POLITECNICO DI MILANO
Scuola di Ingegneria Industriale e dell’ Informazione
Laurea Magistrale in Ingegneria Matematica
Backward Stochastic Riccati Equations driven by
Wiener and Point Processes, with applications to
Optimal Control
Relatore: Prof.ssa Giuseppina Guatteri
Correlatore: Prof. Marco Alessandro Fuhrman
Candidato: Giovanni Genga,
Mat. 836512
Contents
1 Introduction 4
2 Notations 7
3 Point Processes 8
3.1 Marked Point Processes on R+ . . . 8
3.2 Stochastic Integral with respect to an Integer Random Measure . . . . 10
3.3 The Predictable Representation property . . . 12
3.4 Stationary MPP . . . 13
3.5 Invariance Property of the stochastic integral . . . 14
3.6 Predictable Projection of a Process . . . 16
4 Generalised BSDE 17 4.1 Finite horizon case . . . 17
4.2 Infinite horizon case . . . 18
4.3 A little existence results on BSDE driven by MPP . . . 19
5 Optimal LQ Control of Switching Systems 21 5.1 Problem Statement . . . 21
5.2 The Lyapunov Equation . . . 22
5.3 The Riccati Equation . . . 23
6 Infinite Horizon LQ Control of Switching Systems 25 6.1 Infinite horizon Riccati Equation: Existence of a Solution . . . 25
6.2 Synthesis of the optimal control . . . 28
7 The Affine Case on finite and infinite horizon 30 7.1 Dual equation . . . 30
7.2 Finite horizon affine control . . . 32
7.3 Infinite horizon dual equation . . . 32
7.4 Synthesis of the optimal control . . . 37
7.5 Preliminary results for the Ergodic Problem . . . 39
8 The Ergodic Problem 49 8.1 Framework and Preliminaries . . . 49
8.2 Linear Quadratic optimal control in the stationary case . . . 51
8.3 Ergodic Control . . . 54
9 The predictable case 57
A Datko Theorem 59
Prefazione
La seguente tesi riguarda la soluzione di problemi di controllo lineare quadratico, nei quali l’ obiettivo è minimizzare il seguente funzionale
J (t, x, u) = EFt " Z T t (|pSsXs|2+ |us|2)ds + hGXT, XTi #
al variare del controllo u nell’ insieme dei processi progressivi di quadrato integrabile, dove l’ evoluzione dello stato X è regolata da una SDE lineare:
dXt= AtXtdt + Btutdt +P d
i=1CtiXtdWti
X0= x
in cui i coefficienti A, B e C vengono solitamente ipotizzati adattati alla filtrazione generata dal moto browniano. Questo problema è stato ampiamente discusso in let-teratura (vedere ad esempio [4], [16], [17], [19], [20] ). Noi lavoreremo sotto ipotesi più generali di quelle classiche: i coefficienti che compaiono nell’ equazione di stato non saranno adattati alla filtrazione generata dal moto browniano W , bensì a una fil-trazione allargata, generata sia dal processo di Wiener W che da un processo di punto µ. Problemi di controllo ottimale a orizzonte finito, sotto queste ipotesi più generali, sono trattati nel paper (ancora in fase di stesura) di Confortola, Fuhrman,Guatteri e Tessitore [8]. Questo lavoro rappresenterà il nostro punto di partenza: lo scopo principale della tesi è quello di estendere, per questa classe di sistemi, i risultati già esistenti su orizzonte finito a orizzonte infinito e caso ergodico. In questo senso, i risultati principali sono contenuti nei Teoremi 6.3, 8.10, 8.14. Come solito nel caso lineare quadratico, la soluzione del problema di controllo è ricondotta alla soluzione di una equazione stocastica backward di Riccati. In seguito all’ allargamento della filtrazione, questa equazione backward è guidata sia da un moto browniano che da un processo di punto marcato:
−dPt= (A0tPt+ PtAt+ Ct0PtCt+ Ct0Qt+ QtCt− Pt0BtB0tPt+ St) dt −QtdWt− Z K Utµ(ds, dx)˜ PT = G
Così la soluzione del problema su orizzonte infinito è ricondotta a risultati di buona posizione dell’ equazione precedente su tutto il semiasse positivo. Questo risultato è contenuto nel teorema 6.1.
L’ estensione al caso ergodico richiede prima una lunga digressione sul problema di controllo affine, in cui l’ equazione di stato diventa:
dXs= (AsXs+ Bsus)ds + d X i=1 Ci sXsdWsi+ fsds s ∈ [t, T ] Xt= x
Infatti, come si capirà procedendo nella lettura, l’ equazione omogenea porta a un comportamento banale del funzionale di costo ergodico, il cui valore risulta indipen-dente dal controllo scelto. Il problema significativo da affrontare in questo caso è quindi quello in cui la dinamica dello stato viene resa affine con l’ aggiunta di un termine forzante f . Avremo quindi bisogno di studiare questo problema, su orizzonte finito e infinito. L’ intera sezione 7 è dedicata alla presentazione dei nuovi risultati che abbiamo ottenuto relativamente a questo problema. Si noti che, a causa del termine forzante f che compare nell’ equazione, la BSRE da sola non è più sufficiente per caratterizzare il controllo ottimo. Sarà necessario introdurre un’ equazione duale, che congiuntamente all’ equazione di Riccati fornirà il controllo ottimale.
Infine, nella sezione 8 presenteremo la connessione tra il problema affine con fun-zionale costo scontato, il problema stazionario, e quello ergodico. Tale connessione sarà mostrata sotto l’ ipotesi di stazionarietà per i coefficienti dell’ equazione di stato. La soluzione esplicita per il caso ergodico sarà quindi ottenuta per mezzo delle soluzioni relative agli altri due problemi.
I risultati ottenuti trovano applicazione, in modo particolare, nel determinare il con-trollo ottimale per i cosiddetti regime-switching systems. In questi ultimi, si assume che lo stato X possa evolvere in un numero finito di regimi K = {1, ..., m}.
Il sistema viene allora modellato introducendo un processo stocastico ausiliario (It)t≥0
con valori in K: in ogni istante, I rappresenta il particolare regime in cui il sistema sta funzionando. Il sistema dinamico da analizzare risulta quindi
dXs= (A(s, Is)Xs+ B(s, Is)us)ds + d X i=1 Ci(s, Is)XsdWsi s ∈ [t, T ] Xt= x
In letteratura, il processo I viene solitamente assunto essere un processo markoviano (con spazio degli stati K), indipendente dal moto browniano che guida l’ equazione. In questo caso, la coppia (X, I) rappresenta un processo di Markov, e il problema di controllo ottimo associato può essere risolto con tecniche standard. Il presente lavoro mira a creare un framework in cui possano essere analizzati sistemi più generali, in cui il processo I può assumere valori in uno spazio non numerabile K e non possiede una dinamica markoviana. Lo strumento matematico che rende possibile una tale generalizzazione é quello di processo di punto marcato, che verrà introdotto nei capitoli successivi.
Per una rapida sintesi sui contenuti dei capitoli seguenti, il lettore é rinviato all’ introduzione in lingua inglese.
1
Introduction
This work deals with the solution of a Linear-Quadratic optimal control problem, i.e. optimize the functional
J (t, x, u) = EFt " Z T t (|pSsXs|2+ |us|2)ds + hGXT, XTi #
over all the possible controls u belonging to a specific space, where the state variables are governed by a Linear SDE of the following kind:
dXt= AtXtdt + Btutdt +P d i=1C i tXtdWti X0= x
where in the preceding equation the coefficients A, B and C are stochastic processes, that are usually supposed to be adapted to the filtration generated by the Wiener Process W . This problem has been extensively studied in recent years
This is a classical problem, which has been studied extensively (see e.g. [4], [16], [17], [19], [20] ).We will work under a more general set up than the classical one: our coefficients will not be adapted to the natural filtration generated by the Wiener Process W , but to a larger filtration, generated by a Wiener Process W and an inde-pendent Marked Point Process µ. Optimal control problems on finite horizon under this generalised framework are treated in the (not yet submitted) work of Confortola, Fuhrman, Guatteri and Tessitore [8]: we will base our present work precisely on the results presented in their manuscript work [8].
More precisely, we will extend the analysis of the optimal control problem de-scribed above to the infinite horizon case and to the ergodic case. In this sense, the main results of the work are stated in Theorems 6.3, 8.10, 8.14. As usual with the Linear Quadratic case, the optimal control is found by solving a Backward Stochastic Riccati Equation. Due to our generalized framework, the BSRE we’re lead to happens to be driven both by a Wiener Process and a compensated Marked Point Process:
−dPt= (A0tPt+ PtAt+ Ct0PtCt+ Ct0Qt+ QtCt− Pt0BtB0tPt+ St) dt −QtdWt− Z K Utµ(ds, dx)˜ PT = G
where a solution to the equation is to be thought as a triple of processes (P, Q, U ) living in a specific space, such that the above differential formula holds. Thus, the desired extension to the infinite horizon case relies on an existence and uniqueness result for the above BSRE on Infinite Horizon: this new result is stated and proved in Theorem 6.1.
The extension to the ergodic case requires quite a long digression on the Affine control problem, where the state equation is now an affine one:
dXs= (AsXs+ Bsus)ds + d X i=1 Ci sXsdWsi+ fsds s ∈ [t, T ] Xt= x
Indeed it will turn out that the homogeneous state equation gives rise to a banal behaviour of the ergodic functional, independently of the chosen control. The signif-icant ergodic problem to solve is instead the one where a forcing term appears in the state equation. We then need first to study the affine case on finite and infinite horizon. All Section 7 is devoted to the detailed presentation of the new results we
achieved concerning this affine problem. Due to the presence of the forcing term f in the equation of state, the BSRE alone is not sufficient anymore to solve the optimal control problem. Following a classical procedure, we introduce a new Backward SDE called Dual Equation, which together with the BSRE furnishes a representation of the optimal control.
Finally, the last section 8 presents the connection among the Affine Problem with discounted cost functional, the Stationary Problem (in which the coefficients are as-sumed to be stationary) and the Ergodic one. The explicit solution of this last problem is found in terms of the other two, thus characterizing the optimal control for the er-godic case (see Theorem 8.14).
As a particular class of systems than can comfortably fit in our new framework, we find the so called regime-switching diffusions. In these systems the evolution of the state X can take place under a finite set of regimes K = {1, ..., m}. The system can thus be modelled introducing an auxiliary stochastic process (It)t≥0 with values in
K: in every instant, I represents the regime under which the system is working. The dynamical system becomes:
dXs= (A(s, Is)Xs+ B(s, Is)us)ds + d X i=1 Ci(s, I s)XsdWsi s ∈ [t, T ] Xt= x
What is usually supposed for the process I, is being markov. This, jointly with in-dependence from the Wiener process W , allows to consider the couple (X, I) as a new markovian system, and consequently to apply standard, markovian techniques. Thanks to our generalized framework, we will be able to deal, on the other hand, also with cases where I is non markovian and takes values in an uncountable set K. The mathematical concept that allows such a generalization is that of marked point process.
The reader can find below a brief description of each Section of the present work.
Section 3. This section introduces the basic facts regarding Marked Point Processes that will be used throughout the work. In particular we recall the fundamental Rep-resentation Theorem for local martingales (see e.g. [12]), which allows us to represent martingales of the natural filtration of a Point Process as a stochastic integral with respect to the compensated Point Process. We devote the two last subsections of this part to the analysis of stationary MPP. We deduce an Invariance Property for the stochastic integral with respect to this processes, which will be our starting point in Section 8.
Section 4. We present some classical results on BSDE driven by Wiener and Jump Processes. The majority of the results presented in this section will not be used in the rest of the work, but may help the reader to understand how our work is related to the already existing theory on BSDE. We conclude the section with an Existence and Uniqueness result concerning a BSDE driven by MPP and Brownian Motion on a finite horizon. The result is new (see Theorem 4.4), although its proof is just an easy adaptation to our case of already existing results. Nevertheless it is explicitly stated as it will be used in the following Sections.
Section 5. This section collects together the results obtained in the paper [8].The optimal control of a switching system is characterized by means of the solution of a BSRE driven by Wiener and Marked Point Processes. This is the starting point of our work, in the sense that the main objective of this thesis is to extend the results of this section to the infinite horizon and to the ergodic case.
Section 6 . The Infinite Horizon Riccati Equation is introduced. We state an existence (Theorem 6.1) and minimality result (Corollary 6.2)for a particular solution. We next address the optimal control problem on infinite horizon and solve it explicitly thanks to the minimal solution of the Riccati BSDE previously found.
Section 7. In order to treat the Ergodic Problem, a detailed study of the affine case on finite and infinite horizon is in order. This is precisely what is carried out by this chapter. We first introduce the Dual (or costate) equation (7.2). Existence results for this equation on finite and infinite horizon are derived (see Proposition 7.1 and Theorem 7.7). Thanks to these results we are able to solve the optimal control problem for the affine case, for finite and infinite horizon. In the last subsection we introduce a family of discounted optimal control problems, where the state equation is again affine. We next study the behaviour of the optimal couple as the discount α → 0. This family of problems will turn out to be closely linked to the stationary and ergodic control problems. More precisely, we will be able to show the connection among discounted and ergodic problems under the additional assumption of stationarity, which will allow us to exploit the stationary problem as a link between the other two.
Section 8. This last section is devoted to the ergodic case itself. We consider as state equation an affine one, because the homogeneous case would lead to banal results as far as the ergodic analysis is concerned. We introduce another auxiliary control problem, namely the stationary one (see eq.(8.3)). An important result towards the solution of this stationary auxiliary problem is Proposition 8.9, stating the stationarity of the solution to the closed loop state equation. Finally, the last subsection connects the Ergodic, Stationary and Infinite horizon discounted problem: this furnishes a characterization of the optimal ergodic control.
2
Notations
Throughout the work we will use P(Ft) to denote the sigma algebra generated on
Ω×[0, T ] by all left continuous processes adapted toFt, for all t ∈ [0, T ]. This is called
the predictable sigma-field. When considering processes defined on the whole positive line, the finite interval [0, T ] is replaced by R+. Given a stochastic basis (Ω, P, F , Ft)
we will say it satisfy the usual conditions if : • F is P-complete.
• (Ft)t≥0is right continuous.
• F0 contains the P-null sets of Ft, for all t ≥ 0.
Let R be any Euclidean space and letB(R) be the Borel σ-field on R. We denote bySn the space of symmetric matrices of dimension n × n, and bySn+ its subset of
non-negative definite matrices.
The following classes of processes will be used in the paper.
• LpP(Ω × [0, T ]; R), p ∈ [1, ∞] denotes the subset of Lp(Ω × [0, T ]; R), given by
all equivalence classes admitting a predictable version. This space is endowed with the natural norm
|Y |pLp(Ω×[0,T ];R)= E
Z T
0
|Ys|pds < ∞.
Elements of this space are identified up to modification.
• LpP(ν, [0, T ]; R), p ∈ [1, ∞) denotes the set of equivalence classes with respect to the measure ν(ω, dt, de) of mappings H : Ω × [0, T ] × K → R which are predictable (P ⊗K-measurable) and such that
|H|pLp(ν):= E Z T 0 Z K |Ht(x)|pµ(dt, dx) = E Z T 0 Z K |Ht(x)|pν(dt, dx) < ∞.
Here K is a space of marks and µ is a Marked point process with compensator ν. Note that we will sometime just write Lp(ν) instead of Lp(ν, [0, T ]; R) when the meaning is clear from the context.
• We will use a lower "Loc" qualifier (e.g. L2
Loc(Ω×R)) to indicate those processes
Y such that for every compact set D ⊂ R+, Y ∈ L2(Ω × D).
• A , ALoc processes of integrable and locally integrable variation respectively.
• A+,A+
Loc increasing integrable and increasing locally integrable processes. An
increasing process X is said to be integrable if EX∞< ∞. An increasing process
X is locally integrable if there exist a sequence of stopping times Tn→ ∞ such
that XTn∧t∈A
+.
• We use GLoc(µ) for processes integrable with respect to a stochastic integer
valued random measure (see subsection 3.2). Moreover we denote with L2(Ω,F
T; R) the set of the R-valued random variables
3
Point Processes
3.1
Marked Point Processes on R
+Let (K,K) be a Lusin space, i.e. a measurable space with the following property: there exist a compact metric space M and a Borel set B ∈B(M) such that (K,K) and (B,B(B)) are isomorphic as measurable spaces.
We further introduce the following spaces:
• ˜MK+: space of sequences (tn, zn)n∈N, where tn∈ R+ and zn∈ K.
• MK+: space of measures p =P
n∈Nδ(tn,zn).
• M+
K: sigma algebra on MK generated by maps µ 7→ µ(C, L), for C ∈B(R+)
and L ∈K.
Remark 3.1. It could be shown that under the present setting, the mapping p 7→ (tn, zn) is measurable.
Definition 3.1. The measure space (MK+,M+
K) is called canonical space of MPP on
R+
Given now a filtered probability space,(Ω,F , (Ft)t≥0), we can give a definition of
K-Marked point process
Definition 3.2. A K-Marked Point Process µ is a measurable mapping µ : (Ω, P, F ) → (MK+,M+
K). We will set:
Tn(ω) = tn(µ(ω))
Zn(ω) = zn(µ(ω)),
and suppose that the filtration (Ft)t≥0 contains the internal history associated to the
MPPFtµ:
Fµ
t = σ{µ(ω)((0, s] × L), 0 ≤ s ≤ t, L ∈K}.
This implies that the random variables Tn are stopping times w.r.t. Ft (they are
stopping times relative to the internal history) and that each Zn is FTn-measurable
(because Zn isF µ Tn).
We will use the following notation:
Nt(L) := µ(ω)((0, t] × L) ∀L ∈K (3.1)
In what follows, the counting measure µ, the sequence (Tn, Zn) and the processes
Nt(L) will be identified, all being called K-marked point process. On (0, +∞) × Ω × E
we define the σ-field
˜
P(Ft) =P(Ft) ⊗K (3.2)
Any ˜P(Ft)-measurable function H : Ω × R+× E → R is then said to be predictable.
Given the definitions above, it is true that: Z R+ Z K H(s, e)µ(ds, de) = ∞ X n=1 H(Tn, Zn)1Tn<∞,
for anyFtpredictable process H : Ω × R+× E → R.
The next Theorem will be fundamental in what follows. It is a direct consequence of the existence of the Dual Predictable Projection of an increasing process, as was defined by Dellacherie and Meyer.
Theorem 3.2. [5] Let (Ω,F , P, Ft) verify the usual conditions, whereFtis a history
of a K-marked point process µ. Suppose that the mark space (K,K) is Lusin. Then there exist
1. a unique (up to P-indistinguishability) right continuous increasing Ft-predictable
process At with A0≡ 0,
2. a transition measure φt,ω(de) from ((0, ∞) × Ω,P(Ft)) into (K,K), such that,
for all n ≥ 1: E " Z Tn 0 Z K H(s, e)µ(ω, ds, de) # = E " Z Tn 0 Z K H(s, e)φs,ω(de)dAs(ω) # (3.3) for all nonnegative Ft-predictable K-indexed processes H.
Proof Set
µn(ω, ds, de) = µ(ω, ds, de)1(s ≤ Tn(ω))
and then define, the following measure on ˜P(Ft)
Mµn(D) = E " Z Tn 0 Z K 1D(s, e)µn(ω, ds, de) # ∀D ∈ ˜P(Ft). (3.4)
Let mµn be the projection measure of Mµn onP(Ft):
mµn(C) = Mµn(C × K) ∀C ∈P(Ft) (3.5)
Since Mµn is σ-finite, it can be disintegrated (here we need the assumption that
(K,K) is Lusin):
Mµn(dω, dt, de) = mµn(dω, dt)φω,t(de)
where φω,t(de) is a transition measure from ((0, ∞) × Ω,P(Ft)) into (K,K).
Now, mµn(dω, dt) = P(dω)dNt(K). Indeed, mµn(C) = E Z ∞ 0 1C Z K µ(ω, ds, de) = E Z ∞ 0 1CdNt(K)
Therefore, defining Atas the dual predictable projection of Nt(K) we get:
Mµn(dω, dt, de) = P(dω)dAt(ω)φω,t(de)
After applying the monotone class theorem we get the thesis.
Remark 3.3. If the sequence of stopping times is non-explosive (Tn → ∞ a.s.), then
the measure Mµ (defined analogously to Mµn) is already σ-finite, and we get:
E Z ∞ 0 Z K H(s, e)µ(ω, ds, de) = E Z ∞ 0 Z K H(s, e)φs,ω(de)dAs
for all nonnegativeFt-predictable K-indexed processes H.
Remark 3.4. φt,ω(de) is unique in the following sense. If φ0t,ω(de) has the same property
as φt,ω(de), then for any L ∈Kthere exist a P-null set N (L) such that outside N (L):
Z t 0 Z L φω,s(de)dAs(ω) = Z t 0 Z L φ0ω,s(de)dAs(ω). (3.6)
This follows from the uniqueness of the dual predictable projection, together with the fact thatRt 0 R Lφω,s(de)dAs(ω) and Rt 0 R Lφ 0
ω,s(de)dAs(ω) are two versions of the dual
predictable projection of Nt(L).
Note that since (K,K) is Lusin, we know that K is separable (i.e. generated by a countable family of sets). Hence we can find a universal P-null event N such that outside this set (3.6) holds for any L ∈K.
In Theorem 3.2 both sides could take the value +∞. If we assume they’re both finite, we obtain the following corollary:
Corollary 3.5. Under the hypothesis of Theorem 3.2, assume that the quantities on both sides of (3.3) are finite. Then,
Z t∧Tn
0
Z
K
H(s, e)˜µ(ds, de) is a (P, Ft) − martingale
where ˜µ = µ − ν and
ν(ω, dt, de) = φω,t(de)dAt(ω)
For the rest of this work we will make the following assumptions: Hypothesis 1. The sequence Tn is non-explosive, i.e. Tn → ∞ P-a.s.
Hypothesis 2. The process Atis a.s. continuous.
3.2
Stochastic Integral with respect to an Integer Random
Measure
We now pause for a moment our discussion about point processes in order to make a brief digression. At the end of the previous chapter we defined a measure ˜µ = µ−ν: this is always possible, since both µ and ν are finite variation processes, hence the integral with respect to their difference is always defined. Nevertheless, it is possible to define a stochastic integral with respect to an arbitrary integer valued random measure in a different way. We now briefly describe this other integral, and specify under what conditions it coincides with the one defined as the difference of the two separated integrals. We will see that, under the hypothesis we have adopted, the two notions coincide, hence we won’ t have to worry about the meaning of ˜µ. Having shown this equivalence, we will be able to apply deep results coming from semimartingale theory to our case. We remark however that all the results we apply could be derived directly for our case without making use of this new stochastic integral. Stated differently, we could just choose to interpret the integral in d˜µ as the difference dµ − dν and directly derive the properties we need for this integral.
Let µ(ω, dt, de) be an integer valued random measure on R × K, where (K,K) is an enough regular space (e.g. a Lusin Space). Let moreover ν(ω, dt, dx) be its compensator.
Definition 3.3 (Jacod, Shiryaev, [14]). a)We denote with GLoc(µ) the space of all
P⊗K measurable processes U (ω, s, e) such that the process:
|U |2∗ µ1/2 t = Z t 0 Z K U2(ω, s, e)µ(ω, ds, de) 1/2 (3.7)
is locally integrable, i.e. |U |2∗ µ1/2
belongs toALoc+ .
b) For a process U ∈ GLoc(µ) we define the stochastic integral
U ∗ (µ − ν) = Z T 0 Z K U (ω, s, e)˜µ(ω, ds, de) = Z T 0 Z K U (ω, s, e)(µ − ν)(ω, ds, de) (3.8) as any purely discontinuous local martingale X such that the two processes ∆Xs =
Xs− Xs− and
Z
K
U (ω, s, e)µ(ω, {s}, de) are indistinguishable.
Proposition 3.6. Let U be a predictable function on Ω × R+ such that the process
|U | ∗ µ (or equivalently, the process |U | ∗ ν) is locally integrable. In formulae (|U | ∗ µ)t= Z t 0 Z K |U (s, e)|µ(ds, de) ∈A+ Loc.
Then U ∈ GLoc(µ) and moreover U ∗ (µ − ν) = U ∗ µ − U ∗ ν.
Proof. See [Jacod, Shiryaev, [14]].
Remark 3.7. Since the Marked Point Process that generates our random measure is non explosive, we always have that if U ∈ L2
P(ν) then |U | ∗ µ (or equivalently, the
process |U | ∗ ν) is locally integrable. To verify this, take as localizing sequence the Tn
times of the jumps. Since the process is non-explosive, Tn(ω) → ∞ P-a.s.. Now note that
E n X i=1 U2(Ti, Zi) = E Z Tn 0 Z K U2(s, e)µ(ds, de) = E Z Tn 0 Z K U2(s, e)ν(ds, de) < +∞ E n X i=1 |U (Ti, Zi)| = E Z Tn 0 Z K |U |(s, e)µ(ds, de) = E Z Tn 0 Z K |U |(s, e)ν(ds, de) But then : E n X i=1 |U (Ti, Zi)| ≤ n X i=1 EU2(Ti, Zi) 1/2 ≤√n E n X i=1 U2(Ti, Zi) !1/2
and thus we have found a localizing sequence that makes the process (|U | ∗ µ)t∧Tn
integrable.
As last thing, we characterize the predictable quadratic variation of an integrated process. Introduce the process
ˆ Ut(ω) = Z K U (ω, t, e)ν(ω, {t}, e), if Z K |U (ω, t, e)|ν(ω, {t}, e) < +∞, ˆ
Ut(ω) = +∞ otherwise. Then, if we set:
C(U )t= (U − ˆU )2∗ νt+ X s≤t (1 − ν(ω, {t}, K))( ˆUs)2, (3.9) ¯ C(U )t= |U − ˆU | ∗ νt+ X s≤t (1 − ν(ω, {t}, K))| ˆUs|, (3.10)
The following proposition holds:
Proposition 3.8 (Jacod, Shiryaev, [14]). Let U be a P⊗K measurable process on Ω × R+.
a) U belongs to GLoc(µ) and (U ∗ (µ − ν)) is a square integrable martingale if and
only if C(U ) belongs to A+.
b) U belongs to GLoc(µ) and (U ∗ (µ − ν)) is a square integrable local martingale
if and only if C(U ) belongs toALoc+ .
In both cases a) and b) it holds (h·, ·i stands for the predictable quadratic variation): hU ∗ (µ − ν), U ∗ (µ − ν)i = C(U ) (3.11) c) U belongs to GLoc(µ) and U ∗ (µ − ν) belongs to A (processes of integrable
variation) if and only if ¯C(U ) belongs toA+.
d) U belongs to GLoc(µ) and U ∗ (µ − ν) belongs to ALoc (processes of locally
Proof. See [Jacod, Shiryaev, [14]].
Remark 3.9. Note that since we assume the process At(ω) to be a.s. continuous, the
quantity ˆU in the preceeding formulas is always null.
3.3
The Predictable Representation property
We recall the following theorem from [14]:
Theorem 3.10 (Representation Property of Local Martingale w.r.t. a Marked Point Process). Let (Ω, P, F , ¯Ftµ) be a stochastic bases such that F¯
µ
t is the natural
aug-mented filtration generated by a marked point process µ.Then all local martingales have the form
M = M0+ ρ ∗ µ − ρ ∗ ν
where ρ is some ˜P(Ft)-measurable function such that (|ρ| ∗ µ) ∈ALoc+ (i.e. |ρ| ∗ µ is
a locally integrable process).
Remark 3.11. From proposition 3.8 we can derive that if the local martingale is in particular a square integrable martingale, then ρ ∈ L2P(ν, [0, ∞)), i.e.:
Z
R+
Z
K
|ρ(s, e)|2ν(ds, de) < ∞.
At this point, recall that an analogous representation property holds for brownian martingales, i.e. martingales adapted to the natural filtration of a brownian motion. More precisely
Theorem 3.12. Let Wt be a (Ω, P, F , ¯FtW) Wiener process. Then any square
inte-grable martingale M on [0, T ] admits a representation: Mt= M0+
Z t
0
φsdWs
for some process φ ∈ L2P(Ω × [0, T ]).
Putting together the two previous results yields the following:
Theorem 3.13. Let be given a marked point process µ and an independent brownian motion W . Define Fµ,W t = ¯F µ t ∨ ¯F W t
Then any square integrableFtµ,W-martingale Mt on [0, T ] admits a representation of
the form: Mt= M0+ Z t 0 φsdWs+ Z t 0 Z K ρ(s, e)˜µ(ds, de) (3.12) where φ ∈ L2P(Ω × [0, T ]) and ρ ∈ L2P(ν, [0, T ]).
Proof This property is true for all square integrable martingales with final value 1A1B for A ∈ F¯TW and B ∈ F¯
µ
T. Indeed, by standard results, we know that the
two aformentioned augmented filtrations are naturally right continuous, so that one can apply the two previous propositions (this result is standard for Wiener processes, for the analogue in the case of MPP see e.g. the Appendix of [5]). Hence it is easily verified for linear combination of random variables of this type. Hence we have an isometry:
I :M2⊃ D → L2
P(Ω × [0, T ]) × L2P(ν, [0, T ])
where D is the subspace ofM2consisting of martingales whose final value is a linear
combination of random variables of the form 1A1B. Since D is a dense subspace of
M2, there’s a canonical way to extend this Isometry to all M2. Ito Isometry and
proposition 3.8 then tell us that this isometry has the form of equation (3.12) for every M ∈M2.
3.4
Stationary MPP
We will now extend the definition we gave in subsection 3.1 for Marked Point Processes on all the real line. More precisely, let be defined the following:
• ˜MK: space of sequences (tn, zn)n∈Z.
• MK: space of measures p =Pn∈Zδ(tn,zn).
• MK: sigma algebra on MKgenerated by maps µ 7→ µ(C, L), for C ∈B(R) and
L ∈K.
• St: A traslation operator on the space MK. Stµ(C × L) = µ(C + t, L).
Definition 3.4. The measure space (MK,MK) is called canonical space of MPP.
Given now a probability space,(Ω,F , (Ft)), we define a MPP on R as follows:
Definition 3.5. A Marked Point Process µis a measurable mapping µ : (Ω, P, F ) → (MK,MK). Again we will set set:
Tn(ω) = tn(µ(ω))
Zn(ω) = zn(µ(ω)),
We next introduce the notion of a stationary MPP on a filtered probability space on which a measurable flow is defined. Take then a measurable flow (θ)t∈R on (Ω,F ) i.e.
• (t, ω) → θt(ω) is measurable fromB ⊗ F to F .
• θt is bijective for all t.
• θs+t = θsθt.
Definition 3.6. Given a measurable flow (θ)t∈R on (Ω,F ), we say that a MPP µ is stationary if it satisfies:
• µ(θtω) = Stµ(ω).
• P ◦ θt= P
Remark 3.14. Thanks to the form of the traslation operator St, it can be shown that
the above definition implies the following "shadowing property" of the marks: Zn(ω) = Z0(θTnω)
.
Lastly, there’s a canonical way to construct a stationary point process. Take a measureP on (MK,MK) such that:
P◦ St=P ∀t.
Then we can set:
• (Ω,F ) = (MK,MK), with µ = Id.
• θt= St.
• P =P.
3.5
Invariance Property of the stochastic integral
We assume that the flow θtsatisfies the following additional hypothesis:
Hypothesis 3. θtFs=Fs−t
It can be shown, see e.g. [3], that ifFtis the natural filtration associated to the
Marked Point Process, then this property is satisfied.
Proposition 3.15 (Invariance property of the Stochastic Integral). Let ˜µ be the random compensated measure associated to a Stationary MPP satisfying assumption of subsection 3.1. Then for U ∈ L2
P(ν, R) the following property holds:
Z b a Z K U (s, e)˜µ(ds, de) ◦ θr= Z b+r a+r Z K U (s − r, e) ◦ θrµ(ds, de)˜ (3.13)
Lemma 3.16. Let θt satisfy hypothesis 3. Then for any F -random variable X it
holds
EFt[X ◦ θr] = EFt−r[X] ◦ θr (3.14)
Proof Since θrFt=Ft−r the r.h.s. is Ft-measurable. Take now G ∈Ft.
E [(X ◦ θr) 1G] = E [(X ◦ θr) 1G◦ θ−r◦ θr] = E(X ◦ θr) 1θr(G)◦ θr =
EX1θr(G) = E 1θr(G)E
Ft−rX = E 1
GEFt−r[X] ◦ θr
Proof of Proposition 3.13. Since we assume both U ∈ L2
P(ν, R) and that the Tn
are non explosive, we are in the situation described by remark 3.7. Hence, we can write ˜µ = µ − ν and work with the two measures separately. Note however that the proposition is still valid in the case this splitting of the measure ˜µ is not possible: the proof in this case makes use of the Shadowing Property, see remark 3.14. We proceed in steps:
First step: We prove the property for the integral with respect to the measure µ of any predictable process U such that U ∗ µ is locally integrable. Since the MPP µ is stationary we have, (from the definition of stationarity):
µ(ω, C × L) ◦ θr= µ(ω, (C + r) × L)
which expressed in integral terms reads: Z R Z K 1C(s)1L(e)µ(ω, ds, de) ◦ θr= Z R Z K 1C(s − r)1L(e)µ(ω, ds, de) ∀C × L ∈B(R) ⊗K
Take now any process of the form 1A(ω)1C(s)1L(e). Since the integral with respect
to µ is defined in this case trajectory by trajectory, we can bring 1A(ω) outside the
integral and use the previous identity to obtain: Z R Z K 1A(ω)1C(s)1L(e)µ(ω, ds, de) ◦ θr= Z R Z K
1A(θr(ω))1C(s − r)1L(e)µ(ω, ds, de) ∀A × C × L ∈F ⊗ B(R) ⊗K
where the equality is to be intended ω by ω, and we allow the two members to possibly take both the value +∞. In particular this holds for sets of the form: A × C × L with
A ∈Ft, C = (t, u] and L ∈K. Using then a monotone class argument we can extend
this to all bounded processes measurable with respect to P ⊗K. More precisely: letH be the class of processes satisfying the invariance integral property. Take any increasing sequence of positive processes (fn(ω, s, e)) ∈H such that sup fn is finite.
We now define, for fixed ω the processes: gω
n(s, e) = fn(ω, s, e). This sequence is again
increasing, with sup gn bounded. For each of these functions it is true moreover that:
Z R Z K gθr(ω) n (s, e)µ(θr(ω), ds, de) = Z R Z K gθr(ω) n (s − r, e)µ(ω, ds, de)
By monotone convergence this equality holds also for supngω
n = gω(s, e) = supnfn(ω, s, e).
We can thus extend to all bounded positive predictable processes U . It is then easy to extend the property to all predictable processes U such that the two integrals on both sides are a.s. finite: this class contains in particular processes of the form U 1[a,b]
with U ∈ L2 P(ν, R).
Second Step. We show that the same property holds for the compensator measure ν. We want to exploit the uniqueness of the Dual Predictable projection. In order to do this, let’s consider the following process, adapted to the filtration (Ft+r)t∈R=
( ˆFt)t∈R: Z t 0 Z L µ(ω, ds, de) ◦ θr= Nt+r(L) − Nr(L) := ˆNt(L)
Since the composition with θrdoes not alter the predictability of the processes (in the
sense that the new process is again predictable with respect to the shifted filtration), we have that: Z t 0 Z L ν(ω, ds, de) ◦ θr:= ˆAt(L)
is again a predictable process with respect to the filtration ( ˆFt)t∈R. Moreover:
E ˆ Fsh ˆN t(L) − ˆAt(L) i = EFs+r[(N t(L) − At(L)) ◦ θr] = EFs[Nt(L) − At(L)] ◦ θr = (Ns(L) − As(L)) ◦ θr= ˆNs(L) − ˆAs(L).
Thus ˆAt(L) is the predictable compensator of the process ˆNt(L) with respect to the
filtration ( ˆFt)t∈R. We now show that the same is true for
At+r(L) − Ar(L) = Z r+t r Z L ν(ω, ds, de) Since we assume the process to be stationary,
EFs+rh ˆNt(L) − (At+r− Ar) i = EFs+r[(N t+r(L) − Nr(L)) − (At+r− Ar)] = (Ns+r(L) − Nr(L)) − (As+r− Ar) = ˆNs(L) − (As+r− Ar). Hence Z R Z K 1C(s)1L(e)ν(ω, ds, de) ◦ θr= Z R Z K 1C(s − r)1L(e)ν(ω, ds, de) ∀C × L ∈B(R) ⊗K
From this point onward the rest of the proof is identical to the one for µ, hence we leave the details to the reader.
3.6
Predictable Projection of a Process
We now briefly introduce the projection of a process onto the predictable σ field. This will not be heavily used in the following, hence this section could be omitted at a first reading.
Proposition 3.17. Let X(t, ω) be a measurable process defined on a filtered proba-bility space (Ω, P, F , (Ft)), such that for all t > 0:
Z t
0
|Xs|ds ∈ L1(Ω).
Then there exist an Ft-predictable process XP(t, ω) such that for all bounded Ft
-predictable processes Y (t, ω), E Z t 0 Y (s, ω)X(s, ω)ds = E Z t 0 Y (s, ω)XP(s, ω)ds. (3.15) The process XP is defined up to a predictable set of measure zero with respect to: Leb(R+) × P.
Proof. Define XP as the Radon-Nikodym derivative: XP= X(s, ω)ds × dP|P(Ft)
ds × dP|P(Ft)
Then, by definition of Radon Nykodym derivative, it holds for all sets A ∈P(Ft):
Z Z
1A(s, ω)X(s, ω)(ds × dP(ω)) =
Z Z
1A(s, ω)XP(s, ω)(ds × dP(ω)).
We extend this property to all predictable processes using a monotone class argument. The following properties of the predictable projection are easy consequences of equation (3.15):
• If Y is a predictable process, (XY )P = XPY ds × dP-a.e. • If X ∈ L2
P rog(Ω × R+) then XP ∈ L2P(Ω × R+) and moreover:
inf Z∈L2 P(Ω×R+) E Z ∞ 0 |Xs− Zs|2ds = E Z ∞ 0 |Xs− XsP|2ds (3.16) • X ∈ L∞ P rog(Ω × R+) =⇒ XP∈ L∞P(Ω × R+)
• If X is a stationary process in the sense of definition 8.1 then the same holds for XP.
4
Generalised BSDE
We will present in this section some results concerning generalised BSDEs, following very closely [18]. In the first and second subsections we will be concerned with exis-tence and uniqueness results for BSDEs driven by Brownian motion and a Poisson random measure. Equation of this type have been shown to provide probabilistic for-mulas for solution of systems of semi-linear PDEs of second order, where the linear second order operator may differ from one equation to the other. The finite horizon BSDE are linked to parabolic-type PDEs, while infinite horizon BSDEs provide solu-tions to elliptic PDEs problems. The results of these subsecsolu-tions will not be directly used in what follows, (hence the reader can safely skip them if he needs to go quickly to the core of this work). Nevertheless, it will be helpful to see how the material of the next chapters fits in the general theory.
In the last part of this section we will prove an existence and uniqueness result that will be used later on.
Remind finally that adopt the natural filtration generated by the two processes: Ft=Ftµ,W = ¯F
µ t ∨ ¯FtW
4.1
Finite horizon case
For this and the following subsection only we will assume that:
Hypothesis 4. µ is a homogeneous Poisson random measure on R+ × K, where
K = Rl\ {0}. This means in particular that the compensator ν is deterministic and
has the form:
ν(ω, dt, de) = dt × m(de).
We will further suppose that m satisfies the following regularity condition: Z
K
(1 ∧ |e|2)m(de) < ∞
We consider in this section the finite horizon Backward SDE: Yt= ξ + Z T t f (s, Ys, Zs, Us)ds − Z T t ZsdWs− Z T t Z K Us(e)˜µ(ds, de) (4.1) where, setting Λ2:= L2(K,K , m; Rk): f : Ω × R+× Rk× Rk×d× Λ2→ Rk
The natural conditions under which the above problem is well posed are a combination of monotonicity with respect to the first component of the solution and a Lipschitz condition with respect to the second component. More precisely, let the following hypothesis on the generator f and the final data ξ hold true:
Hypothesis 5. There exist some constant α ∈ R, β < 0, L > 0 and some R+ valued
adapted process {φt, 0 ≤ t ≤ T }, such that, for all (t, y, z, u) ∈ R+× Rk× Rk×d× Λ2
i. f (·, y, z, u) is progressively measurable.
ii. y 7→ f (t, y, z, u) is continuous for all z, u, (t, ω). iii. ER∞
0 φ 2
sds < ∞
v. hy − y0, f (t, y, z, u) − f (t, y0, z, u)i ≤ α|y − y0|2
vi. |f (t, y, z, u) − f (t, y, z0, u0)| ≤ L(kz − z0k + ku − u0k Λ2)
vii. E|ξ|2< ∞
In particular, note that α can be any real number: hence the case of lipschitz condition with respect to the first component of the solution is trivially comprised in this setting.
Under these hypothesis the following result hold ([18]):
Theorem 4.1. If the conditions of hypothesis 5 are satisfied, there exists a unique process {(Yt, Zt, Ut), 0 ≤ t ≤ T with values in Rk× Rk×d× Λ2, satisfying:
E sup 0≤t≤T |Yt|2+ Z T 0 kZtk2+ kUtk2Λ2 dt ! < ∞
and solves equation (4.1).
4.2
Infinite horizon case
Our next aim is to extend the results of the previous section to the infinite horizon case. Before proceeding, let’s clarify what it means to solve a BSDE on an infinite horizon.
Take a (not necessarily finite) stopping time τ . We now cast equation (4.1) in a new form: (i). Yt= YT + Z T ∧τ t∧τ f (s, Ys, Zs, Us)ds − Z T ∧τ t∧τ ZsdWs− Z T ∧τ t∧τ Z K Us(e)˜µ(ds, de)
(ii). Yt= ξ on the set {t ≥ τ }. (4.2)
In the case τ (ω) = ∞ the second condition disappears. We will assume this time that ξ is anFτ-measurable random vector.
Remark 4.2. Thanks to the Martingale representation theorem 3.13, if ξ is a Fτ
-measurable k-dimensional random vector, there exist two progressively -measurable processes, ηt and ρt, with values in Rk×d and Λ2respectively, such that
ξ = Eξ + Z τ 0 ηtdWt+ Z τ 0 Z K ρ(t, e)˜µ(ds, de) and E Z τ 0 kηtk2+ kρtk2Λ2 dt < ∞
To treat this new problem, we strengthen some of the hypothesis of the previous section. For some λ > 2α + L2:
(iii’)
E Z ∞
0
eλt φ2t+ |f (t, ξt, ηt, ρt)|2 dt < ∞
where ξt= EFtξ, and ηt, ρt are the two processes of the previous remark.
(vii’) E 1 + eλτ ξ < ∞.
Theorem 4.3. Let ξ be an Fτ-measurable k-dimensional random vector, and let
hypothesis 5 hold, with (iii’) and (vii’) in place of (iii) and (vii). Then there exist a unique progressively measurable process {(Ys, Zs, Us); 0 ≤ t ≤ τ } with values in
Rk× Rk×d× Λ2, such that for some λ > 2α + L2, E sup 0≤t≤τ eλt|Yt|2+ Z τ 0 eλt |Yt|2+ kZtk2+ kUtk2Λ2 dt < ∞ and (Y, Z, U ) solves (4.2)
Proof. See [18].
4.3
A little existence results on BSDE driven by MPP
In this section we will present a very little result on BSDE driven by a marked point process and brownian motion. This is a very little extension to some of the results presented in the aforementioned paper by Pardoux [18].
Let then go back to our setting, letting µ be a generic marked point process satis-fying the assumption of subsection 3.1. We want to prove an existence and uniqueness result for the following BSDE:
dYt= f (t, Yt, Zt)dt − ZtdWt−
Z
K
Ut(e)˜µ(dt, de)
YT = ξ (4.3)
We make the following assumption on the generator and on the final data: Hypothesis 6. There exist some constant C and R+valued adapted process {φt, 0 ≤
t ≤ T }, such that, or all (t, y, z) ∈ R+× Rn× Rn×d
i. f (·, y, z) is progressively measurable. ii. ER0∞φ
2
sds < ∞
iii. |f (t, y, z, u)| ≤ φt+ L(|y| + |z|)
iv. |f (t, y, z) − f (t, y0, z0)| ≤ L(|z − z0| + |y0− y|)
v. E|ξ|2< ∞
Theorem 4.4. Under hypothesis 6 there exist a unique solution (Y, Z; U ) to the BSDE (4.3) which moreover satisfies:
E " sup 0≤t≤T |Yt|2+ Z T 0 |Zt|2dt + Z T 0 Z K |Ut|2ν(dt, de) # < ∞ Proof. Define B= L2P rog(Ω × [0, T ]; Rn) × L2P(Ω × [0, T ]; Rn×d) × L2P(ν, [0, T ])
We now define a map Φ : B → B as follows. We set (Y, Z, U ) = Φ(X, V, Q), where (Y, Z, U ) is a triple satisfying:
Yt= ξ + Z T t f (s, Xs, Vs)ds − Z T t ZsdWs− Z K Z T t Us(e)˜µ(ds, de) (4.4)
More precisely, this triple is constructed as follows. Define the square integrable mar-tingale: Mt = EFt
h
ξ +R0Tf (s, Xs, Vs)ds
i
theorem (see 3.13), we can find Z ∈ L2P(Ω × [0, T ]; Rn×d) and U ∈ L2P(ν, [0, T ]) such that: Mt= M0+ Z t 0 ZsdWs+ Z K Z t 0 U (s, e)˜µ(ds, de) (4.5) Next, define: Yt= EFt " ξ + Z T t f (s, Xs, Vs) # = Mt− Z t 0 f (s, Xs, Vs)
Now using the representation (4.5) of M in the previous relation we get that Y satisfies (4.4).
Note moreover that thanks to Burkholder-Davis-Gundy’s Inequality we have that: E sup
0≤t≤T
|Yt|2dt < ∞
So that in particular, Y ∈ L2P rog(Ω × [0, T ]; Rn) and the map Φ is formB toB. Let (X, V, Q), (X0, V0, Q0) ∈B, (Y, Z, U ) = Φ(X, V, Q), (Y0, Z0, U0) = Φ(X0, V0, Q0). We use an upper bar to indicate the difference processes, e.g. ¯X = X0− X. Using our Lipschitz assumption, Ito’s Formula and Young’s Inequality, we get, for any real number γ: E| ¯Y0|2+ E Z T 0 eγs γ| ¯Ys|2+ | ¯Zs|2 ds + E Z T 0 eγs Z K | ¯Us(e)|2ν(ds, de) ≤ 4C2 E Z T 0 eγs| ¯Ys|2ds + 1 2E Z T 0 eγs| ¯Xs|2+ | ¯Vs|2ds Choosing γ = 1 + 4C2we obtain E Z T 0 eγs | ¯Ys|2+ | ¯Zs|2 ds + E Z T 0 eγs Z K | ¯Us(e)|2ν(ds, de) ≤ 1 2E Z T 0 eγs| ¯Xs|2+ | ¯Vs|2ds
This shows that Φ is a strict contraction on the Hilbert spaceB with the norm k(Y, Z, U )k2 γ= E Z T 0 eγs |Ys|2+ |Zs|2 ds + E Z T 0 eγs Z K |Us(e)|2ν(ds, de)
5
Optimal LQ Control of Switching Systems
For the rest of the work the following set up will be used: there shall be given a brownian motion W and a MPP µ on a probability space (Ω, P, F ). eventually, the following filtration will be taken:
Ft= ¯FtW∨ ¯F µ t.
5.1
Problem Statement
In this section we will consider the following minimum problem: inf u∈L2 P rog(Ω×[0,T ];Rk) J (0, x, u). (5.1) where J (t, x, u) = EFt " Z T t (|pSsXs|2+ |us|2)ds + hGXT, XTi # (5.2) and X is the state variable, governed by the following equation:
dXt= (AtXtdt + Btut)dt + CtXtdWt
X0= x
(5.3) Form now on we will work under the following assumptions:
Hypothesis 7.
(A1) (i) A ∈ L∞P rog(Ω × [0, T ]; R
n×n). We denote by M
A a positive constant such
that
|A(t, ω)| ≤ MA, P-a.s. and for a.e. t ∈ (0, T ). (ii) B ∈ L∞P rog(Ω × [0, T ]; Rn×k). We denote by M
B a positive constant such
that
|B(t, ω)| ≤ MB, P-a.s. and for a.e. t ∈ (0, T ). (iii) C is of the form: C =Pd
j=1C jh·, e
ji where {ej : j = 1, ...d} is the canonical
base of Rd. We suppose that
Cj ∈ L∞
P rog(Ω × [0, T ]; R n×n),
and we set MC= maxj:1,...,d|Cj|L∞ P rog
(A2) G ∈ L∞(Ω,FT, P;Sn+), with MG= |G|L∞.
(A3) S ∈ L∞P rog(Ω × [0, T ],S +
n), with MS = |S|L∞.
Before proceeding, we briefly recall the following classical result from the theory of SDE, that will be frequently used in the following chapters:
Theorem 5.1. Let A, C ∈ L∞P rog(Ω × [0, T ]; Rn×n) and B ∈ L∞
P rog(Ω × [0, T ]; R n×k).
Then for any p ≥ 1, given any x ∈ Rn and progressive control u with
E Z T 0 |us|2ds !p/2 < ∞,
problem (5.3) has a unique solution X ∈ LpP(Ω; [0, T ]; Rn)) and
EFs sup t∈[0,T ] |Xt|p≤ Cp |x|p + EFs Z T s |ur|2dr !p/2 (5.4)
We will link the solution to the optimum problem (5.1) to the solution of the fol-lowing BSDE, called Riccati BSDE or Backward Stochastic Riccati Equation (BSRE)
−dPt= (A0tPt+ PtAt+ Ct0PtCt+ Ct0Qt+ QtCt− Pt0BtBt0Pt+ St) dt −QtdWt− Z K Utµ(ds, dx)˜ PT = G (5.5)
Unfortunately we cannot prove, using the results from standard theory of BSDE with jumps, that the above equation has a solution. The presence of a quadratic makes our equation not satisfy the monotonicity or lipshitz condition required by the theory. We will thus devote the main part of this chapter to establish existence and uniqueness for this equation. All the results will be taken from the paper [8].
5.2
The Lyapunov Equation
We will start our analysis of the Riccati equation by studying its linear part, which give rise to a so called Lyapunov equation:
−dPt= (A0tPt+ PtAt+ Ct0PtCt+ Ct0Qt+ QtCt+ Lt) dt −QtdWt− Z K Utµ(ds, dx)˜ PT = G (5.6) where L ∈ L2
P rog(Ω × [0, T ];Sn+) and G ∈ L2(Ω,FT, P;Sn+). Thanks to Theorem 4.4,
we know that there exist a unique solution (P, Q, U ) to this equation, which belongs to the space:
L2P rog(Ω × [0, T ]; Rn×n) × L2P(Ω × [0, T ]; Rn×k×d) × L2P(ν, [0, T ]; Rn×n). With a little more effort, it could be possible also to give a stability estimate: the argument can be easily adapted from the one used by Pardoux in [18].
Proposition 5.2. Assume Hypotheses 7. Moreover, assume that L ∈ L2
P rog(Ω ×
[0, T ];S+
n). Then problem (5.6) has a unique solution (P, Q, U ) such that:
E sup t∈[0,T ] |Pt|2+ E Z T 0 |Qs|2ds + E Z T 0 Z K |Us(x)|2ν(ds, dx) ≤ CE " |G|2+Z T 0 |Ls|2ds # (5.7) Thanks to this existence result, we can proceed proving the following result, a key step in order to get the fundamental relation (5.5):
Theorem 5.3. Assume Hypotheses 7. Moreover, assume that L ∈ L∞P rog(Ω×[0, T ],S+ n).
Let (P, Q, U ) be the unique solution to (5.6) and let Xt,x,u be the solution to (5.3). Then for all t ∈ [0, T ], x ∈ Rn, u ∈ L2P(Ω × [0, T ]; Rk) it holds, P-a.s., that
hPtx, xi = EFthPTX t,x,u T , X t,x,u T i + EFt Z T t [hLrXrt,x,u, X t,x,u r i − 2hPrBrur, Xrt,x,ui]dr (5.8) Moreover, for all t ∈ [0, T ],
|Pt| ≤ C2 h |G|L∞(Ω,F T,P;S+n)+ (T − t)|Ls|L∞(Ω×[0,T ],Sn+) i P − a.s (5.9) where C2 is the positive constant in (5.4).
5.3
The Riccati Equation
We will now move to the existence and uniqueness for the Riccati Equation. Due to the non linear term, we cannot prove immediately the existence of a solution on the whole [0, T ]. We will proceed first showing existence on a small time interval and then showing that this solution can be extended step by step to cover the whole interval of interest. We will thus start showing:
Proposition 5.4. (local existence). Under Hypotheses 7 there exists a δ ∈]0, T ] such that problem (5.5) has a unique solution in the interval [T −δ, T ], where δ only depends on the constants
Proof. See the paper [8].
Applying again the theorem recursively, the quantity δ will in general depend only on T, MA, MB, MC, MS and on the maximal value that P has reached in the previous
subinterval. Hence, if we can bound the solution from above, we will be able to fix an inferior bound for δ greater than zero, and thus extend the solution on the whole [0, T ] by applying the precedent theorem a finite number of times.
In order to find a bound for the solution P , we exploit the connection of the Riccati equation with the optimal control problem.
Proposition 5.5. . Assume hypothesis 7 and let (P, Q, U ) be the solution of (5.5) in an interval [T0, T ]. Then, for all t ≥ T0, x ∈ Rn, u ∈ L2P rog(Ω × [t, T ]; U ) it holds
hPtx, xi = J (t, x, u) + EFt
Z T
t
|us+ B0PsXst,x,u|
2 (5.10)
Proof. (P, Q, U ) is the solution of the Lyapunov equation 5.6 with L = S − P BB0P . Hence by (5.8) hPtx, xi = EFthGXTt,x,u, X t,x,u T i + EFt Z T t |pSsXst,x,u| 2ds − EFt Z T t hPsBsBs0PsXst,x,u, X t,x,u s i − EFt Z T t hPsBsus, Xst,x,uids (5.11)
Adding and subtracting 12EFtRT
t |us|
2ds we obtain the thesis.
Thanks to this characterization, it’s now easy to find the aforementioned bound for P . Further, we will be able to show that the solution is always nonnegative. Proposition 5.6. (positivity and a priori estimate). Let (P, Q, U ) be the solution to (5.5) in [T0, T ]; then
1. for every t ∈ [T0, T ] and x ∈ Rn, hPtx, xi ≥ 0 P-a.s.;
2. for every t ∈ [T0, T ], |Pt| ≤ C2[MG + T MS] P-a.s. where C2 is the constant
defined in Theorem 5.1.
Proof. The (5.10) with u = 0, give for all x ∈ Rnwith |x| ≥ 1 and for all t ∈ [T 0, T ] hPtx, xi = EFthGX t,x,0 T , X t,x,0 T i + EFt Z T t hpSsXst,x,0| 2ds ≤ |G|L∞(Ω,F,P)EFt|XTt,x,0|2+ Z T t EFt|Ss|2|Xst,x,0| 2ds
and by (5.4) hPtx, xi ≤ C2 h |G|L∞(Ω,F,P)+ T |Ss| L∞(Ω×[0,T ],S+ n)) i P − a.s.∀x : |x| ≤ 1. (5.12) Then consider the following closed loop equation, starting at a certain instant t ≥ T0
with an arbitrary initial data x ∈ Rn
d ¯Xr = [A ¯Xr− BrB 0 rPrX¯r]dr + CrX¯rdWr ¯ Xt = x. (5.13) The assumptions of Theorem 5.1 still hold. So there exists a unique solution ¯X ∈ LpP(Ω, C([t, T ]; Rn)) for every p ≥ 2. Applying then the fundamental relation (5.10)
to the control ¯u = −B0P ¯X and to ¯Xt,x,¯u= ¯X we get
hPtx, xi = EFthG ¯XT, ¯XTi + EFt
Z T
t
|pSrX¯r|2+ |B0rPrX¯r|2dr; (5.14)
This implies that hPtx, xi ≥ 0, P-a.s. for all x ∈ Rn and this together with (5.124)
gives the claim.
It is now possible to extend the solution to the whole interval [0, T ], proving global existence.
Theorem 5.7. Assume (7). Problem (5.5) has a unique solution (P, Q, U ) with the following regularity: P ∈ L∞P rog(Ω × [0, T ];S+
n), Q ∈ L2P(Ω × [0, T ]; ) and U ∈
L2
P(ν, [0, T ]).
Proof. We refer again to [8]
Now that we have proven the Riccati Equation to admit a unique solution on the whole [0,T], it is straightforward to extend Proposition 5.5 to the whole interval and get the synthesis of the optimal control.
Theorem 5.8. Fix T > 0 and x ∈ Rn. Then:
1. There exists a unique optimal control. That is a unique control ¯u ∈ L2(Ω ×
[0, T ]; Rk) such that
J (0, x, ¯u) = inf
u∈L2
P(Ω×[0,T ];Rk)
J (0, x, u).
2. If ¯X is the solution of the state equation corresponding to ¯u (that is the optimal state), then ¯X satisfies the closed loop equation
d ¯Xs = [AsX − B¯ sB 0 sPsX¯s]ds + CsX¯sdWs, ¯ X0 = x. (5.15)
3. The following feedback law holds P-a.s. for almost every s ∈ [0, T ]: ¯
us= −B
0
sPsX¯s. (5.16)
4. Moreover, the following Foundamental Relation holds P-a.s. : J (t, x, u) = hPtX¯t,x,u, ¯Xt,x,ui + EFt Z T t |us+ B 0 PsX¯t,x,u|2ds. (5.17)
6
Infinite Horizon LQ Control of Switching Systems
In this section we will extend the results of the previous one to the infinite horizon case. Namely, we will study the infinite horizon control problem of finding:inf u∈L2 P rog(Ω×R+;Rk) J (0, x, u) with J (0, x, u) = E Z ∞ 0 |pStXt|2+ |us|2ds .
In the rest of this chapter we will adopt techniques which have been already used in the work of Guatteri and Tessitore [9] in an infinite dimensional brownian setting.
6.1
Infinite horizon Riccati Equation: Existence of a Solution
In this and the following sections we will strengthen our hypothesis, namely we will assume that:
Hypothesis 8.
(A1) (i) A ∈ L∞P rog(Ω × R+; Rn×n). We denote by MAa positive constant such that
|A(t, ω)| ≤ MA, P-a.s. and for a.e. t ∈ R+
(ii) B ∈ L∞P rog(Ω × R+; Rn×k). We denote by MB a positive constant such that
|B(t, ω)| ≤ MB, P-a.s. and for a.e. t ∈ R+
(iii) C is of the form: C = Pd j=1C
j < ·, e
j > where {ej : j = 1, ...d} is the
canonical base of Rd. We suppose that
Cj∈ L∞P rog(Ω × R+; Rn×n),
and we set MC= maxj:1,...,d|Cj|L∞ P rog
(A2) S ∈ L∞P rog(Ω × R+,Sn+), with MS= |S|L∞ P rog.
Definition 6.1. We say that (A, B, C) is stabilizable relatively to the observations √
S (or√S-stabilizable) if there exists a control u ∈ L2
P([0, +∞) × Ω; Rk) such that
for all t ≥ 0 and all x ∈ Rn
EFt Z +∞ t [SsXst,x,u, X t,x,u s + |us|2]ds < Mt,x. (6.1)
for some positive constant Mt,x.
Hypothesis 9. We assume that (A, B, C) is√S-stabilizable. Definition 6.2. Given a triple (P, Q, U ) with
P ∈ L∞P rog,Loc(Ω × R+,Sn+), Q ∈ L 2
P,Loc(Ω × R+; (Sn)d), U ∈ L2P,Loc(ν, R+)
we say that it solves equation 5.5 on R+ if for every t ≤ T it holds P a.s.:
Pt= PT + RT t Ss+ A 0 sPs+ PsAsds + RT t C 0 sPsCs+ Cs0Qs+ QsCsds + RtTQsdWs+ RT t R KUs(x)˜µ(ds, dx) − RT t PsBsB 0 sPsds. (6.2)
Theorem 6.1 (Existence). Assume Hypothesis 8 and 9, then there exist a solution (P, Q, U ) of the Riccati Equation in the sense of definition 6.2.
Proof. We first construct the P component of the solution by a limit argument. For every positive integer N > 0 let PN the generalized solution of the finite horizon Riccati equation (5.5) with final data PN = 0. We extend each PN on R+ setting
PtN = 0 for t > N .
Note that for a fixed t the sequence PtN is increasing: indeed by Theorem 5.8:
hPtN +1x, xi = inf u∈L2 P rog([t,N +1]×Ω;Rk) EFt Z N +1 t (|√SrXrt,x,u| 2+ |u r|2) dr ≥ inf u∈L2 P rog([t,N ]×Ω;Rk) EFt Z N t (| √ SrXrt,x,u| 2 + |ur|2) dr = hPtNx, xi.
Since Rn is separable, we can select a dense countable subset (say Qn ) such that almost everywhere the above relation holds ∀x ∈ Qn. Thanks then to the density of
Qn in Rn and the continuity of the scalar product we have that: PhPtN +1x, xi ≥ hP
N
t x, xi ∀N ∈ N, ∀x ∈ R
n = 1. (6.3)
Indicating with ¯u the stabilizing control (which exists thanks to the Hypothesis), thanks to Theorem 5.8 we have that:
hPN t x, xi = | q PN t x| 2 ≤ EFt Z N t (|√SrXrt,x,¯u| 2+ |¯u r|2) dr ≤ EFt Z +∞ t (| √ SrXrt,x,¯u| 2+ |¯u r|2) dr ≤ Mt,x, P − a.s.; for a suitable constant Mt,x. We can see
√
PN as a linear (and continuous) operator
from Rn to L∞
P rog(Ω,Ft, P, Rn). Thus by the previous inequality, taking a base in Rn,
it is easy to find a constant Mt independent of x such that:
| q
PN
t |L(Rn;L∞
P rog(Ω,Ft;Rn))≤ Mt.
(take for instance the 1-norm in Rn and then set M
t= maxi[Mt,ei] with (ei)i=1,...,n
the canonical base in Rn
. Again, thanks to the separability of Rn this means that:
PhPtNx, xi ≤ Mt|x|2, ∀N ∈ N, ∀x ∈ Rn = 1 (6.4)
or, equivalently,
P|PtN| ≤ Mt, ∀N ∈ N = 1. (6.5)
For fixed x, y, t the sequence hPN
t x, yi is thus P-a.s. increasing and bounded, and we
can set: φ(t, x, y) = lim N →∞hP N t x, yi =1 2 h lim N →∞hP N t x + y, x + yi − lim N →∞hP N t x, xi − lim N →∞hP N t y, yi i
the limit holding P a.s. For every t, φ(t, x, y) is bilinear in x, y, so there exists a matrix ¯
Ptsuch that:
φ(t, x, y) = h ¯Ptx, yi ∀x, y ∈ Rn.
¯
P is moreover symmetric and non-negative, thanks to the properties of the PN.
We claim that for each T ≥ 0 there exists a positive constant CT such that:
Indeed, if we fix N > T , by Theorem 5.8 we have: hPN t x, xi ≤ EFthP N T X t,x,0 T , X t,x,0 T i + EFt Z T t |√SrXrt,x,0| 2dr (6.6)
where Xt,x,0 is the solution to the state equation corresponding to a control u ≡ 0.
Thanks to the standard estimate 5.4 we know that there exist a constant KT such
that:
hPN
t x, xi ≤ KT(MT + T · MS)|x|2:= CT|x|2, P − a.s.. Using again the separability of Rn we have:
PhPtNx, xi ≤ CT|x|2, ∀N ∈ N, ∀x ∈ Rn = 1 (6.7)
and by construction
Ph ¯Ptx, xi ≤ CT|x|2, ∀x ∈ Rn = 1. (6.8)
Now choose 0 ≤ t ≤ T ≤ N and write the foundmental relation from Theorem 5.8: hPN t x, xi = EFthP N T X t,x,u T , X t,x,u T i +EFt Z T t (| √ SsXst,x,u|2+ |us|2) ds −EFt Z T t |us+ B0PsNX t,x,u s | 2ds, P − a.s.. (6.9)
At this point, thanks to the monotone convergence of PN → ¯P and to the bound 6.8 we can pass to the limit in the previous relation by dominated convergence and get:
h ¯Ptx, xi = EFth ¯PTX t,x,u T , X t,x,u T i +EFt Z T t (|√SsXst,x,u| 2+ |u s|2) ds −EFt Z T t |us+ B0P X¯ st,x,u| 2ds, P − a.s.. (6.10)
Thus having defined:
b JT(t, x, u) := EFt " Z T t |√SsXst,x,u| 2+ |u s|2 ds + h ¯PTXTt,x,u, XTt,x,ui # . Relation (6.10) yields (choosing u = −B0P y)¯
h ¯Ptx, xi = inf u∈L2
P([t,T ]×Ω;Rk)
b
JT(t, x, u).
Consequently, by Theorem 5.8, ¯Pt, for 0 ≤ t ≤ T must be the first component of the
solution to the finite horizon BSRDE with final datum ¯PT. Theorem 5.7 gives us the
existence, for a fixed final time T , of the other two component of the solution as well, (QT, UT).
Take now two distinct final times T < S. We know, from the previous steps, that ( ¯P , QS, US) and ( ¯P , QT, UT) are solution to the BSRDE on the intervals [0, S] and [0, T ] respectively, and with respective final data ¯Ps and ¯PT. Thanks to the
uniqueness of the solution to the BSRE it is easily seen that (QS, US)|[0,T ]= (QT, UT)
in L2
P(Ω × R+; (Sn)d) × L2P(ν, R+). We can then construct two processes ( ¯Q, ¯U ) on
R+such that on each finite interval [0, T ], ( ¯Q, ¯U )|[0,T ]= (QT, UT) without ambiguity.
The triple ( ¯P , ¯Q, ¯U ) is then a solution to the BSRE in the sense of definition 6.2. In general the solution is not unique. Nevertheless, we can prove that the con-structive procedure of the previous proof furnishes us a very special one: indeed it is the minimal solution to the Riccati Equation.
Corollary 6.2. ¯P is the minimal solution of equation (5.5) on the infinite horizon, in the sense that if there exists another solution P\
on R+ of (5.5), then P \ t ≥
¯
Pt P − a.s. for every t ≥ 0.
Proof. By the Fundamental Relation in Theorem (5.8) for N > t we have that: hP\(t)x, xi = inf u∈L2 P rog([t,N ]×Ω);Rk) [EFthP\ NX t,x,u N , X t,x,u N i + EFt Z N t (|√SrXrt,x,u| 2+ |u r|2) dr] ≥ inf u∈L2 P rog([t,N ]×Ω);Rk) EFt Z N t (| √ SrXrt,x,u|2+ |ur|2) dr = hPN(t)x, xi,
thus hP\(t)x, xi ≥ hPN(t)x, xi, P-a.s. for every x ∈ Rn, since the PN converges a.s. to ¯P .
6.2
Synthesis of the optimal control
We ’re now in the position to characterize the optimal control for the infinite horizon problem. Note that having just proved the existence of a solution to the Riccati equation on R+ is not enough: the characterization that will be given shortly is valid
only for the minimal solution ¯P . Besides, since a Fundamental relation (similar to the one given in Theorem 5.8), would hold on a finite horizon for every solution P , the following characterization of ¯P would allow us to characterize it also by its different behaviour at infinity with respect to the other solutions.
Theorem 6.3. Assume Hypothesis 8, 9. Fix x ∈ Rn, then the following holds: 1. there exists a unique control ¯u ∈ L2
P rog(Ω × [0, +∞); Rk) such that:
J∞(0, x, ¯u) = inf u∈L2
P rog(Ω×[0,+∞);Rk)
J∞(0, x, u);
2. if ¯X is the solution of the state equation corresponding to ¯u (that is the optimal state), then ¯X is the unique solution to the closed loop equation:
d ¯Xr= [A ¯Xr− BrBr0P¯rX¯r] dr + C ¯XrdWr, r > t > 0
¯ Xt= x
(6.11) and the following feedback law holds P-a.s. for almost every s:
¯
us= −Bs0P¯sX¯s; (6.12)
3. the optimal cost is given by ¯J∞(t, x, ¯u) = h ¯Ptx, xi.
Proof. Thanks to the Fundamental Relation in Theorem 5.8, hPN t x, xi ≤ Z N t |pSrX¯rt,x,u| 2+ |u r|2dr ≤ J∞(t, x, u) for every u ∈ L2
P rog Ω × [t, +∞] , Rk letting N → +∞ we get:
h ¯Ptx, xi ≤ J∞(t, x, u) and so h ¯Ptx, xi ≤ inf u∈L2 P rog(Ω×[t,+∞],Rn) J∞(t, x, u)
On the other hand, applying Ito’s Formula to the process h ¯PtX¯tt,x,u, ¯X t,x,u t i, and setting us= −Bs0PsXswe get: h ¯Ptx, xi = EFth ¯PTX¯ t,x,u T , ¯X t,x,u T i + EFt Z T t |√S ¯Xt,x,u|2+ |u|2ds ≥ EFt Z T t |√S ¯Xt,x,u|2+ |u|2ds
Let now T tend to infinity and get, by monotone convergence: h ¯Ptx, xi ≥ EFt
Z +∞
t
|√S ¯Xt,x,u|2+ |u|2ds = J
∞(t, x, u) (6.13)
7
The Affine Case on finite and infinite horizon
Our next aim would be to further extend the results of the previous chapter to the ergodic case. However, it turns out that the ergodic functional (which will be intro-duced in the last chapter) has a trivial behaviour when the state dynamics are the one in (5.3). In fact, the ergodic functional would be null, independently of the cho-sen control. The ergodic functional becomes interesting when the state dynamics are added a forcing term f , which makes the state SDE an affine equation. Hence, for studying the ergodic case a study of the affine control problem is in order.In the rest of this Section we will closely follow the paper of Guatteri and Masiero [10], extending some of their results to our generalized framework.
7.1
Dual equation
We thus turn to the finite horizon control problem related to the following controlled affine equation: dXs= (AsXs+ Bsus)ds + d X i=1 Ci sXsdWsi+ fsds s ∈ [t, T ] Xt= x (7.1)
The cost functional to be minimized is the same as the one in equation (5.2), where we suppose in this section to take the final cost G = 0. We make the following additional assumption on the process f :
Hypothesis 10. f : [0, T ] × Ω → Rn ∈ L∞
P rog([0, T ] × Ω).
To deal with this problem we introduce the so-called dual equation: drt= −Ht0rtdt − ¯Ptftdt − d X i=1 Cti0 gitdt + d X i=1 gi tdWti+ R KVs(e) ˜µ(ds, de) rT = ξ. t ∈ [0, T ] (7.2) where: Ht= At− BtB0tP¯t (7.3)
This is an affine BSDE driven by a brownian motion and a MPP. We note that due to the hypothesis and the previous results on ¯P , its coefficients are all uniformly bounded in [0, T ]. Thanks to Theorem 4.4 we know that the following Proposition holds: Proposition 7.1. Assume Hypothesys 7 and ξ ∈ L2(Ω; Rn). Then equation (7.2)
admits a unique solution
(r, g, V ) ∈ L2P rog(Ω × [0, T ]; Rn) × L2P(Ω × [0, T ]; Rn×d) × L2P(ν, [0, T ]; Rn×n) We go on exploiting a duality argument to obtain more information on the rt
process. Let Xt,x,ηand r
tbe the solutions to the following stochastic equations:
dXt,x,η s = HsXst,x,ηds + d X i=1 CsiXst,x,ηdWsi+ ηsds, s ∈ [t, T ] , Xtt,x,η= x, (7.4)
where x ∈ L2(Ω,Ft) and η ∈ L2P rog(Ω × [0, T ], R n),
Proposition 7.2. Let Hypothesis 7 and 10 hold and assume moreover that ξ ∈ L∞(Ω; Rn). Then rt ∈ L∞P rog([0, T ] × Ω; R
n). Moreover, the following duality
rela-tion holds: EFtξ, XTt,x,η − hrt, xi = −EFt Z T t ¯ Psfs, Xst,x,η ds + EFt Z T t hηs, rsi ds. (7.5) Proof. Let η ∈ L4
P rog(Ω × [0, T ]; Rn). This implies, thanks to Theorem 7.29 X ∈
L4P rog(Ω × [0, T ]; Rn).
Take a function Ψ ∈ C2(Rn) with Ψ(y) = 1 for |y| ≤ 1, Ψ(y) = 0 for |y| ≥ 2 and Ψ(y) ∈ [0, 1], ∀y ∈ Rn. Note that, with this choice, Ψ(N−1y) = 0, Ψ0(N−1y) = 0 if |y| > 2N and Ψ(Xs/N ) converges to 1, P-a.s.
Then, applying the Ito Formula to the product hXtt,x,η, rtiΨ(XNs) we get (we suppress
for a moment the upperscript): d[hXt, rtiΨ( Xs N )] = N −1F N(s)ds + GN(s)dW s + Ψ(Xs N) R KhVs(e), Xsi ˜µ(ds, de) − Ψ(Xs N)[h ¯Ptft, Xti − hηt, rti]ds, (7.6) where: FN(s) = hΨ0(XNs), [HsXs+ ηs]ihXt, rti +2N1 P d i=1hΨ 00 (Xs N )C i sXs, CsiXsihXs, rsi +Pd i=1hΨ0( Xs N ), C i sXsihCsiXs, rsi +Pd i=1hΨ0( Xs N ), C i sXsihgti, Xsi and for i = 1, ...d GiN(s) = 1 NhΨ 0(Xs N ), C i sXsihXs, rsi + Ψ( Xs N ) hC i sXs, rsi + hgis, Xsi
As it can be easily verified, ERtT|FN(s)|ds ≤ k for all N ∈ N, where k is a
constant independent of N (this is precisely where we need to use the fact that X ∈ L4P rog(Ω × [0, T ]; Rn)). Moreover d X i=1 E Z T t |Gi N(s)|2ds ≤ cN2 MC2E Z T t |rs|2ds + E Z T t |gs|2ds ! < ∞,
with c a constant independent of N , and
E Z T t Z K |Ψ(Xs N )hVs(e)Xs, Xsi| 2ν(ds, de) ≤ N4 E Z T t Z K |Vs(e)|2ν(ds, de) < ∞.
Thus first integrating, taking conditional expectation, and then letting N → ∞ we get by dominated convergence:
EFtξ, Xt,x,η T − hrt, xi = −EFt Z T t ¯ Psfs, Xst,x,η ds + EFt Z T t hηs, rsi ds. (7.7)
Taking η = 0 and thanks to Theorem 5.4 (applied with p = 1) we get:
