A de Finetti-type theorem for random-rotation-invariant continuous semimartingales
aa r X i v : . [ m a t h . P R ] D ec A de Finetti-type theorem forrandom-rotation-invariant continuoussemimartingales
Francesco C. De Vecchi ∗ Abstract
We provide a characterization of continuous semimartingales whose law is invariant withrespect to predictable random rotations. In particular we prove that all such semimartingalesare obtained by integrating a predictable process with respect to an independent n dimensionalBrownian motion. Keywords : Invariant stochastic processes, Random rotations, Continuous semimartingales
MSC numbers : 60F17, 60G44, 60G46
The problem of characterizing random objects invariant with respect to some group of (determin-istic or random) transformations was faced for the first time by de Finetti for describing the formof a sequence of infinite random variables invariant with respect to finite permutations (see [5]).This result has been generalized in many ways, for example considering different settings (con-tinuous time processes, non-commutative probability etc.) or new types of transformations (timetranslations, rotatability predictable transformations for processes etc.) so that this topic is nowa classical research field in probability (see, e.g., [2, 13] for some reviews on the subject).In this paper we characterize the continuous semimartingales invariant with respect to predictablerandom rotations. Denoting by F Zt the natural filtration generated by the semimartingale Z andby O ( n ) the group of n × n orthogonal matrices and using Einstein notation, we introduce thefollowing definition. Definition 1
Let Z be a semimartingale taking values in R n . We say that Z is invariant withrespect to (predictable) random rotations if, for any process B predictable with respect to F Zt andtaking values in O ( n ) , we have that the R n semimartingale Z ′ , given by Z ′ it = Z t B ij,s dZ js , has the same law of Z . ∗ Dipartimento di Matematica, Universit`a degli Studi di Milano, via Saldini 50, Milano, Italy, email:[email protected], [email protected] n dimensionalBrownian motion. Indeed this kind of invariance is an easy consequence of L´evy characterizationof Brownian motion (see [7]). Nevertheless this property is not peculiar of Brownian motion but isshared by many other semimartingales, such as homogeneous α -stable L´evy processes, Hermetianrandom matrices and other Markovian and non-Markovian semimartingales (see [1]).This invariance property is useful for explaining the relationship between Brownian motion andRiemannian geometry (for example the well-known relationship between R n Brownian motionand the stochastic development of Brownian motion on a Riemannian manifold, see [10, 8, 9])or more generally for studying L´evy processes taking values in Riemannian manifolds (see [3]).Furthermore the above invariance with respect to random rotations is a particular case of gaugesymmetry introduced in [1] for extending the Lie symmetry analysis from the deterministic to thegeneral stochastic setting (see also [6, 7, 11] for the particular case of Brownian motion).Although the invariance property stated in Definition 1 has many similarities with the invariancewith respect to rotations already studied in the literature, there is a fundamental difference. Indeed,in the traditional theorems of rotatability of stochastic processes, the rotations usually act bothon the space variables of R n processes and on the time variable t (see, e.g., [13]). For example, aclassical result of this form is that, if a real process X is such that for any h > { X nh } n ∈ N is invariant with respect to finite deterministic rotations, then thereexists a random variable σ and an independent Brownian motion W t such that X t = σW t . InDefinition 1 the action of the rotations group is only on the space variable and the previous resultdoes not hold. Nevertheless we are able to prove a generalization of this result for continuoussemimartingales, which is a de Finetti type representation theorem. Theorem 2
Under suitable hypotheses on the continuous semimartingale Z taking values in R n (more precisely hypotheses A , A1 and B below) if Z is invariant with respect to random rotations(according to Definition 1) then there exists an n dimensional Brownian motion W and a pre-dictable process f t , independent of W , and both adapted with respect to F Zt , such that R t f s ds < + ∞ almost surely and Z it = Z t f s dW is . In order to describe more precisely the hypotheses of Theorem 2 we start by recalling the notionof characteristics of a R n semimartingale (see [12]): the two continuous predictable processes ofbounded variation b t ∈ R n and A t ∈ Mat( n, n ) are the characteristics of the continuous semi-martingale Z with respect to its natural filtration F Zt if Z it − b t is a local ( F Zt ) martingale and A ijt = [ Z i , Z j ] t . It is clear that A t is a symmetric semidefinite positive matrix increasing withrespect to t , i.e. A t − A s is semidefinite positive whenever t ≥ s .We introduce the following hypothesis on Z • hypothesis A : A t is almost surely absolutely continuous, i.e. there exists a predictable process˜ A t taking value in the set of symmetric semidefinite positive matrices almost surely finitewith respect the measure P ⊗ dt such that A ijt = Z t ˜ A ijs ds ; • hypothesis A1 : the process ˜ A ijt of the hypothesis A is such that ˜ A ijt is a symmetric definitepositive (and not only semidefinite positive) matrix almost surely with respect to P ⊗ dt ; • hypothesis B : b t is almost surely absolutely continuous, i.e. there exists a predictable process˜ b t such that b it = b + R t ˜ b is ds . 2e briefly discuss the relationship between the hypotheses of Theorem 2, namely the invarianceof Z with respect to random rotations, the request that Z is a continuous semimartingale and theabove hypotheses A , A1 and B .Obviously, when the thesis of Theorem 2 holds, Z is a continuous semimartingale and hypotheses A and B are satisfied. It is well known (see [1]) that there are non-continuous semimartingaleswhich are invariant with respect to random rotations, so the continuity of Z is a necessary hy-pothesis. Indeed there is a deeper reason for the continuity hypothesis: since the discontinuousprocesses have not the martingales representation property if we use only Itˆo integration of theform R t H s dZ s , we think that a trivial equivalent of Theorem 2, replacing the integral with respectto Brownian motion with some integral with respect to some L´evy process, does not hold in thediscontinuous case. Finally the continuity of the process Z , and so of the Brownian motion, is akey ingredient in proving Lemma 5 below.We remark that Hypothesis A1 is not necessary if we change the thesis, not requesting that theBrownian motion W is measurable with respect to the natural filtration F Zt of Z . Furthermore ifwe require an invariance property with respect to random rotations stronger than Definition 1 weare able to modify the proof of Theorem 2 without using hypothesis A1 and suitably enlarging theprobability space where Z is defined. This stronger invariance needs the law of Z to be invariantwith respect to any random rotation predictable with respect to any filtration F t generated by F Zt and by an other filtration G t independent of F Zt . Until now, all the semimartingales for which weare able to prove the invariance according to Definition 1 satisfy this stronger notion of invariance.For this reason Theorem 2 might hold without hypothesis A1 and requesting (only) the invariancewith respect to Definition 1.Finally under suitable hypotheses on the process f t it is possible to prove that, if Z satisfies thethesis of Theorem 2, then Z is also invariant with respect to random rotations. For example if f t is measurable with respect to a filtration F Ht generated by a R h semimartingale H whose law isuniquely characterized by its characteristics and independent of W , we are able to use Theorem3.18 of [1] proving that R t f t dW t is invariant with respect to random rotations.The paper is organized into two sections. In Section 2 we introduce some notations, concepts andresults useful in the proof of Theorem 2. Section 3 contains some lemmas and the proof of ourmain result. In this section in order to fix the setting, we provide some notations and results about the randomrotations invariance of semimartingales.We consider the probability space Ω given by the Fr´echet space C ( R + , R n ) with the usual σ -algebra ˜ F of the Borel sets. In order to have a filtration on Ω we fix a probability measure P suchthat the coordinate process ω ( t ) = Z t , where ω ∈ Ω, is a semimartingale. In the following whenwe consider σ -algebra generated by some random variables or some processes we always mean theusual completed σ -algebra generated by these random variables or by these processes.Let B t be a predictable process with respect to the filtration F Zt taking values in O ( n ) and definethe process Z ′ it = Z t B ij,s dZ js . (1)3he process Z ′ t defines a measurable map Λ B : Ω → Ω ′ (where Ω ′ = C ( R + , R n )) in the followingway Λ B ( ω )( t ) = Z ′ t ( ω ) . We denote by P ′ = Λ B ∗ ( P ) the pushforward of the measure P with respect to Λ B . The canonicalprocess ω ′ ( t ) (where ω ′ ∈ Ω ′ ) with respect to the probability measure P ′ has exactly the same lawof Z ′ t . For this reason, in the following, with a slight abuse of notation, we identify the canonicalprocess ω ′ ( t ) with the process Z ′ t .If Z is invariant with respect to random rotations, for any B as above we have P ′ = P . We saythat Λ B is almost surely invertible if there exists a predictable measurable map Λ ′ B : Ω ′ → Ω,such that Λ B ◦ Λ ′ B = id Ω ′ almost surely with respect to P ′ and Λ ′ B ◦ Λ B = id Ω almost surely withrespect to P .Using the notion of characteristics of a semimartingale introduced in Section 1 we state the followingtheorem. Theorem 3
Let ( b t , A t ) be the characteristics of a continuous semimartingale Z . If Z is invariantwith respect to random rotations (according to Definition 1) then b it ( ω ) = Z t B ik,s (Λ ′ B ( ω )) db ks (Λ ′ B ( ω )) , (2) A ijt ( ω ) = Z t B ik,s (Λ ′ B ( ω )) B jℓ,s (Λ ′ B ( ω )) dA kℓs (Λ ′ B ( ω )) , (3) almost surely with respect to the measure P . Proof.
This is a special case of Theorem 3.8 in [1].In the following if Λ B is almost surely invertible and K ( ω ) is a random variable defined on Ωwe define Λ B ∗ ( K )( ω ′ ) = K (Λ ′ B ( ω ′ )) . A random variable K is said to be invariant with respect to the action of Λ B if Λ B ∗ ( K ) = K almostsurely with respect to P .If Z is a local martingale, under the hypothesis A1 it is simple to construct the Brownianmotion whose existence is stated in Theorem 2. Indeed, consider the square root p ˜ A t of thematrix ˜ A t . Since the matrix ˜ A t is almost surely invertible with respect to the measure P ⊗ dt , thematrix C = ( p ˜ A ) − is defined almost surely with respect to the measure P ⊗ dt . Furthermore C is integrable with respect to Z and the integral W it = Z t C ik,s dZ ks is a Brownian motion. Indeed W i are local martingales and (cid:2) W i , W j (cid:3) t = Z t C ik,s C jℓ,s ˜ A kℓs ds = δ ij t, thus, by L´evy characterization, W is an n dimensional Brownian motion. If B is a predictableprocess such that Λ B is invertible it is simple to study the action of Λ B on W i . In particular if W ′ is the Brownian motion obtained with the previous procedure from Z ′ we have thatΛ B ∗ ( W it ) = Z t Λ B ∗ ( B − ,ij,s ) dW ′ js . Proof of the main theorem
We start by proving the following two lemmas.
Lemma 4
Under the hypothesis A and B , if a continuous semimartingale Z with characteristics ( b, A ) is invariant with respect to random rotations then b t = 0 almost surely and there exists apredictable process F t such that A ijt = F t δ ij . Proof.
Under the hypothesis A for any ǫ > A ij + ǫI n is almost surely (with respectto the measure P ⊗ dt ) a symmetric strictly positive definite matrix. Using Proposition 1.8 of[4], there exists a predictable process B t taking values in O ( n ) such that B t · ( ˜ A t + ǫI n ) · B Tt is adiagonal matrix for any t and almost surely. Thus B t · ˜ A t · B Tt is a diagonal matrix for any t andalmost surely. By exploiting (1) this means that A ′ ijt = [ Z ′ i , Z ′ j ] t = Z t B ik,s B jℓ,s dA kℓs = Z t B ik,s B jℓ,s ˜ A kℓs ds, is diagonal. Since Z is invariant with respect to random rotations, the semimartingale Z has thesame law of Z ′ given by (1). In particular the quadratic variation matrix A of Z has the same lawof the quadratic variation A ′ of Z ′ . This means that A ijt = 0 = A ′ ij whenever i = j , i.e. A ijt isdiagonal.On the other hand if we choose a constant process B and A iit are not all almost surely equals wehave that A ′ ijt = R t B ik B jk ˜ A kks ds is not almost surely identically equal to zero when i = j . Thismeans that A ′ has not the same law of A and so Z ′ has not the same law of Z . Thus A ijt = F t δ ij for some increasing, positive and absolutely continuous process F .Suppose that b it is not almost surely zero: this means that Z it is not a local martingale with respectto the natural filtration F Zt . Under the hypothesis B there exists a predictable random process B t such that ( B t · ˜ b t ) i = 0 almost surely. Since the characteristics of Z ′ under the filtration F Zt (andnot, in general, under the natural filtration F Z ′ t of Z ′ ) are given by ˆ b t = R t B s · ˜ b s ds , this meansthat Z ′ i is a local martingale with respect to the filtration F Zt . On the other hand since Z ′ is F Z ′ t adapted, and since the filtration F Zt contains the filtration F Z ′ t , Z ′ i is a local martingale also withrespect to F Z ′ t . This means that the characteristic b ′ t of Z ′ with respect to F Z ′ t is such that b ′ it = 0almost surely and Z ′ cannot have the same law of Z . Thus we must have b t = 0 almost surely. Lemma 5
Under the hypotheses A , A1 and B let K be a random variable defined on Ω invariantwith respect to the action Λ B , for any B such that Λ B is almost surely invertible. Then K isindependent of the Brownian motion W constructed in Section 2. Proof. If Z is invariant with respect to random rotations and hypotheses A and B hold, byLemma 4 Z must be a local martingale with respect to the filtration F Zt , and so the process W defined in Section 2 is a well defined Brownian motion.We define a sequence of stopping times τ hk depending on the real parameter h >
0. Setting τ h = 0, τ h is defined as follows τ h = inf { t | k W t k ≥ h } , where k·k is the Euclidian norm of R n , while the stopping times τ hk ( k ≥
2) are defined by recursionas τ hk = τ hk − + inf { t | k W τ hk − + t − W τ hk − k > h } . B = ( B , B , ... ) ∈ O ( n ) ∞ be a sequence of (deterministic) rotations in O ( n ) and define B B ,ht = X k ∈ N B k I ( τ hk − ,τ hk ] ( t ) . Since τ hk are predictable stopping times, the process B B ,ht is predictable, and Λ B ,h := Λ B B ,h isinvertible with inverse Λ B − ,h defined by the random rotation˜ B B ,ht = X k ∈ N B − k I ( τ ′ hk − ,τ ′ hk ] ( t )where τ ′ hk are the stopping times defined on Ω ′ with the same definition of τ hk but using thetransformed Brownian motion W ′ instead of W .In order to prove that Λ B − ,h , defined above, is actually the inverse of Λ B ,h we exploit the factthat the map Λ B ′ , defined by the random rotation B ′ , is the inverse of the map Λ B ,h if and only if (cid:18)Z t B ′ ij,s dZ ′ js (cid:19) ◦ Λ B ,h = Z it (4) P almost surely. In particular Λ B − ,h is the inverse of Λ B ,h if and only if the relation (4) holdswith B ′ s = ˜ B B ,ht . If we are able to prove that τ ′ hk ◦ Λ B ,h = τ hk , using the fact that Z ′ t ◦ Λ B ,h = Z t B B ,hs · dZ s = X k ∈ N B k · ( Z τ ik ∧ t − Z τ ik − ∧ t ) , we obtain (cid:18)Z t ˜ B B ,hs · dZ ′ s (cid:19) ◦ Λ B ,h = X k ∈ N B − k · ( Z ′ ( τ ′ hk ◦ Λ B ,h ) ∧ t − Z ′ ( τ ′ hk − ◦ Λ B ,h ) ∧ t )= X k ∈ N B − k · B k · ( Z τ hk ∧ t − Z τ hk − ∧ t ) = Z t , proving in this way that equation (4) holds and thus that Λ B − ,h is the inverse of Λ B ,h .We now prove that τ ′ hk ◦ Λ B ,h = τ hk . Using the fact that W ′ t ◦ Λ B ,h = Z t B B ,hs · dW s = X k ∈ N B k · ( W τ hk ∧ t − W τ hk − ∧ t ) , we have that τ ′ h ◦ Λ B ,h ≤ τ h since k ( W ′ ◦ Λ B ,h ) τ h k = k B · W τ h k = k W τ h k = h. Using this result we have τ ′ h ◦ Λ B ,h = inf { t ≤ τ h | k W ′ t ◦ Λ B ,h k ≥ h } = inf { t ≤ τ h | k B · W t k ≥ h } = inf { t ≤ τ h | k W t k ≥ h } = τ h , and, with analogous reasoning we can prove that τ ′ hk ◦ Λ B ,h = τ hk . Therefore Λ B − ,h is the inverseof Λ B ,h and the stopping times τ hk are invariant with respect to Λ B ,h for any B ∈ O ( n ) ∞ .In order to prove the lemma we introduce a σ -algebra G h ⊂ F Z generated by G h = _ k ∈ N σ (cid:16) W τ hk (cid:17) = _ k ∈ N σ (cid:16) W τ hk − W τ hk − (cid:17) . B B ,h and the invariance properties of τ hk we haveΛ B ,h ∗ ( W iτ k − W iτ k − ) = B − ,ik,j ( W jτ ′ hk − W jτ ′ hk − ) . Since B i are invertible matrices we have that Λ B ,h ( G h ) = G ′ h where G ′ h = W k ∈ N σ (cid:16) W ′ τ ′ hk (cid:17) .Given a bounded continuous function f : R → R , we define K f,h = E [ f ( K ) |G h ] = K f,h (∆ h W , ∆ h W , ... )where K : R ∞ → R is a measurable map and ∆ h W i = W τ hi − W τ hi − . By the explicit expression ofthe inverse of Λ B ,h and the invariance property of τ hk we have thatΛ B ,h ( K f,h ) = K f,h ( B − · ∆ h W ′ , B − · ∆ h W ′ , ... ) . On the other hand Λ B ,h ∗ ( K f,h ) = Λ B ,h ∗ ( E P [ f ( K ) |G h ])= E Λ B ,h ∗ ( P ) [Λ B ,h ∗ ( f ( K )) | Λ B ,h ( G h )] . If Z is invariant with respect to random rotations and thus Λ B ,h ( P ) = P , and K satisfies thehypotheses of the lemma we haveΛ B ,h ∗ ( K f,h ) = E P [ f ( K ) |G ′ h ] = K f,h (∆ h W ′ , ∆ h W ′ , ... ) . This means that, for any B ∈ O ( n ) ∞ , K f,h (∆ h W ′ , ∆ h W ′ , ... ) = K f,h ( B − · ∆ h W ′ , B − · ∆ h W ′ , ... ) . Since the previous equality holds for any B ∈ O ( n ) ∞ the random variable K f,h depends only onthe random variables k ∆ h W ′ k k . On the other hand, by definition of τ ′ hk we have k ∆ h W ′ k k = h , andthus the random variable K f,h depends only on the deterministic parameter h , i.e. it is a constant.Therefore the random variable f ( K ) is independent in mean of the σ -algebra G h , but since f isany continuous function we have that K is independent of G h .The last step is to prove that we can approximate the Brownian motion W using G h measurablerandom variables with respect to almost surely convergence. This is sufficient for proving thelemma. Indeed, suppose that g : R k → R is a continuous bounded function, and let f be as above;if W h n r is a suitable sequence of G h n measurable random variables such that W h n r → W t r almostsurely, then E [ f ( K ) g ( W t , ..., W t r )] = E [ lim n → + ∞ f ( K ) g ( W h n , ..., W h n r )]= lim n → + ∞ E [ f ( K ) g ( W h n , ..., W h n r )] = E [ f ( K )] lim n → + ∞ E [ g ( W h n , ..., W h n r )]= E [ f ( K )] E [ lim n → + ∞ g ( W h n , ..., W h n r )] = E [ f ( K )] E [ g ( W t , ..., W t r )] , and this ensures that K is independent of the Brownian motion W .In the following we prove that W can be approximated by G h measurable random variables. Thegeneral case is a simple extension. First of all we note that τ hk − τ hk − form a sequence of independentidentically distributed random variables, since they depend all in the same way on the increments W τ hk − + t − W τ hk − which are independent and identically distributed. Furthermore by the rescalingproperty of Brownian motion, we have that the law of τ hk − τ hk − coincides with the law of h τ .7n the other hand (see [14]) τ is an L random variable with mean µ n = E [ τ ] = n and variance σ n = var( τ ) = n ( n +2) . Let k h = j µ n h k ∈ N be the maximum integer less then µ n h . Thus wehave E [ τ hk h ] = k h X i =1 E [ τ hi − τ hi − ] = µ n h (cid:22) µ n h (cid:23) h → −→ τ hk h ) = k h X i =1 var( τ hi − τ hi − ) = σ n h (cid:22) µ n h (cid:23) h → −→ . Since τ hk h converges in probability to 1 as h →
0, choosing a suitable subsequence h k → τ hk h converges to 1 almost surely. Since the Brownian motion is almost surely continuous wehave that W τ hnkhn → W almost surely. So W can be approximated by an almost surely convergentsequence of G h random variables and the thesis is proved. Proof of Theorem 2.
By Lemma 4 the matrix A of quadratic covariation of Z is of the form A ijt = F t δ ij , where F t = R t f s ds for some predictable random process f t . If B t is any predictableprocess taking values in O ( n ) and such that Λ B is invertible, the quadratic variation matrix A ′ of Z ′ is A ′ ijt = Z t Λ B ∗ ( B ik,s B jℓ,s δ kℓ f s ) ds = Λ B ∗ ( F t ) δ ij . By Theorem 3 we have F t ( ω ′ ) = Λ B ∗ ( F t )( ω ′ ) , therefore the random process F is invariant with respect to the action of Λ B for B as above. Ifwe fix some times t , ..., t k and a continuous function g : R k → R , we can apply Lemma 5 to therandom variable g ( F t , ..., F t k ) proving that it is independent of the Brownian motion W . Since k ∈ N , t i ∈ R + and the continuous function g are arbitrary we have proved that the process F is independent of the Brownian motion W . This implies that the process f is independent of theBrownian motion W . Since, by the construction of W , Z it = R t f s dW is , the theorem is proved. Acknowledgements
The author would like to thank Prof. Paola Morando and Prof. Stefania Ugolini for their usefulcomments, suggestions and corrections of the first draft of the paper. This work was supportedby Gruppo Nazionale Fisica Matematica (GNFM-INdAM) through the grant: “Progetto Giovani,Symmetries and reduction for differential equations: from the deterministic to the stochastic case”.
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