A Vector Girsanov Result and Its Applications to Conditional Measures Via the Birkhoff Integrability

In memory of Domenico Candeloro who is for us, Master, Mentor and Friend, † May 3◦ 2019

A VECTOR GIRSANOV RESULT AND ITS APPLICATIONS TO CONDITIONAL MEASURES VIA THE BIRKHOFF INTEGRABILITY.

DOMENICO CANDELORO, ANNA RITA SAMBUCINI, AND LUCA TRASTULLI

Abstract. Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and viceversa) are investigated in order to establish abstract versions of classical theorems of Probability and Stochastic Processes. In particular the Girsanov Theorem is extended and used with the treated methods.

1. Introduction The theory of stochastic processes plays a very important role in modelling various phenomena, in a large class of disciplines such as physics, economics, statistics, finance, biology and chemistry. Then it is crucial to extend as much as possible the tools and results regarding this theory, in order to make them available even in abstract and general contexts. A very important tool in Measure Theory is the Girsanov Theorem, strictly linked to the well-known Wiener stochastic process called the standard Brownian motion (wt)t∈[0,∞), defined on a probability space (Ω, A, P). A classic formulation of this result in the real case can be found in [31] and it allows to change the underlying probability measure P, through the definition of Radon-Nikodým derivative, in order to obtain an equivalent measure Q. This turns out to be useful, for example, in mathematical finance when a neutral risk measure must be determined, in the Black-Scholes model. Here we generalize the Girsanov Theorem to the case of vector measure

arXiv:1811.04597v4 [math.FA] 17 Feb 2020 spaces following the idea formulated in [31] for the real Brownian motion and using the Birkhoﬀ vector integral studied in [12,14,16,19,20,22–25,30,34–38], in [17] for non additive-measures and in [3–9, 13, 18, 21] for the multivalued integration. Other results on the Brownian motion subject are given also in [10, 26, 29]. This paper is inspired by [11, 39], in particular some of the results were announced at the ICSSA 2018 Conference.

2010 Mathematics Subject Classiﬁcation. 28B20, 58C05,28B05, 46B42, 46G10, 18B15. Key words and phrases. Birkhoﬀ integral, vector measure, Girsanov Theorem. -The Fondo Ricerca di Base 2018 University of Perugia - and the GNAMPA – INDAM (Italy) Project "Dinamiche non autonome, analisi reale e applicazioni" (2018) supported this research. 1 2 CANDELORO, SAMBUCINI, AND TRASTULLI

Now, we give a plan of the paper. In Section 2, after an introduction of the Birkhoﬀ integrals (Deﬁnitions 2.2 and 2.3) the properties of such integrals are studied togheter with a link between them (Theorem 2.7). Moreover, the notions of conditional expectation in this framework is given togheter with a tower property for martingales and Theorem 2.13. In section 3 the main result: a vector version of the Girsanov result (Theorem 3.6) is pre- sented after having introduced the equivalent martingale measure and under Assumptions A1 and A2. At the end of this section an example is given satisfying Theorem 3.6. Section 4 is devoted to applications of Theorem 3.6 such as conditional measures (Proposition 4.1 for C([0,T ])-valued measures) and to extensions of the Itô representation of stochastic X-valued processes, using vector-stochastic integral and the classical Itô formulas. In particular, in Theorem 4.8, a process of the type Z t Z t ? Ct = (Bi1) Ψ(s)ds + (Bi1) Φ(s)dws, (under P), t ∈ [0,T ] 0 0 is considered and it is proved that it is possible to eliminate the drift term in order to obtain a local martingale.

2. The Birkhoff integrals and their properties Let (Ω, A, ν) denote a measure space and (X, k·k) a Banach space. From + now on, with the letters µ, ν we refer to scalar measures, that is ν : A → R0 while we use letters as N,M to denote vector measures, that is N : A → X. We also use capital letters like X,Y to denote arbitrary Banach spaces, X∗,Y ∗ to denote their dual spaces and x∗, y∗ the elements of the dual spaces. With letters like φ, ψ we refer to scalar functions, and with Φ, Ψ to vector valued functions, defined on the measure space (Ω, A, ν). We also denote with B(R), (B(I)) as usual the Borel σ-algebra on the real line (on the interval I = [0,T ], T > 0) and with B(X) the Borel σ-algebra on X. Finally, we shall denote by at, bt, zt, scalar-valued stochastic processes, while At,Bt,Zt denote X-valued stochastic processes. We recall that an X-valued stochastic process is a strongly measurable function Z :(I × Ω, B(I) ⊗ A) → (X, B(X)). There are many different versions of the Birkhoff integral (see [1]). We shall use the following one, that provides two different kinds of integrals. In order to do this we recall some basic definitions.

Definition 2.1. We say that a finite or countable family of non-empty measurable sets P := (Uj)j∈J is a partition of Ω if Uj and Uk are pairwise S disjoint and they cover Ω, that is j∈J Uj = Ω. We denote by P the set of all partitions of the set Ω. Given two different partitions P, P 0 we say that the partition P is finer than P 0 if for every U ∈ P there exists a U 0 ∈ P 0 such that U ⊂ U 0. We distinguish between two different kinds of Birkhoff integral. The first one is about the integration of an X-valued function with respect to a scalar A VECTOR GIRSANOV RESULT ... 3 measure. The second one is the notion of integral of a scalar function with respect to a vector measure. Definition 2.2 (First type Birkhoff integral). Let Φ:Ω → X be a vector + space-valued function and ν : A → R0 be a scalar, countably additive measure. Then Φ is said to be first type Birkhoff integrable with respect to ν, briefly Φ ∈ Bi1 (Ω, ν) (or simply Bi1 if there is not ambiguity about the space and the measure), if there exists I ∈ X such that for all ε > 0 there exists a partition of Ω, Pε, such that, for every countable partition (Un)n∈ of Ω, finer Pn N than Pε and for all ωn ∈ Un, it is lim supn k( k=1 Φ(ωk)ν(Uk) − I)k < ε. We call I ∈ X the Birkhoff integral of Φ with respect to ν, and we denote it Z by (Bi1) Φdν. Ω Now we are going to define the second type of Birkhoff integral.

Definition 2.3. Let φ :Ω → R and N : A → X be a countably additive measure. Then φ is said to be second type Birkhoff integrable with respect to N, briefly φ ∈ Bi2 (Ω,N) (or simply Bi2), if there exists I ∈ X such that for all ε > 0 there exists a partition of Ω, Pε, such that, for every countable partition (Un)n∈ of Ω, finer than Pε and for all ωn ∈ Un, it is Pn N lim sup k( φ(ωk)N(Uk) − I)k < ε. We call I ∈ X the Birkhoff integral n k=1 Z of φ with respect to N and we denote it by (Bi2) φdN. Ω The first (second) type Birkhoff integrability/integral of a function on a set A ∈ A is defined in the usual manner since, thanks to a Cauchy criterion the integrability of the function restricted to A follows immediately.

Remark 2.4. If µ is σ-finite then the Bi1 integrability is correspondent to the classic Birkhoff integrability for Banach space-valued mappings (see also [5, Theorem 3.18]). Moreover, since the Birkhoff integral is stronger than the Pettis integral, it is clear that, as soon as F is first type Birkhoff Z integrable with respect to m, the mapping M := A 7→ (Bi1) F dm is a A countably additive measure. Z For the second type, the mapping M := A 7→ (Bi2) fdN is weakly count- A ably additive, since for each x∗ in the dual space X∗ the scalar mapping f is integrable with respect to the scalar measure x∗(N) (and to its variations). Then, thanks to the Orlicz-Pettis Theorem (see [33]), M turns out to be also strongly countably additive. We want to show a link between these two types of Birkhoff integral. Firstly we recall a result concerning the first type Birkhoff integrability.

Theorem 2.5. [7, Th 3.14] Let Φ be a strongly measurable and Bi1 (Ω, ν)- integrable function. Then, for every ε > 0 there exists a countable partition ∗ P := {Un, n ∈ N} of measurable subsets of Ω, such that, for every ﬁner 4 CANDELORO, SAMBUCINI, AND TRASTULLI

0 ∗ partition P := {Vm, m ∈ N} of P and for every ωm ∈ Vm, we have Z X Φ(ωm)ν(Vm) − (Bi1) Φdν ≤ ε. m Vm Lemma 2.6. If φ is a scalar measurable function and Φ is an X-valued strongly measurable function in Bi1 (Ω, ν) then, given

(Gn)n := ({ω ∈ Ω: n − 1 ≤ |φ(ω)| ≤ n})n ∈ A, n j o for every ε > 0 there exists a measurable countable partition Un, j ∈ N of n j o j j Gn such that for every ﬁner partition Vn , j ∈ N and for every ωn ∈ Vn ,

X X j j j j j (1) Φ(ωn)φ(ωn)ν(Vn ) − φ(ωn)N(Vn ) ≤ 2ε. n j Proof. By hypothesis the product function φ(ω)Φ(ω) is strongly measurable. Then, thanks to Theorem 2.5, for every ε > 0 and for every n there exists a n j o measurable countable partition Un, j ∈ N of Gn such that, for every ﬁner n j o j j partition Vn , j ∈ N and for every ωn ∈ Vn , we obtain Z X j j ε Φ(ωn)ν(Vn ) − Φdν ≤ . j n2n j Vn Then it follows that X X j j j j j Φ(ωn)φ(ωn)ν(Vn ) − φ(ωn)N(Vn ) ≤ 2ε. n j

Theorem 2.7. Let φ :Ω → R be a measurable function and Φ ∈ Bi1(Ω, ν) be a vector valued strongly measurable function. We denote by N(A) = Z (Bi1) Φdν. Then φ(·)Φ(·) ∈ Bi1 (Ω, ν) ⇐⇒ φ(·) ∈ Bi2 (Ω,N) and A Z Z (2) (Bi1) φ(ω)Φ(ω)dν = (Bi2) φ(ω)dN. Ω Ω Z Proof. Suppose that φ(ω)Φ(ω) ∈ B1 (Ω, ν) and let J = (Bi1) φ(ω)Φ(ω)dν. Ω ∗ Then, fixed arbitrarily ε > 0, we can find a measurable partition P := {Un : n ∈ N} of Ω such that Z X 0 Φ(ωn)φ(ωn)ν(Un) − (Bi1) Φ(ω)φ(ω)dν ≤ ε 0 n Un 0 0 for every finer partition {Un, n ∈ N} and for every ωn ∈ Un. So, taking a ∗ k partition {Vm, m ∈ N} finer than P and a partition {Vm, m, k ∈ N} given A VECTOR GIRSANOV RESULT ... 5 by Lemma 2.6, using (1) we infer that

Z X X X φ(ωm)N(Vm) − J = φ(ωm)N(Vm) − (Bi1) Φ(ω)φ(ω)dν ≤ m m m Vm Z X ≤ φ(ωm)N(Vm) − (Bi1) φ(ω)Φ(ω)dν ≤ m Vm X ≤ kφ(ωm)N(Vm) − φ(ωm)Φ(ωm)ν(Vm)k + m Z X + φ(ωm)Φ(ωm)ν(Vm) − (Bi1) φ(ω)Φ(ω)dν ≤ 3ε m Vm for every choice of ωm ∈ Vm. Then φ ∈ Bi2(Ω,N), as we wanted. Conversely, we assume that φ ∈ Bi2(Ω,N). Then, for every ε > 0 there exists a countable # measurable partition P := {Un : n ∈ N} such that n X Z lim sup k φ(ωi)N(Ui) − (Bi2) φdNk ≤ ε. n i=1 Ω # k So, considering {Vk, k ∈ N} a ﬁner partition than P and {Vj , j, k ∈ N}, as in Lemma 2.6, we get, using (1),

n X Z lim sup φ(ωk)Φ(ωk)ν(Vk) − (Bi2) φdN ≤ n k=1 Ω n n X X Z ≤ lim sup φ(ωk)Φ(ωk)ν(Vk) − φ(ωk)(Bi1) Φdν + n k=1 k=1 Vk n X Z + lim sup φ(ωk)N(Vk) − (Bi2) φdN ≤ 3ε. n k=1 Ω

This shows that φ(·)Φ(·) ∈ Bi1(Ω, ν) and the equality (2) is true. In order to ﬁnd applications in stochastic processes, we need to extend notions like distribution of a function with respect to a vector measure.

Deﬁnition 2.8. Let φ :Ω → R be a measurable function and N : A → X be a countably additive measure. We deﬁne the measure induced by φ −1 as Nφ(B) = N(φ (B)) for every B ∈ B (R) . This measure is countably additive and we call it the distribution of φ with respect to N.

We give now an integration by substitution result for the (Bi2) integral. Theorem 2.9. Given two measurable functions φ :Ω → , ψ : → , Z Z R R R then the following relation holds: (Bi2) ψ(φ)dN = (Bi2) ψ(x)dNφ under Ω R the assumption that the integrals involved exist. 6 CANDELORO, SAMBUCINI, AND TRASTULLI

Proof. Suppose that the two integrals above exist as second type Birkhoff integrals. Now if we fix x∗ ∈ X∗, we can consider the real measures x∗(N) ∗ ∗ ∗ and x (Nφ) = (x (N))φ. Then ψ(φ) is integrable with respect to x (N) and ∗ ψ is integrable with respect to (x (N))φ. So we obtain that Z Z Z ∗ ∗ ∗ x (Bi2) ψ(φ)dN = ψ(φ)dx (N) = ψ(ω)dx (N)φ = Ω Ω R Z ∗ = x (Bi2) ψ(ω)dNφ . R ∗ By the arbitrariness of x , the assertion follows. In order to introduce in this setting the definition of a martingale the notions of conditional expectation and filtration are needed.

Definition 2.10. Let φ :Ω → R be a scalar function and Bi2(Ω,N) integrable. We denote with σφ the sub-σ-algebra of A, obtained by taking −1 all pre-images φ (B), for all B ∈ B(R). In the special case of φ = zt, with t ∈ [0,T ] fixed, then we define the natural filtration of zt, denoted by

Fz = (Ft)t, since Ft = σzt for every t. Now we give the notion of conditional expectation. Remark 2.11. Given a sub σ-algebra F of A, we say that an X-valued function is F-measurable if it is strongly measurable as a function from the measurable space (Ω, F) to the measurable Banach space (X, B(X)).

Deﬁnition 2.12. Let Φ ∈ Bi1 (Ω, ν) and F be a sub σ-algebra of A. We deﬁne, provided that it exists, the conditional expectation of Φ with respect ν to F, indicated by E (Φ|F) (E (Φ|F) if there is no ambiguity about the measure), as the strongly F-measurable function Ψ such that Ψ ∈ Bi1(Ω, ν) and for every E ∈ F it holds Z Z (Bi1) Φdν = (Bi1) Ψdν. E E From this the classic tower property follows, i.e. for every sub σ-algebras of A, such that F ⊂ G ⊂ A, we have that

(3) E(Φ|F) = E(E(Φ|G)|F). Another important property of the conditional expectation that we can extend is the following. Theorem 2.13. Let Φ:Ω → X be a strongly measurable function and F be a sub σ-algebra of A, having conditional expectation E (Φ|F). Then, given a F measurable function φ :Ω → R so that the product function Φ(·)φ(·) ∈ Bi1(Ω, ν), it holds: E (Φ(ω)φ(ω)|F) = φ(ω)E (Φ|F) . A VECTOR GIRSANOV RESULT ... 7

Proof. We claim that, for every E ∈ F the following relation is satisﬁed: Z Z (4) (Bi1) Φ(ω)φ(ω)dν = (Bi1) φ(ω)E(Φ(ω)|F)(ω)dν. E E ∗ ∗ ∗ ∗ For every x ∈ X it holds x (E (Φ|F)) = E (x (Φ) |F), then we obtain that Z Z Z ∗ ∗ ∗ x (Bi1) Φ(ω)φ(ω)dν = x (Φ(ω)) φ(ω)dν = x (E (Φ|F)) φ(ω)dν E E E Z ∗ = x (Bi1) E (Φ|F) φ(ω)dν . E So the function ω 7→ E (Φ|F) φ(ω) is Pettis integrable with respect to µ; since the product is strongly measurable, this means that the product is Bi1(Ω, µ)-integrable. Moreover, by the Hahn-Banach Theorem, the formula (4) follows. Theorems 2.7, 2.9 and 2.13 of this section were announced in [11, Theorems 2,3,4] respectively without any proof.

3. The Girsanov Theorem for vector measures To the aim of extending the Girsanov Theorem to the Banach-valued measures we shall find the distribution of a scalar valued stochastic process under a vector measure N, make a transformation of this process, compute its new distribution and then define a new measure using these two density functions. First, we need to define the concept of martingale when we use a vector + measure N. Let a scalar measure ν : A → R0 be fixed and Φ: A → X be an X-valued function strongly measurable and B1(Ω, ν) integrable. We Z define the vector measure N : A → X as follows: N(A) := (Bi1) Φdν. A A measure Q is equivalent to a measure N (Q ∼ N) if there exists a Bi2(Ω,N) integrable and positive function ϕ such that dQ/dN = ϕ. So the definition of martingale for a scalar process (zt)t can be given.

Deﬁnition 3.1. Let (zt :Ω → R)t∈[0,T ] be a scalar stochastic process on the probability space (Ω, A,N), where N is as above. We say that zt is a N- martingale in itself, that is a martingale with respect to its natural ﬁltration N Fz = (Ft)t, if for every s < t, s, t ∈ [0,T ], we have that E (zt|Fs) = zs. N Remark 3.2. The identity E (zt|Fs) = zs in terms of integrals means that, Z Z for every E ∈ Fs, it is (Bi2) ztdN = (Bi2) zsdN. E E

Deﬁnition 3.3. Let (zt :Ω → R)t∈[0,T ] be a scalar stochastic process on the probability space (Ω, A, P, F) which is adapted to F. A vector-valued measure Q is called an equivalent martingale measure for (zt)t with respect to P if (zt)t is Bi2-integrable, it is a martingale with respect to Q and Q ∼ P. 8 CANDELORO, SAMBUCINI, AND TRASTULLI

In ﬁnancial market the equivalent martingal measure is called also the risk-neutral measure. Now we give some basic assumptions on stochastic processes.

Assumptions A1. Let us assume that (zt)t∈[0,T ] is a stochastic scalar process deﬁned as usual on the space (Ω, A,N) such that these conditions are satisﬁed:

A1.a) The function ω 7→ z(t, ω) belongs to the space Bi2(Ω,N), for every t ∈ [0,T ], with null integral and admits a density function, that means that its distribution under the vector measure N, denoted by −1 Nt(B) := Nzt (B) = N(zt (B)), for every B(R) could be written as a Birkhoff first type integral of some vector function Ft : R → X ∈ Bi1(Ω, λ) (where λ is the Lebesgue measure), namely: Z Nt(B) = (Bi1) Ft(x)dx. B A1.b) Let z˜t = zt + θ(t), where θ : [0,T ] → R is a measurable function. Suppose that for every t ∈ [0,T ] there exists a scalar positive function gt(x) such that the following factorization holds Ft(x) = gt(x)F˜t(x) = gt(x)Ft(x − θ(t)). A1.c) The process defined as yt = (gt(˜zt))t is a N-martingale in itself. Remark 3.4. Conditions A1.a) and A1.b) allow us to work with the distributions of zt and z˜t under the measure N. We note that by A1.a) it follows that the process z˜t defines again a Bi2(Ω,N) random variable, for every t ∈ [0,T ] and it admits a density function too, of type F˜t := Ft(x − θ(t)). In fact, by Lebesgue integral translation invariance, substituting r = x + θ(t), we obtain Z Z ˜ Nt(B) := Nt(B + θ(t)) = (Bi1) Ft(x)dx = (Bi1) Ft(r − θ(t))dr. B+θ(t) B

(Bi )R So, thanks to Theorem 2.7, we have Nt(B) = 1 B gt(x)dNz˜t . It is important to note that gt is a positive function, so, by Assumptions A1.c) we know that yt = gt(˜zt) is a N-martingale.

Finally, we recall that, when {zt}t is the classical (scalar) Brownian Mo- 1 2 tion, then it turns out that gt(x) = exp(−qx + 2 q t) and so the process 1 (5) {g (z )} = {exp(−qz − q2t)} t et t t 2 t is a martingale with respect to the natural ﬁltration of {wt}t (see e.g. [31, (4.20)], [32]).

So, we deﬁne the change of measure setting the new vector measure Q : A → X as follows: Z Z (6) Q(A) = (Bi2) yT dN = (Bi1) yT Φdν, A A A VECTOR GIRSANOV RESULT ... 9 for every A ∈ A. Observe that Q ∼ N. Under Assumptions A1, the marginal distribution of the stochastic process zt is preserved when we change the underlying measure. This means that the distribution of zt under N is the same of the process z˜t under the new measure Q.

Theorem 3.5. Let N, Q, zt, z˜t be the vector measures and the scalar stochastic processes deﬁned before. Under Assumptions A1, the marginal distributions of these two processes are preserved under the change of measure, that is, for every t ∈ [0,T ], Nzt = Qz˜t . Proof. We ﬁx B ∈ B(R) and t ∈ [0,T ]. By the Assumptions A1.a) and A1.b) Z −1 we have that: Qz˜t (B) = Q (˜zt) (B ) = (Bi2) yT dN. Now, using −1 (˜zt) (B) the fact that yt is a martingale, that is the Assumption A1.c) with respect to N, and by Theorem 2.9, we write Z Z Z (Bi2) yT dN = (Bi2) ytdN = (Bi2) gt(˜zt)dN = −1 −1 −1 (˜zt) (B) (˜zt) (B) (˜zt) (B) Z

= (Bi2) gt(r)dNz˜t . B Since this holds for every B ∈ B(R), the proof is complete.

Our aim is to prove that Q is an equivalent martingale measure for (zet)t with respect to N. In order to do this, we assume

Assumptions A2. The product process (˜ztyt)t is a martingale in itself with respect to N. Then, we are ready to formulate the main theorem, that is the Girsanov Theorem for vector measures Theorem 3.6 (Girsanov Theorem). Under Assumptions A1 and A2 the process (˜zt)t is a Q-martingale in itself, where Q is the vector measure, defined in (6). Proof. By assumptions A1.a) and A1.b), we can define the positive real pro- R cess yt and by means of A1.c), we have that Q ∼ N. Then it is (Bi2) z˜tdQ = R E (Bi2) E z˜tyT dN. Now we fix s, t ∈ [0,T ], with s ≤ t and E ∈ Fs. Using the martingale property of the process yt, we have Z Z Z Z (Bi2) z˜tdQ = (Bi2) z˜tyT dN = (Bi2) ztytdN + (Bi2) θ(t)yT dN = E E E E Z Z = (Bi2) ztytdN + (Bi2) θ(t)ysdN = E E Z Z = (Bi2) ztytdN + θ(t)(Bi2) ysdN = E E Z = (Bi2) ztytdN + θ(t)Q(E). E 10 CANDELORO, SAMBUCINI, AND TRASTULLI

Now, from Assumptions A2 and A1.c) we deduce Z Z Z Z (Bi2) z˜tdQ = (Bi2) (zsys + θ(s)ys)dN = (Bi2) zsysdN + θ(s)(Bi2) ysdN E E E E but using the tower property (3), we have Z Z (Bi2) zsyT dN = (Bi2) zsysdN, E E and then we get Z Z Z (Bi2) z˜tdQ = (Bi2) zsysdN + θ(s)(Bi2) ysdN = E E E Z Z = (Bi2) zsyT dN + θ(s)(Bi2) yT dN = E E Z Z = (Bi2) (zs + θ(s))dQ = (Bi2) z˜sdQ. E E The last relation is the martingale property of the process z˜t and the proof is completed. Theorems 3.5 and 3.6 were announced in [11, Theorems 5, 6]; the last one with also a brief sketch of the proof.

If we consider, for example, a Brownian motion (wt)t≥0, we know that this process is a martingale on this space with respect to this ﬁltration F. However, if we condition it on an expiration time T > 0 ﬁxed, the distribution of this process changes and in general, it doesn’t preserve some of its properties, such as the martingale property (see for example [15, Theorem 5.4]). Let 1 (7) N := P (·|wT ): A → L (Ω) . (i.e. N(A) = P(A|wT ) = E(1A|wT )). If we study the future time process (wt+T )t>0 under this new measure, we can observe that the stochastic process w˜t = wt+T + qt is an N-martingale. In fact, since N(A) = P (A|wT ) = 1 E (1A|wT ) , the last term is a L random variable, depending on wT and it 1 shows that N is a vector L measure. We observe that the process wt+T has the same distribution of wt+wT , where here we consider wt as independent of wT , and, under the measure N, it has a Gaussian distribution of parameters N (wT , t). Then, it admits the following density function: 2 1 (x − wT ) ft(x) = √ exp − . 2πt 2t

Now, if we consider the transformed process (w ˜t)t = (wt + wT + qt)t, its marginal distribution admits a conditional density given by 2 1 −(x − qt − wT ) f˜t(x) = √ exp . 2πt 2t A VECTOR GIRSANOV RESULT ... 11

Their quotient is 2 ft(x) q t gt(x) = = exp − q(x − wT ) . f˜t(x) 2

Then, evaluating gt(w ˜t+T ), we obtain the process q2t y := g (w ˜ ) = exp − − qw , t t t 2 t again considering wt, and then the process yt, independent of wT . Then the process yT , that we know to be a martingale under P (by [31, (4.20)]), since it is independent by wT is a martingale even under the conditional probability N = P|wT . Furthermore, we have that the process wetYt = (wt + qt)gt(wet) + wT gt(wet) is the sum of two martingales under the conditional probability N, again considering wt as independent by wT , so it is a martingale too. Then the process (wt+T )t>0, under the probability N obtained conditioning with respect to wT , satisﬁes all the condition of the Girsanov Theorem 3.6 and then the new process (w ˜t)t is a martingale under N = P|wT .

4. Some applications Since conditional measures can be seen as vector measures, using the tools obtained previously about the Birkhoﬀ integral, we give some examples of applications of the Girsanov Theorem 3.6. This could be useful for example when conditioning of random variables to future (or past times) is considered. Now consider a C([0,T ])-valued stochastic process Φ deﬁned as t + τ − τ ∧ t Φ(ω, t) = exp −w − w + w − = τ t t∧τ 2 ( exp{−w − τ/2} if t ≤ τ (8) = τ exp{−wt − t/2} if t > τ with τ ∈ [0,T ] and Z ◦ (9) N (A) = Φ(ω, T )dP, A be a C([0,T ])-valued measure.

Proposition 4.1. Let (wt)t∈[0,T ] be a Brownian motion on the ﬁltered prob- ˜ ability space (Ω, A, P, F). Let Ft and Ft be the density functions of wt and ◦ w˜t = wt + t under N , respectively. Then, for every t ∈ [0,T ] there ex- ˜ ists a real function gt(x) such that Ft(x) = gt(x)Ft(x) for every x ∈ R and dQ the vector measure Q, deﬁned by = g (w ˜ ), is an equivalent martingale dN ◦ t t measure for the process w˜t. 12 CANDELORO, SAMBUCINI, AND TRASTULLI

Proof. We are going to prove that for every s ≤ t in [0,T ] 1 (10) (Φ(ω, t)|F ) = exp{−w − s}, E s s 2 namely it is independent of τ. We distinguish three cases: 1 (τ ≤ s ≤ t): in this case we have Φ(·, t) = exp{−w − t} so, being t 2 a martingale process, it is trivial to deduce that E (Φ(ω, t)|Fs) = 1 exp{−w − s}. s 2 1 (s ≤ τ ≤ t): again Φ(ω, t) = exp{−w − t}, then, as before, we get t 2 1 (Φ(ω, t)|F ) = exp{−w − s}. E s s 2 1 (s ≤ t ≤ τ): in this case Φ(ω, t) = exp{−w − τ}, and since it is a τ 2 1 martingale, it follows: (Φ(ω, t)|F ) = exp{−w − s}. E s s 2 Now, considering the vector measure N ◦, it is Z Z Z ◦ 1 wtdN = wtΦ(ω, T )dP = wt exp{−wt − t}dP, E E E 2 for every E ∈ Fs, where we have used the tower property (3), that is P P P E (wtΦ(ω, T )|Fs) = E E (wtΦ(ω, T )|Ft) |Fs = P P = E wtE (Φ(ω, T )|Ft) |Fs = 1 = P w exp{−w − t}|F . E t t 2 s

This shows that wtΦ(ω, t) is not a martingale under P and this is equivalent to say that it is not a martingale under N ◦. Since the marginal distributions of the processes wt and w˜t under P are Gaussian with parameters respectively N (0, t) and N (t, t), the marginal distributions under N ◦ admit a density function obtained in this way: Z Z ◦ ◦ Nt (B) = dN = Φ(ω, T )dP = −1 −1 (wt) (B) (wt) (B) Z = E(Φ(ω, T )|Ft)dPwt = B Z 1 1 x2 = exp{−x − t}√ exp − dx. B 2 2πt 2t Then we have the following densities function under N ◦: 1 1 x2 Ft(x) = √ exp −x − t − , 2πt 2 2t 1 1 (x − t)2 F˜t(x) = F˜t(x − t) = √ exp −x − t − t − , 2πt 2 2t A VECTOR GIRSANOV RESULT ... 13 and then it is easy to see that the function gt we are looking for is given by ft(x) 1 = gt(x) = exp − t − x . f˜t(x) 2 3 Then we have that y = g (w + t) = exp − t − w and we want to prove t t t 2 t that it is a martingale under the measure N ◦, which is equivalent to prove that the vector process Φ(ω, T )yt is a P-martingale. Again, using the tower property (3), we ﬁnd that P P P E (ytΦ(ω, T )|Fs) = E E (ytΦ(ω, T )|Ft) |Fs = P P = E ytE (Φ(ω, T )|Ft) |Fs = 1 = P y exp{−w − t}|F = E t t 2 s P = E (exp {−2wt − 2t} |Fs) = P = exp{−2ws − 2s} = E (ysΦ(ω, T )|Fs), ◦ because exp{−2ws − 2s} is a martingale under P. Since yt is a positive N - martingale satisfying Assumptions A1 with respect to N ◦ and with θ(t) = Z ◦ t, we deﬁne a new vector measure as Q (A) := yT dN ,A ∈ A. Now, A recalling the increment invariance of the Brownian motion and the expected value of the log normal random variable, we prove that Q N ◦ N ◦ E (wt + t|Fs) = E (wtyt|Fs) + tE (yt|Fs) = P P = E (wtytΦ(ω, T )|Fs) + tys = E (wt exp{−2wt − 2t}|Fs) + tys = P = E ((wt − ws) exp{−2wt + 2ws}|Fs) exp {−2ws − 2t} + tys + P + ws exp {−2ws − 2t} = E ((wt − ws) exp{−2wt + 2ws}) exp {−2ws − 2t} + + tys + wsys = −(t − s) exp{−2s + 2t} exp{−2ws − 2t} + (t + ws)ys =

= (s − t)ys + (t + ws)ys = (ws + s)ys =w ˜sys. ◦ This means that the process w˜tyt is an N -martingale and this proves As- sumptions A2 with respect N ◦. So, using the Girsanov Theorem 3.6, the process w˜t is a Q-martingale and the proof is completed. A Birkhoff integral representation Using the classical Girsanov The- orem, it is possible to change the drift term of an Itô integral and to obtain a local martingale. We want to prove an analogous result for vector processes that have an integral representation in terms of Birkhoff integrals (Theorem 4.8). The main problem is to adapt to this theory the expression Z t Z t (11) At = (Bi1) Ψ(s)ds + (·) Φ(s)dws, (under P), 0 0 where P is the underlying probability. The first integral could be seen as a Birkhoff integral of first type. We can observe that the function Φ integrated 14 CANDELORO, SAMBUCINI, AND TRASTULLI with respect to the Brownian motion is a vector one and the Brownian motion could also be seen as a vector-valued function, taking values for example in L1 (Ω). Then, the second integral in (11) cannot be defined as a first type and neither as a second type Birkhoff integral. To solve this problem, we recall the Itô formula. If we consider as usual the standard Brownian Motion wt and the stochastic process zt = h(t, wt), where the function h : [0,T ]×R → R 2 is in the class C ([0,T ] × R), then it is: 1 d (z ) = d (h(t, w )) = h0 (t, w )dt + h0 (t, w )dw + h00 (t, w )dt t t t t x t t 2 xx t

Obviously, in the special case that the function h is linear with respect to wt 00 0 0 0 and it is of type h(t, x) = xr(t), we get hxx = 0, ht = r (t)x and hx = r(t). 0 Then we obtain d (h(t, wt)) = wtr (t)dt + r(t)dwt. This is equivalent to the following integral relation Z t Z t 0 (12) r(s)dws = r(t)wt − r (s)wsds. 0 0 It is interesting to note that, if X = R, then the equation (12) can be deduced simply applying the classic Itô formula. If we focus on the expression in (12), we observe that it should be used as definition of a stochastic integral in the Birkhoff sense. In fact, for every fixed t ∈ [0,T ], the Brownian motion wt :Ω → R is a scalar random variable and given a vector function Φ: [0,T ] → X, we have that Φ(t)wt :Ω → X is an X-valued random variable. Thus, the first term on the right side of the equation (12) is an X-valued random variable. For the second one we need to recall the definition of Fréchet derivative, for other definitions see for example [2,27, 28]. Definition 4.2. A function Φ : [0,T ] → X is said to be differentiable at a 0 point t ∈ [0,T ] if exists Φt ∈ L (R,X), (where L (R,X) denotes the space of all continuous and linear operators from R to X) such that kΦ(t + h) − Φ(t) − Φ0 (h)k lim t X = 0. h→0 |h| 0 Φt is said to be the Fréchet differential of Φ at t ∈ [0,T ]. We say that Φ is differentiable on [0,T ] if it is differentiable at every t ∈ [0,T ]. Remark 4.3. Now, if Φ is differentiable, we consider the X-valued function 0 0 0 t 7→ Φ (t) := Φt(1). Let (ws)s be a Brownian process. If Φ w it is first type Birkhoff integrable with respect to the Lebesgue measure, we can denote, R t 0 with the integral notation (Bi1) 0 Φ (s)wsds, an X-valued random variable defined by Z t Z t 0 0 (Bi1) Φ (s)wsds (ω) = (Bi1) Φ (s)(w(s, ω)) ds. 0 0 Now we are ready to define the stochastic integral of the second summand in (11). A VECTOR GIRSANOV RESULT ... 15

Definition 4.4. A function Φ: R → X is said to be stochastic Bi1-integrable ? ((Bi1) for short) with respect to a Brownian motion wt on the filtered space, (Ω, A, P, F) if • Φ is differentiable on [0,T ] ( Φ0 being its differential); 0 • for every fixed ω ∈ Ω the function Φ (·)w(·)(ω) : [0,T ] → X is Bi1- integrable with respect to the Lebesgue measure. In this case, for every [a, b] ⊂ [0,T ], we define Z b Z ? 0 (Bi1) Φ(r)dwr = Φ(b)wb − Φ(a)wa − (Bi1) Φ (r)wrdr. a [a,b] Assumptions A3. Suppose that Φ is strongly measurable and differentiable on [0,T ] and such that for every x∗ ∈ X∗, the function < x∗, Φ(·) >∈ L2 (Ω) and satisfies d (13) (< x∗, Φ(t) >) =< x∗, Φ0(t) > . dt Remark 4.5. When X = R, Definition 4.4 agrees with the classic Itô for- R t mula. In fact, given a scalar function r(t), it is 0 r(s)dws = r(t)wt − R t 0 0 r (s)wsds. Note that in this case, the Fréchet differential of r is the classic 0 0 differential given by Dt(r)(t) = r (t)t, then Dt(r)(1) = r (t) and Assump- tions A3 hold. ? From now on, we suppose that Φ, Ψ are Bi1-integrable and satisfy As- sumptions A3. So, we get the following Proposition 4.6. For every x∗ ∈ X∗, the process < x∗, Φ > is integrable with respect to the Brownian motion (wt)t and moreover Z t Z t ∗ ? ∗ (14) < x , (Bi1) Φ(s)dws >= < x , Φ(s) > dws. 0 0 Proof. Fixed x∗ ∈ X∗, it is Z t Z t ∗ ? ∗ ∗ 0 < x , (Bi1) Φ(r)dwr > = < x , Φ(t)wt > − < x , (Bi1) Φ (r)wrdr > 0 0 Z t ∗ ∗ 0 = < x , Φ(t) > wt − < x , Φ (r)wr > dr 0 Z t ∗ d ∗ = < x , Φ(t) > wt − (< x , Φ(t) >) wrdr 0 dt Z t ∗ = < x , Φ(r) > dwr, 0 where, in the last equalities, we have used the Assumptions A3 and the ∗ formula (12) applied to h(t, wt) = wt < x , Φ(t) >. Then, the assertion follows. Moreover, the next result holds. 16 CANDELORO, SAMBUCINI, AND TRASTULLI ? R t Theorem 4.7. The process (At)t∈[0,T ] := (Bi1) Φ(s)dws is a mar- 0 t∈[0,T ] tingale with respect to the natural filtration of the Brownian motion (wt)t. Proof. Let s ≤ t in [0,T ] and fix x∗ ∈ X∗. By Proposition 4.6 we have that ∗ < x , Φ(r) > is integrable with respect to the Brownian motion (wt)t and Z t Z t ∗ ? ∗ < x , E (Bi1) Φ(r)dwr|Fs > = E < x , Φ(r) > dwr|Fs = 0 0 Z s Z s ∗ ∗ = E < x , Φ(r) > dwr =< x , (Bi1) Φ(r)dwr > . 0 0 So, by arbitrariness of x∗, we have Z t Z s ? ? E (Bi1) Φ(r)dwr|Fs = (Bi1) Φ(r)dwr 0 0 so the proof is completed. Now, we consider a process as follows: Z t Z t ? Ct = (Bi1) Ψ(s)ds + (Bi1) Φ(s)dws, (under P), t ∈ [0,T ] 0 0 (in the scalar case it is an Itô process). We would like to eliminate the drift term so that we obtain a local martingale. To obtain it, we need a hypothesis of connection between the drift term Ψ and the diffusion term Φ: namely we need to know if there exists a stochastic process r(t) such that Ψ(t) = r(t)Φ(t), for every t ∈ [0,T ] (and this is the process that we use to define the change of measure). The main peculiarity of this kind of result is that we could change the drift term, without changing the diffusion term.

Theorem 4.8. ([Change of a drift term]) Let (ws)s be a Brownian motion ? on (Ω, A, P, F), Φ, Ψ be two processes that are Bi1-integrable and satisfy Assumptions A3. Let Ct : [0,T ] × Ω → X be the stochastic process defined by Z t Z t ? Ct = (Bi1) Ψ(s)ds + (Bi1) Φ(s)dws, (under P). 0 0 2 If there exists r : [0,T ] → R be in L ([0,T ]) such that Ψ(t) = r(t)Φ(t) n 1 R t 2 R t o and the process yt = exp − 2 0 r (s)ds − 0 r(s)dws is a martingale with ◦ R R t respect to F, denoted by Q = (Bi1) yT dP and w˜t = wt + 0 r(s)ds, then Z t ? ◦ Ct = (Bi1) Φ(s)dw˜s, (under Q ). 0 ◦ Therefore the process Ct is a martingale under Q . Proof. By construction and using the classical Girsanov Theorem the process ◦ w˜t is a Brownian motion under the new probability Q . Now we claim that Z t Z t Z t ? ? (15) (Bi1) Φ(s)dw˜s = (Bi1) Φ(s)dws + (Bi1) r(s)Φ(s)ds 0 0 0 A VECTOR GIRSANOV RESULT ... 17 as a vector equivalence between the Birkhoff stochastic integral and the Birkhoff integral. To prove this we consider, for every x∗ ∈ X∗, Z t Z t ∗ ? ∗ < x , (Bi1) Φ(s)dw˜s >= < x , Φ(s) > dw˜s = 0 0 Z t ∗ = < x , Φ(s) > (dws + r(s)ds) = 0 Z t Z t ∗ ∗ = < x , Φ(s) > dws + < x , Φ(s) > r(s)ds = 0 0 Z t Z t ∗ ? ∗ ? = < x , (Bi1) Φ(s)dws > + < x , (Bi1) Φ(s)r(s)ds >= 0 0 Z t Z t ∗ ? = < x , (Bi1) Φ(s)dws + (Bi1) Φ(s)r(s)ds > . 0 0 So, by the arbitrariness of x∗ ∈ X∗, the equation (15) holds. Thus, Z t Z t ? ? Ct = (Bi1) Ψ(s)ds + (Bi1) Φ(s)dws = 0 0 Z t Z t Z t ? ? ? = (Bi1) Ψ(s)ds + (Bi1) Φ(s)dw˜s − (Bi1) r(s)Φ(s)ds = 0 0 0 Z t ? = (Bi1) Φ(s)dw˜s, 0 and then we have that, Z t ? ◦ Ct = (Bi1) Ψ(s)dw˜s (under Q ) 0 and it turns out to be a martingale, thanks to Theorem 4.7.

Acknowledgements This is a post-peer-review, pre-copyedit version of an article published in Mediterranean Journal of Mathematics. The ﬁnal authenticated version is available online at: http://dx.doi.org/10.1007%2Fs00009-019-1431-x.

Udine ? 10/18/1951, Rome † 05/03/2019. His research was devoted to Measure Theory and Integration, with particular concern for: many different types of integrals, with applications to Calcu- lus of Variations, Stochastic Integration, infinite-dimensional integrals; multivalued integration with applications to the properties of measurable and integrable multifunctions and to the search of Equilibria; finitely additive measures, concerning structural prob- lems (non-atomicity, extensions and restrictions, decompositions) and applications to Probability, in particular Stochastic integration for infinite-dimensional Processes. 18 CANDELORO, SAMBUCINI, AND TRASTULLI

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Department of Mathematics and Computer Sciences, University of Peru- gia, Via Vanvitelli, 1 - 06123 Perugia (Italy), Orcid Id: 0000-0003-0526-5334 E-mail address: [email protected]

Department of Mathematics and Computer Sciences, University of Perugia, Via Van- vitelli, 1 - 06123 Perugia (Italy) , Orcid Id: 0000-0003-0161-8729 E-mail address: [email protected]

Department of Mathematics and Computer Sciences, University of Perugia, Via Van- vitelli, 1 - 06123 Perugia (Italy) , Orcid Id: 0000-0002-7722-4008 E-mail address: [email protected]