Representations of Gaussian measures that are equivalent to Wiener measure

Representations of Gaussian measures that are equivalent to Wiener measure Patrick Cheridito Departement für Mathematik, ETHZ, 89 Zürich, Switzerland. E-mail: dito@math.ethz.ch Summary. We summarize results on the representation of Gaussian measures that are equivalent to Wiener measure and discuss consequences for the law of the sum of a Brownian motion and an independent fractional Brownian motion. Introduction Let < T. For < T <, we set I T = [, T ], and I = [,. By CI T we denote the space of real-valued, continuous functions on I T. The coordinates process X t t IT on CI T is given by It generates the σ-algebra X t ω = ωt, ω CI T, t I T. B T := σ { X 1 t B : t I T, B an open subset of R }. By W we denote Wiener measure on CI T, B T. We call a probability measure Q on CI T, B T a Gaussian measure if X t t IT is a Gaussian process with respect to Q. Such a measure is determined by its mean M Q t := E Q [X t ], t I T, and its covariance Γ Q [ ts := E Q Xt M Q t Xs Ms Q ], t, s IT. We need some properties of integral operators induced by L -kernels. The proofs of the following facts can be found in Smithies 1958. We denote by L I T and L IT the Hilbert spaces of equivalence classes of real-valued, square-integrable functions on I T and IT, respectively. An L - kernel is an element f L IT. It induces a Hilbert Schmidt operator

Patrick Cheridito given by F at = F : L I T L I T, ft, s as ds, t I T, a L I T. The spectrum SpecF consists of at most countably many points. Every non-zero value in SpecF is an eigenvalue of finite multiplicity. If λ j N, N N { }, is the family of non-zero eigenvalues of F, repeated according to their multiplicity, then N λ j <. Let f, g L IT with corresponding Hilbert Schmidt operators F and G, respectively. Then, the kernel ft, u gu, s du, t, s I T, is again in L IT and induces the operator product F G. If f L IT and 1 / SpecF, then there exists a unique kernel f L IT such that the corresponding operator F satisfies Id F = Id F 1. Since f is usually called the resolvent kernel of f for the value 1, we call f the negative resolvent kernel of f. It is the unique L -kernel f that solves the equation ft, s + ft, s = It is also the unique L -kernel that solves the equation ft, s + ft, s = ft, u fu, s du, t, s I T. 1 ft, u fu, s du, t, s I T. If f is symmetric, F is self-adjoint. Therefore, all eigenvalues λ j are real, there exists a sequence e j N of orthonormal eigenfunctions in L I T, and f can be represented as ft, s = N λ j e j t e j s, where the series converges in L I T. It follows that if f is symmetric and 1 / SpecF, then

ft, s = Representations of Gaussian measures 3 N In particular, f is again symmetric. We set λ j 1 λ j e j t e j s. S 1 T := { f L I T : f is symmetric and SpecF, 1 }. It can be seen from that if f ST 1, then f ST 1 as well. A kernel g L IT is called a Volterra kernel if gt, s = for all s > t. In this case the corresponding operator G is quasi-nilpotent, that is, the spectral radius sup{ λ : λ SpecG} = lim inf n Gn 1/n is zero. Hence, the negative resolvent kernel g exists, and it can be shown that g is also a Volterra kernel. We set V T := { g L I T : g is a Volterra kernel }. 1 The representations of Shepp and Hitsuda In the following theorem we recapitulate the statements of Theorems 1 and 3 of Shepp 1966. For a, b L I T, we set a, b := as bs ds. Theorem 1 Shepp. i Let f S 1 T and a L I T. Then E W [exp ] fs, u dx u dx s + as dx s 1 exp a, Id F a = N expλ j/ <, 1 λ j where λ j N, N N { }, are the non-zero eigenvalues of the Hilbert Schmidt operator induced by f and F is the Hilbert Schmidt operator corresponding to the negative resolvent kernel f of f. Furthermore, the probability measure N Q = expλ j/ 1 λ j 1 exp a, Id F a exp fs, u dx u dx s + as dx s W 3

4 Patrick Cheridito is a Gaussian measure on CI T, B T with mean and covariance M Q t = Γ Q ts = t s Id F as ds, t IT, 4 fu, v dv du, t, s I T. ii Let Q be a Gaussian measure on CI T, B T that is equivalent to W. Then there exist unique f S 1 T and a L I T such that Q has the representation 3. Remark 1. a We call 3 the Shepp representation of the Gaussian measure Q. b Let k L IT be symmetric. Then, k can be written as kt, s = L -lim λ j e j t e j s, n λ j 1/n where N N { }, λ j N is a sequence of real numbers such that N λ j <, and the e j s are orthonormal in L I T. Hence, under W, ks, u dx u dx s = L -lim n = L -lim n λ j 1/n λ j 1/n λ j λ j e j s e j u dx u dx s e j s dx s 1. Since the random variables e js dx s are independent standard normal, it follows that ] E W [exp ks, u dx u dx s < if and only if k ST 1. c Let k L IT be symmetric with corresponding Hilbert Schmidt operator K. Then the map Γ ts = t s ku, v dv du, t, s I T, 5 is the covariance function of a Gaussian process if and only if it is positive semi-definite, that is, n n n c j 1 [,tj], Id K c j 1 [,tj] = c j Γ tjt l c l, 6 j,l=1

Representations of Gaussian measures 5 for all n N, {t 1,..., t n } I T and c R n. Since the functions of the form n c j 1 [,tj], n N, {t 1,..., t n } I T, c R n, are dense in L I T, condition 6 is equivalent to c, Id Kc for all c L I T. Hence, 5 is the covariance function of a Gaussian process if and only if SpecK, 1]. Corollary 1. i Let B t t IT be a Brownian motion and Z t t IT an independent Gaussian process. Then the law of B t + Z t t IT is equivalent to W if and only if there exist m L I T and k L IT such that and E[Z t ] = CovZ t, Z s = ms ds, t I T, 7 ku, v dv du, t, s I T. 8 ii Let Q be a Gaussian measure on CI T, B T that is equivalent to W and f ST 1 the kernel that satisfies 3. Then Q is the law of the sum of a Brownian motion and an independent Gaussian process if and only if Spec F, ]. Proof. i Denote the law of B t + Z t t IT and by Q. Then, M Q t = E[B t + Z t ] = E[Z t ], t I T, Γ Q ts = CovB t + Z t, B s + Z s = t s + CovZ t, Z s, t, s I T. It follows from Theorem 1 that Q is equivalent to W if and only if 7 and 8 hold and SpecK 1,. If 8 holds, then it can be shown as in Remark 1.c that SpecK [,. This proves i. ii If Q is the law of the sum of a Brownian motion and an independent Gaussian process, then it follows as in Remark 1.c that Spec F, ]. If Spec F, ], then the function fu, v du dv, t, s I T, 9 is symmetric and positive semidefinite. Hence, there exists a Gaussian process Z t t IT with covariance 9 and mean M Q. Let B t t IT be an independent Brownian motion. Then, Q is the law of B t + Z t t IT.

6 Patrick Cheridito Example 1. For fixed T, and α R \ {}, let Q α H be the law of the Gaussian process B t + αbt H, t [, T ], where B t t [,T ] is a Brownian motion and B H t t [,T ] an independent fractional Brownian motion with Hurst parameter H, 1], that is, B H t t [,T ] is a Gaussian process with mean and covariance Cov B H t, B H s Since for H 1/, 1], 1 t H + s H t s H = HH 1 1 = t H + s H t s H, t, s [, T ]. u v H dv du, t, s [, T ], it follows from Corollary 1.i that Q α H is equivalent to W if and only if H 3/4, 1]. This assertion is part of Theorem 1.7 in Cheridito 1b, which was proved differently. For H 3/4, 1], let λ H be the largest eigenvalue of the operator K H corresponding to the L -kernel k H t, s = HH 1 t s H, t, s [, T ]. Since K H is positive semi-definite, λ H is equal to the operator norm K H > of K H. If β, 1/λ H, then t s β t H + s H t s H, t, s [, T ], is the covariance function of a centred Gaussian process equivalent to Brownian motion which cannot have the same law as the sum of a Brownian motion and an independent Gaussian process. t s 1 λ H t H + s H t s H, t, s [, T ], is the covariance function of a centred Gaussian process that is neither equivalent to Brownian motion nor equal in distribution to the sum of a Brownian motion and an independent Gaussian process. In the following theorem we reformulate the statements of Theorems 1 and of Hitsuda 1968 note that in the last line of Theorem in Hitsuda 1968 X t should be replaced by Y t. Theorem Hitsuda. i Let g V T and b L I T. Then E W [exp gs, u dx u + bs dx s 1 ] s gs, u dx u + bs ds = 1,

and the probability measure Q = exp gs, u dx u + bs dx s 1 Representations of Gaussian measures 7 s gs, u dx u + bs ds W 1 is a Gaussian measure on CI T, B T. Furthermore, the process B t = X t gs, u dx u + bs ds, t I T, 11 is a Brownian motion with respect to Q, and X t = B t gs, u db u ds + Id G bs ds, t IT, 1 where g is the negative resolvent kernel of g and G the corresponding Hilbert Schmidt operator. ii Let Q be a Gaussian measure on CI T, B T that is equivalent to W. Then there exist unique g V T and b L I T such that Q has the representation 1. Remark. a We call 1 the Hitsuda representation of the Gaussian measure Q. b It follows from 11 and 1 that F B t = σ{b s : s t} = σ{x s : s t} = F X t, t I T. Therefore, 1 is the canonical semimartingale decomposition of X in its own filtration. We call it the Hitsuda representation of the Gaussian process X t t IT, Q. Relations between the representations of Shepp and Hitsuda Theorem 3. Let Q be a Gaussian measure on CI T, B T that is equivalent to W and f, f, g, g, a, b the corresponding objects from Theorems 1 and. Then the following relations hold: Id F a = Id G b; 13 E W [exp ft, s = gt, s ft, s = gt, s ] 1 fs, u dx u dx s = exp t gs, u du ds ; 14 gu, t gu, s du, s t T ; 15 gt, u gs, u du, s t T ; 16

8 Patrick Cheridito ft, s + gt, s = ft, s + gt, s = s ft, u gu, s du, s t T ; 17 gt, u fu, s du, s t T. 18 Proof. Relation 13 follows by comparing 4 and 1. To prove the other relations we let Q be the Gaussian measure on CI T, B T with mean and the same covariance as Q. It follows from Theorems 1 and that and Q = Q = exp N exp λj 1 λj exp gs, u dx u dx s 1 fs, u dx u dx s W gs, u dx u ds W. Now, relation 16 follows from the only if part of the proof of Proposition in Hitsuda 1968 note that in the corresponding equation in Hitsuda 1968 a variable u should be replaced by v. The relations 14, 15, 18 are equivalent to the equations 31d, 38, 1 in Kailath 197, respectively in equation 38 of Kailath 197 there is a wrong sign. Let G denote the adjoint of G and Id G the adjoint of Id G. Then, relation 17 can be deduced from relation 16 as follows: 16 Id F = Id G Id G Id = Id F Id G Id G Id G = Id F Id G 17. Remark 3. a Relation 15 is equivalent to Id F = Id G Id G. Relation 18 is equivalent to Id G = Id GId F. b In all four equations 15 18, either kernel is uniquely determined by the other. Example 1 continued. Let T,, H 3/4, 1], α R \ {} and Q α H be the mean zero Gaussian measure from Example 1. Hence, Γ Qα H ts = t s + α t H + s H t s H, t, s [, T ]. ft, s = α HH 1 t s H, t, s [, T ]. 19

For the special case H = 1, 19 reduces to Representations of Gaussian measures 9 ft, s = α, t, s [, T ], and the equations 1, 18 and 16 can easily be solved. One obtains: ft, s = α α α 1 + α, gt, s = T 1 + α, gt, s = t 1 + α, t, s [, T ]. s If H 3/4, 1, it is less obvious how to find explicit expressions for the functions f, g, and g. The equations 1, 18 and 16 take the forms and ft, s + α HH 1 ft, u u s H du = α HH 1 t s H, t, s [, T ], gt, s + α HH 1 α HH 1 t s H = gt, s + gt, u u s H du = α HH 1 t s H, t, s [, T ], 1 gt, u gs, u du, s t T, respectively. For certain values of H and α, equations and are solved in Sections 4.7 and 4.8 of Cheridito 1a. Equation 1 can be solved similarly. References 1. Cheridito, P. 1a. Regularizing Fractional Brownian Motion with a View towards Stock Price Modelling. doctoral dissertation. http://www.math.ethz.ch/~dito. Cheridito, P. 1b. Mixed fractional Brownian motion. Bernoulli 76, p. 913 934. 3. Hitsuda, M. 1968. Representation of Gaussian processes equivalent to Wiener process. Osaka Journal of Mathematics 5, p. 99 31. 4. Kailath, T. 197. Likelihood ratios for Gaussian processes. IEEE Transactions on Information Theory IT-16, No. 3, p. 76 88. 5. Shepp, L.A. 1966. Radon Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37, p. 31 354. 6. Smithies, F. 1958. Integral Equations. Cambridge Univ. Press, London and New York.