shows ha for almos every s [,T, lim ε E F Z,Y T ξ s =, P a.s.

31 2.4. THE ZAKAI EQUATIONS 21 Furhermore, by Equaion (2.4.9) [ 2ds (ξ s ) 4 f 2 [ρ s h 2 ds <, P a.s.. E F Z,Y T The value of he inegral lim ε [E (ξ F Z,Y s ) 2 ds is no modified wih he change of he funcion T lim ε [E (ξ F Z,Y s ) 2 on he se of Lebesgue zero measure. The dominaed convergence heorem T implies [ 2ds (ξ s ) =, P a.s.. lim ε E F Z,Y T By cenral limi heorem for sochasic inegrals, see Revuz and Yor (1999), page 152, he inegral in Equaion (2.4.22) convergence o, P a.s. Then, we ge he desired resul for E 4, The desired resul of E 5 lim E 4 = ε ρ s (fh )dz s, P a.s.. (2.4.23) lim E 5 =, P a.s. (2.4.24) ε is obain as a consequence of dominaed convergence heorem by he poinwise limi, ( lim f(x ελ 2 s s) ε (1 + ελ s ) 3 h(x s) 2) = and he dominaing, E [ ελ 2 s f(x s )( (1 + ελ s ) 3 h(x s) 2 ) ds f E [ Λ s h(x s ) 2 ds [ = f E h(x s ) 2 ds <. Now we consider E 6. Since, λ is bounded, we ge he following boundedness, E [ Λ s f(x s ) (1 + ελ s ) 2 (λ(x s) 1) ds f E [ Λ s λ(x s ) 1 ds ( ) = f E λ 1 ds Guaraneed by Fubini s heorem, we change he order of condiional expecaion and inegral, and rewrie E 6 equivalenly as [ E 6 = E Λ s f(x F Z,Y s ) T (1 + ελ s ) 2 (λ(x s) 1) ds. Since he poinwise limi Λ s <. lim f(x s) ε (1 + ελ s ) 2 (λ(x s) 1) = f(x s )Λ s (λ(x s ) 1),

32 22 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION and, in view of bounded, he dominaing E [ E F Z,Y T Λ s f(x s ) (1 + ελ s ) 2 (λ(x s) 1) ds = f E( λ 1 )ds <, dominaed convergence heorem shows he desired resul for E 6, lim E 6 = ε I remains o sudy E 7. For sake of simpliciy, define ρ s (λ 1)ds, P a.s.. (2.4.25) H ε := f(x s )Λ s (λ(x s ) 1) (1 + ελ s λ(x s ))(1 + ελ s ), hen E 7 can be rewrien as, ( E 7 = E F Z,Y T H ε s dy s ). Noice he boundedness H ε f ε 2 <, change he order of sochasic inegral and condiional expecaion, which is guaraneed by sochasic Fubini s heorem, see Proer (25), Theorem 64, page 27, E 7 can be rewrien as E 7 = E (H F Z,Y s ε )dy s. T Combining he poinwise limi ( ) lim ε Hε = f(x )Λ λ(x ) 1, P a.s, and he dominaing E (E F Z,Y T ) H ε ) < E( f λ 1 E (Λ F Z,Y ) = f λ 1 <, T sochasic dominaed convergence heorem, see Proer (25), Theorem 32, page 174, implies ha for almos every s [,T, ( ) lim ε E H ε F Z,Y T = E F Z,Y T ( ) f(x )Λ (λ(x ) 1). On he oher hand ) ( ) ρ (f(λ 1) = lim ρ s f(λ 1) s = lim s E ( f(x s )(λ(x s ) 1)Λ s F Z,Y ) =E ( f(x (λ(x s ) 1)Λ ) F Z,Y ),

33 2.4. THE ZAKAI EQUATIONS 23 where he las equaliy follows from Lipser and Shiryaev (1974a), Theorem 1.6, page 17. Hence we ge ) lim (H ε )=ρ (f(λ 1). ε E F Z,Y T By Equaion (2.4.1), ogeher wih boundedness of f and λ, dominaed convergence heorem shows ha lim E 7 = ε Summing up, he Zakai equaion is obained. ) ρ s (f(λ 1) dy s, P a.s.. (2.4.26)

34 24 CHAPTER 2. FILTERING MODEL AND THE ZAKAI EQUATION

35 Chaper 3 Numerical mehods We are ineresed in he condiional disribuion of he sae process X given he pas observaion. In general, his is an infinie dimensional problem, i.e. we canno obain he condiional disribuion by a finie se of parameers. Therefore, i is normal o look for a finie dimensional approximaion which can be used in pracical applicaions. There are various numerical mehods for his, some of which we lis below. For a more comprehensive lising of hese mehods, we refer o Budhiraja, Chen, and Lee (27), Bain and Crisan (29), page To begin, we have he specral approach for SPDEs which is based on he Cameron-Marin version of he Wiener Chaos expansion. The main advanage of he specral approach, as compared o mos oher nonlinear filering algorihms, is ha he ime consuming compuaions, including solving parial differenial equaions and evaluaion of inegrals, are performed in advance. The real-ime par is relaively simple, even when he dimension of he sae process is large. For furher deails, see Loosky (26). Unforunaely, he specral approach only works for SPDEs driven by whie noise and is no useful for Model (2.2.9) wih jump. Nex, we have he exended Kalman filer which is a linearized approximaion of he original problem. The EKF does no always perform well and, in fac, performs poorly if he nonlineariies are srong, see Bain and Crisan (29), page 196. I will give a good esimae only when he coefficiens are slighly nonlinear, see Bain and Crisan (29), Theorem 8.5, page 195. The accuracy of resuls obained using EKF is neiher verifiable, nor reliable. I only works for SPDEs driven by whie noise and is no useful for Model (2.2.9) wih addiional jump observaions. In Secion 2.1.3, we show he filer for a finie-sae Markov chain is finie dimensional. Therefore, i is logical o consider approximaions of he sae process X by a sequence X n which are finiesae Markov chains, see for insance Frey and Runggaldier (21). Similar o he Markov chain approximaion mehod, he paricle mehods are a finie dimensional approximaion of unnormalized measure ρ by discreisaion of he sae variable, see for insance Carpener, Clifford, and Fearnhead (1999), Crisan, Moral, and Lyons (1999). However, paricle mehods are more flexible and easier o implemen. The basic idea is o approximae he condiional expecaion by Mono Carlo mehods. In Chaper 5, we will inroduce an addiional numerical mehod. We will show ha he unnormalized condiional densiy is he soluion of a parial differenial equaion, alhough a sochasic one. Therefore, we can apply classical PDE mehods, such as he Galerkin mehod (see for insance Germani and Piccioni (1984)), o he sochasic PDEs and obain a densiy approximaion. A deailed inroducion of Galerkin mehod follows in Chapers 6 and 7. 25

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