Generalized Gaussian Bridges

Transcription

1 TOMMI SOTTINEN ADIL YAZIGI Generalized Gaussian Bridges PROCEEDINGS OF THE UNIVERSITY OF VAASA WORKING PAPERS 4 MATHEMATICS 2 VAASA 212

2 Vaasan yliopisto University of Vaasa PL 7 P.O. Box 7 (Wolffintie 34 FI 6511 VAASA Finland ISBN ISSN = Proceedings of the University of Vaasa. Working Papers University of Vaasa

3 III Publisher Date of publication Vaasan yliopisto June 212 Author(s Type of publication Tommi Sottinen Working Papers Adil Yazigi Name and number of series Proceedings of the University of Vaasa Contact information University of Vaasa Department of Mathematics and Statistics P.O. Box 7 FI 6511 Vaasa, Finland ISBN ISSN Number Language of pages 27 English Title of publication Generalized Gaussian Bridges Abstract A generalized bridge is the law of a stochastic process that is conditioned on linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. In the canonical representation the filtrations and the linear spaces generated by the bridge process and the original process coincide. In the orthogonal representation the bridge is constructed from the entire path of the underlying process. The orthogonal representation is given for any continuous Gaussian process but the canonical representation is given only for so-called prediction-invertible Gaussian processes. Finally, we apply the canonical bridge representation to insider trading by interpreting the bridge from an initial enlargement of ltration point of view. Keywords Canonical representation, enlargement of ltration, fractional Brownian motion, Gaussian process, Gaussian bridge, Hitsuda representation, insider trading, orthogonal representation, prediction-invertible process, Volterra process.

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5 V Contents 1. INTRODUCTION ABSTRACT WIENER INTEGRALS AND RELATED HILBERT SPACES ORTHOGONAL GENERALIZED BRIDGE REPRESENTATION CANONICAL GENERALIZED BRIDGE REPRESENTATION APPLICATION TO INSIDER TRADING REFERENCES... 2

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7 GENERALIZED GAUSSIAN BRIDGES TOMMI SOTTINEN AND ADIL YAZIGI Abstract. A generalized bridge is the law of a stochastic process that is conditioned on linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. In the canonical representation the filtrations and the linear spaces generated by the bridge process and the original process coincide. In the orthogonal representation the bridge is constructed from the entire path of the underlying process. The orthogonal representation is given for any continuous Gaussian process but the canonical representation is given only for so-called prediction-invertible Gaussian processes. Finally, we apply the canonical bridge representation to insider trading by interpreting the bridge from an initial enlargement of filtration point of view. Mathematics Subject Classification (21: 6G15, 6G22, 91G8. Keywords: Canonical representation, enlargement of filtration, fractional Brownian motion, Gaussian process, Gaussian bridge, Hitsuda representation, insider trading, orthogonal representation, prediction-invertible process, Volterra process. 1. Introduction Let X = (X t t [,T be a continuous Gaussian process with positive definite covariance function R, mean function m of bounded variation, and X = m(. We consider the conditioning, or bridging, of X on N linear functionals G T = [G i T N i=1 of its paths: (1.1 G T (X = [ g(t dx t = N g i (t dx t. i=1 We assume, without any loss of generality, that the functions g i are linearly independent. Indeed, if this is not the case then the linearly dependent, or redundant, components of g can simply be removed from the conditioning (1.2 below without changing it. The integrals in the conditioning (1.1 are the so-called abstract Wiener integrals defined properly in Definition 2.5 later. Informally, the generalized Gaussian bridge X g;y is (the law of the Gaussian process X conditioned on the set (1.2 { } g(t dx t = y = N { i=1 g i (t dx t = y i }. The rigorous definition is given in Definition 1.3 later. Date: August 2, 212. A. Yazigi was funded by the Finnish Doctoral Programme in Stochastics and Statistics. 1

8 2 Working Papers For the sake of convenience, we will work on the canonical filtered probability space (Ω, F, F, P, where Ω = C([, T, F is the Borel σ -algebra on C([, T with respect to the supremum norm, and P is the Gaussian measure corresponding to the Gaussian coordinate process X t (ω = ω(t: P = P[X. The filtration F = (F t t [,T is the intrinsic filtration of the coordinate process X that is augmented with the null-sets and made right-continuous Definition. The generalized bridge measure P g;y is the regular conditional law [ P g;y = P g;y T [X = P X g(t dx t = y. A representation of the generalized Gaussian bridge is any process X g;y satisfying [ P [X g;y = P g;y T [X = P X g(t dx t = y. Note that the conditioning on the P-null-set (1.2 in Definition 1.3 above is not a problem, since the canonical space of continuous processes is small enough to admit regular conditional laws. Also note that as a measure P g;y the generalized Gaussian bridge is unique, but it has several different representations X g;y. Indeed, for any representation of the bridge one can combine it with any P-measure-preserving transformation to get a new representation. In this paper we provide two different representations X g;y. The first representation, given by Theorem 3.1, is called the orthogonal representation. This representation is a simple consequence of orthogonal decompositions of Hilbert spaces associated with Gaussian processes and it can be constructed for any continuous Gaussian process for any conditioning functionals. The second representation, given by Theorem 4.24, is called the canonical representation. This representation is more interesting but also requires more assumptions. The canonical representation is dynamically invertible in the sense that the linear spaces L t (X and L t (X g;y (see Definition 2.1 later generated by the process X and its bridge representation X g;y coincide for all times t [, T. Moreover, the bridge X g;y can be interpreted as the original process X with an added information drift that bridges the process at the final time T. This dynamic drift interpretation should turn out to be useful in applications. We give one such application in connection to insider trading in Section 5. This application is, we must admit, a bit classical. On earlier work related to stochastic bridges, we would like to mention first Alili [1, Baudoin [2 and Gasbarra et al. [7. In [1 generalized Brownian bridges were considered. It is our opinion that our article extends [1 considerably, although we do not consider the non-canonical representations of [1. The article [2 is, in a sense, more general than this article, since we condition on fixed values y, but in [2 the conditioning is on a probability law. However, in [2 only the Brownian bridge was considered. In that sense our approach is more general. In [7 bridges were studied in

9 Working Papers 3 the same general Gaussian setting as in this article. In this article, however, we generalize the results of [7 to generalized bridges. Secondly, we would like to some related works on Markovian and Lévy bridges: [5, 6, 8, 1. This paper is organized as follows. In Section 2 we recall some Hilbert spaces related to Gaussian processes. In Section 3 we give the orthogonal representation for the generalized bridge in the general Gaussian setting. Section 4 deals with the canonical bridge representation. First we give the representation for Gaussian martingales. Then we introduce the so-called prediction-invertible processes and develop the canonical bridge representation for them. Then we consider Gaussian Volterra processes, such as the fractional Brownian motion, as examples of prediction-invertible processes. Finally, in Section 5 we apply the bridges to insider trading. Indeed, the bridge process can be understood from the initial enlargement of filtration point of view. 2. Abstract Wiener Integrals and Related Hilbert Spaces In this section X = (X t t [,T is a continuous (and hence separable Gaussian process with positive definite covariance R, mean zero and X =. Definitions 2.1 and 2.2 below give us two central separable Hilbert spaces connected to separable Gaussian processes Definition. Let t [, T. The linear space L t (X is the Gaussian closed linear subspace of L 2 (Ω, F, P generated by the random variables X s, s t. The linear space is a Gaussian Hilbert space with the inner product Cov[,. Note that since X is continuous, R is also continuous, and hence L t (X is separable, and any orthogonal basis (ξ n n=1 of L t(x is a collection of independent standard normal random variables. (Of course, since we chose to work on the canonical space, L 2 (Ω, F, P is itself a separable Hilbert space Definition. Let t [, T. The abstract Wiener integrand space Λ t (X is the completion of the linear span of the indicator functions 1 s := 1 [,s, s t, under the inner product, extended bilinearly from the relation 1 s, 1 u = R(s, u. The elements of the abstract Wiener integrand space are equivalence classes of Cauchy sequences (f n n=1 of piecewise constant functions. The equivalence of (f n n=1 and (g n n=1 means that where =,. f n g n, as n, 2.3. Remark. (i The elements of Λ t (X cannot in general be identified with functions as pointed out e.g. by Pipiras and Taqqu [14 for the case of fractional Brownian motion with Hurst index H > 1/2.

10 4 Working Papers However, if R is of bounded variation one can identity the function space Λ t (X Λ t (X: { t } Λ t (X = f R [,t ; f(sf(u R (ds, du <. (ii While one may want to interpet that Λ s (X Λ t (X for s t it may happen that f Λ t (X, but f1 s Λ s (X. Indeed, it may be that f1 s > f. See Bender and Elliott [3 for an example in the case of fractional Brownian motion. The space Λ t (X is isometric to L t (X. Indeed, the relation (2.4 I X t [1 s := X s, s t, can be extended linearly into an isometry from Λ t (X onto L t (X Definition. The isometry It X : Λ t (X L t (X extended from the relation (2.4 is the abstract Wiener integral. We denote f(s dx s := It X [f. 3. Orthogonal Generalized Bridge Representation Denote by g the matrix [ g ij := g i, g j := Cov g i (t dx t, g j (t dx t. Note that g does not depend on the mean of X nor on the conditioned values y: g depends only on the conditioning functions g = [g i N i=1 and the covariance R. Also, since g i s are linearly independent and R is positive definite, the matrix g is invertible Theorem. The generalized Gaussian bridge X g;y can be represented as ( (3.2 X g;y t = X t 1 t, g g 1 g(u dx u y. Moreover, any generalized Gaussian bridge X g;y is a Gaussian process with E [ ( X g;y T t = m(t 1 t, g g 1 g(u dm(u y, Cov [ X g;y t, Xs g;y = 1 t, 1 s 1 t, g g 1 1 s, g. Proof. It is well-known (see, e.g., [15, p. 34 from the theory of multivariate Gaussian distributions that conditional distributions are Gaussian

11 Working Papers 5 with T E [X t g(udx u = y = m(t + 1 t, g g 1 (y T Cov [X t, X s g(u dx u = y = 1 t, 1 s 1 t, g g 1 1 s, g. g(u dm(u, The claim follows from this Corollary. Let X be a centered Gaussian process with X = and let m be a function of bounded variation. Let X g := X g; be a bridge where the conditional functionals are conditioned to zero. Then (X + m g;y t = X g t + (m(t 1 t, g g 1 g(u dm(u + 1 t, g g 1 y Remark. Corollary 3.3 tells us how to construct, by adding a deterministic drift, a general bridge from a bridge that is constructed from a centered process with conditioning y =. So, in what follows, we shall almost always assume that the process X is centered, i.e. m(t =, and all conditionings are to y = Example. Let X be a zero mean Gaussian process with covariance R. Consider the conditioning on the final value and the average value: 1 T X T =, X t dt =. This is a generalized Gaussian bridge. Indeed, 1 T X T = X t dt = 1 dx t =: T t T dx t =: g 1 (t dx t, g 2 (t dx t.

12 6 Working Papers Now, and 1 t, g 1 = E [X t X T = R(t, T, [ 1 T 1 t, g 2 = E X t X s ds = 1 R(t, s ds, T T g 1, g 1 = E [X T X T = R(T, T, [ 1 T g 1, g 2 = E X T X s ds = 1 R(T, s ds, T T [ 1 T g 2, g 2 = E X s ds 1 X u du = 1 T T T 2 g = 1 T 2 Thus, by Theorem 3.1, g 1 = 1 [ g R(s, u dsdu, R(T, T R(s, u R(T, sr(t, u ds du, g 2, g 2 g 1, g 2 g 1, g 2 g 1, g 1 X g t = X t 1 t, g 1 g 2, g 2 1 t, g 2 g 1, g 2 g 1 t, g 2 g 1, g 1 1 t, g 1 g 1, g 2 g = X t. g 1 (t dx t g 2 (t dx t R(t, T R(s, u R(t, sr(t, sds du R(T, T R(s, u R(T, sr(t, u ds dux T T R(T, T R(t, s R(t, T R(T, sds R(T, T R(s, u R(T, sr(t, u ds du T t T dx t Remark. (i The conditioning on N conditions can also be done iteratively as follows: Let X n := X g 1,...,g n;y 1,...,y n and let X := X be the original process. Then the orthogonal generalized bridge representation X N can be constructed from the rule Xt n = Xt n 1 1 [ t, g n T n 1 g n (u dxu n 1 y n, g n, g n n 1 where, n 1 is the inner product in L T (X n 1. (ii If the conditioning variables g j are indicator functions 1 tj then the corresponding generalized bridge is a multibridge. That is, it is pinned down to values y j at points t j. For the multibridge X N = X 1t 1,...,1t N ;y 1,...,y N the orthogonal bridge decomposition can be constructed from the iteration X t = X t, X n t = X n 1 t R n 1(t, t n R n 1 (t n, t n [ X n 1 t n y n,

13 where R (t, s = R(t, s, R n (t, s = R n 1 (t, s R n 1(t, t n R n 1 (t n, s. R n 1 (t n, t n Working Papers 7 4. Canonical Generalized Bridge Representation The problem with the orthogonal bridge representation (3.2 of X g;y is that in order to construct it at any point t [, T one needs the whole path of the underlying process X up to time T. In this section we construct a bridge representation that is canonical in the following sense: 4.1. Definition. The bridge X g;y is of canonical representation if, for all t [, T, X g;y t L t (X and X t L t (X g;y Remark. Since the conditional laws of Gaussian processes are Gaussian and Gaussian spaces are linear, the assumptions X g;y t L t (X and X t L t (X g;y of Definition 4.1 are the same as assuming that X g;y t is Ft X -measurable and X t is Ft Xg;y -measurable (and, consequently, Ft X = Ft Xg;y. This fact is very special to Gaussian processes. Indeed, in general conditioned processes such as generalized bridges are not linear transformations of the underlying process. We shall require that the restricted measures P g,y t := P g;y F t and P t := P F t are equivalent for all t < T (they are obviously singular for t = T. To this end we assume that the matrix [( (4.3 g ij (t := E G i T (X G i t(x (G j T (X Gj t (X [ = E g i (s dx s g j (s dx s is invertible for all t < T. t 4.4. Remark. On notation: in the previous section we considered the matrix g, but from now on we consider the function g (. Their connection is of course g = g (. We hope that is overloading of notation does not cause confusion to the reader. Gaussian Martingales. We first construct the canonical representation when the underlying process is a continuous Gaussian martingale M with strictly increasing bracket M and M =. Note that the bracket is strictly increasing if and only if the covariance R is positive definite. Indeed, for Gaussian martingales we have R(t, s = Var(M t s = M t s. Define a Volterra kernel (4.5 l g (t, s := g (t g 1 (t g(s. Note that the kernel l g depends on the process M through its covariance,, and in the Gaussian martingale case we have g ij (t = t t g i (sg j (s d M s.

14 8 Working Papers The following Lemma 4.6 is the key observation in finding the canonical generalized bridge representation. Actually, it is a multivariate version of Proposition 6 of [ Lemma. Let l g be given by (4.5 and let M be a continuous Gaussian martingale with strictly increasing bracket M and M =. Then the Radon-Nikodym derivative dp g t /dp t can be expressed in the form { dp g s t = exp l g (s, u dm u dm s 1 ( s } 2 l g (s, u dm u d M s dp t 2 for all t [, T. Proof. Let p(y; µ, Σ := be the Gaussian density on R N { 1 (2π N/2 exp 1 } Σ 1/2 2 (y µ Σ 1 (y µ and let α g t (dy := P [G T (M dy F M t be the conditional law of the conditioning functionals G T (M = g(s dm s given the information F M t. First, by the Bayes formula, we have dp g t = dαg t dp t dα g (. Second, by the martingale property, we have ( dα g p ; G t (M, g (t t dα g ( = (, p ; G (M, g ( where we have denoted G t (M = g(s dm s. Third, denote ( p ; G t (M, g (t ( =: p ; G (M, g ( ( 1 g ( 2 exp {F (t, Mt F (, M }, g (t with F (t, M t = 1 ( ( g(s dm s g 1 ( g(s dm s. 2 Then, straightforward differentiation yields F s (s, M s ds = 1 ( s 2 l g (s, u dm u d M s, F x (s, M s dm s = s 2 F x 2 (s, M s d M s = log l g (s, u dm u dm s, ( g (t g ( 1 2

15 Working Papers 9 and the form of the Radon Nikodym derivative follows by applying the Itô formula Corollary. The canonical bridge representation M g satisfies the stochastic differential equation (4.8 dm t = dm g t l g (t, s dms g d M t, where l g is given by (4.5. Moreover M = M g. Proof. The claim follows by using the Girsanov s theorem Remark. (i Note that s l g (s, u 2 du ds <. In view of (4.8 this means that the processes M and M g are equivalent in law on [, T. Indeed, equation (4.8 can be viewed as the Hitsuda representation between two equivalent Gaussian processes, cf. Hida and Hitsuda [9. Also note that s l g (s, u 2 du ds = meaning that the measures P and P g are singular on [, T. (ii In the case of y, the formula (4.8 takes the form ( (4.1 dm t = dm g;y t + g (t g 1 (ty l g (t, s dms g;y d M t. Next we solve the stochastic differential equation (4.8 of Corollary 4.7. In general, solving a Volterra Stieltjes equation like (4.8 in a closed form is difficult. Of course, the general theory of Volterra equations suggests that the solution will be of the form (4.13 of Theorem 4.11 below, where l g is the resolvent kernel of l g determined by the resolvent equation (4.14 given below. Also, the general theory suggests that the resolvent kernel can be calculated implicitly by using the Neumann series. In our case the kernel l g is a quadratic form that factorizes in its argument. This allows us to calculate the resolvent l g explicitly as (4.12 below Theorem. Let s t [, T. Define the Volterra kernel (4.12 l g (t g(t, s := l g (t, s g (s = g (tg (t g 1 (t Then the bridge M g has the canonical representation (4.13 dm g t = dm t i.e., (4.13 is the solution to (4.8. g(s g (s. l g(t, s dm s d M t,

16 1 Working Papers Proof. Equation (4.13 is the solution to (4.8 if the kernel l g satisfies the resolvent equation (4.14 l g (t, s + l g(t, s = s l g (t, ul g(u, s d M u. Indeed, suppose (4.13 is the solution to (4.8. This means that ( dm t = dm t l g(t, s dm s d M t l g (t, s ( dm s s or, in the integral form, by using the Fubini s theorem, M t = M t + s s s s l g(u, s d M u dm s l g (u, s d M u dm s u l g(s, u dm u d M s d M t, l g (s, vl g(v, ud M v d M u dm s. The resolvent criterion (4.14 follows by identifying the integrands in the d M u dm s -integrals above. Finally, let us check that the resolvent equation (4.14 is satisfied with l g and l g defined by (4.5 and (4.12, respectively: s since l g (t, ul g(u, s d M u = s g (t g 1 (tg(u g (ug (u g 1 g(s (u g (s d M u = g (t g 1 g(s (t g (s s g(u g (ug (u g 1 (u d M u = g (t g 1 g(s (t g 1 (u g (ud g (u g (s s = g (t g 1 g(s ( (t g (t g (s g (s = g (t g 1 g (t (tg(s g (s g (t g 1 (tg(s = l g(t, s + l g (t, s, d g (t = g (tg(td M t. So, the resolvent equation (4.14 holds. Gaussian Prediction-Invertible Processes. Let us now consider a Gaussian process X that is not a martingale. For a Gaussian process X its prediction martingale is the process ˆX defined as ˆX t = E [ X T Ft X.

17 Working Papers 11 Since for Gaussian processes ˆX t L t (X, we may write, at least formally, that ˆX t = p(t, s dx s, where the abstract kernel p depends also on T (since ˆX depends on T. In Definition 4.15 below we assume that the kernel p exists as a real, and not only formal, function. We also assume that the kernel p is invertible Definition. A Gaussian process X is prediction-invertible if there exists a kernel p such that its prediction martingale ˆX is continuous, can be represented as ˆX t = p(t, s dx s, and there exists an inverse kernel p 1 such that, for all t [, T, p 1 (t, L 2 ([, T, d ˆX and X can be recovered from ˆX by X t = p 1 (t, s d ˆX s Remark. In general it seems to be a difficult problem to determine whether a Gaussian process is prediction-invertible or not. In the discrete time non-degenerate case all Gaussian processes are prediction-invertible. In continuous time the situation is more difficult, as Example 4.17 below illustrates. Nevertheless, we can immediately see that if the centered Gaussian process X with covariance R is prediction-invertible, then the covariance must satisfy the relation R(t, s = s p 1 (t, u p 1 (s, u d ˆX u, where ˆX u = Var (E [X T F u. However, this criterion does not seem to be very helpful in practice Example. Consider the Gaussian slope X t = tξ, t [, T, where ξ is a standard normal random variable. Now, if we consider the raw filtration Gt X = σ(x s ; s t, then X is not prediction invertible. Indeed, then ˆX = but ˆX t = X T, if t (, T. So, ˆX is not continuous. On the other hand, the augmented filtration is simply Ft X = σ(ξ for all t [, T. So, ˆX = X T. Note, however, that in both cases the slope X can be recovered from the prediction martingale: X t = t ˆX T t. In order to represent abstract Wiener integrals of X in terms of Wiener Itô integrals of ˆX we need to extend the kernels p and p 1 to linear operators: Definition. Let X be prediction-invertible. Define operators P and P 1 by extending linearly the relations P[1 t = p(t,, P 1 [1 t = p 1 (t,. Now the following lemma is obvious.

18 12 Working Papers Lemma. Let f be such a function that P 1 [f L 2 ([, T, d ˆX and let ĝ L 2 ([, T, d ˆX. Then (4.2 (4.21 f(t dx t = ĝ(t d ˆX t = P 1 [f(t d ˆX t, P[ĝ(t dx t Remark. (i Equation (4.2 or (4.21 can actually be taken as the definition of the Wiener integral with respect to X. (ii The operators P and P 1 depend on T. (iii If p 1 (, s has bounded variation, we can represent P 1 as P 1 [f(s = f(sp 1 (T, s + s (f(t f(s p 1 (dt, s. Similar formula holds for P also, if p(, s has bounded variation. (iv Let g X (t denote the remaining covariance matrix with respect to X, i.e., [ g X ij (t = E g i (s dx s g j (s dx s. t Let ĝ ˆX(t denote the remaining covariance matrix with respect to ˆX, i.e., Then g ˆX ij (t = g X ij (t = P 1 [g ˆX ij (t = t t g i (sg j (s d ˆX s. t P 1 [g i (sp 1 [g j (s d ˆX s. Now, let X g be the bridge conditioned on g(s dx s =. By Lemma 4.19 we can rewrite the conditioning as (4.23 g(t dx t = P 1 [g(t d ˆX(t =, With this observation the following theorem, that is the main result of this article, follows Theorem. Let X be prediction-invertible Gaussian process. Assume that, for all t [, T and i = 1,..., N, g i 1 t Λ t (X. Then the generalized bridge X g admits the canonical representation (4.25 X g t = X t p 1 (t, up [ˆl ĝ(u, (s d ˆX u dx s, where s ĝ i = P 1 [g i, ˆl ĝ (u, v = ĝ ˆX (uĝ (u( ĝ ˆX 1 (u ĝ ˆX ij (t = t ĝ i (sĝ j (s d M s = g X ij (t. ĝ(v ĝ ˆX (v,

19 Working Papers 13 Proof. Since ˆX is a Gaussian martingale and because of the equality (4.23 we can use Theorem We obtain d ˆXĝs = d ˆX s s ˆl ĝ (s, u d ˆX u d ˆX s. Now, by using the fact that X is prediction invertible, we can recover X from ˆX, and consequently also X g from ˆXĝ by operating with the kernel p 1 in the following way: (4.26 X g t = p 1 (t, s d ˆXĝs ( s = X t p 1 (t, s ˆl ĝ (s, u d ˆX u d ˆX s. In a sense the representation (4.26 is a canonical representation already. However, let us write it in terms of X instead of ˆX. We obtain s X g t = X t p 1 (t, s P [ˆl ĝ(s, (u dx u d ˆX s = X t s p 1 (t, up [ˆl ĝ(u, (s d ˆX u dx s Remark. Recall that, by assumption, the processes X g and X are equivalent on F t, t < T. So, the representation (4.25 is an analogue of the Hitsuda representation for prediction-invertible processes. Indeed, one can show, just like in [16, 17, that a zero mean Gaussian process X is equivalent in law to the zero mean prediction-invertible Gaussian process X if it admits the representation where f(t, s = X t = X t s f(t, s dx s p 1 (t, up [v(u, (s d ˆX u for some Volterra kernel v L 2 ([, T 2, d ˆX d ˆX. An important class of prediction-invertible processes is the class of the so-called invertible Gaussian Volterra processes: Definition. V is an invertible Gaussian Volterra process if it is continuous and there exist Volterra kernels k and k 1 such that (4.29 (4.3 V t = W t = k(t, s dw s, k 1 (t, s dv s. Here W is the standard Brownian motion, k(t, L 2 ([, t = Λ t (W and k 1 (t, Λ t (V for all t [, T.

20 14 Working Papers Remark. (i The representation (4.29, defining a Gaussian Volterra process, states that the covariance R of V can be written as R(t, s = s k(t, uk(s, u du. So, in some sense, the kernel k is the square root, or the Cholesky decomposition, of the covariance R. (ii The inverse relation (4.3 means that the indicators 1 t, t [, T, can be approximated in L 2 ([, t with linear combinations of the functions k(t j,, t j [, t. I.e., the indicators 1 t belong to the image of the operator K extending the kernel k linearly as discussed below. Precisely as with the kernels p and p 1, we can define the operators K and K 1 by linearly extending the relations K[1 t := k(t, and K 1 [1 t := k 1 (t,. Then, just like with the operators P and P 1, we have f(t dv t = g(t dw t = K 1 [f(t dw t, K[g(t dv t. The connection between the operators K and K 1 and the operators P and P 1 are K[g = k(t, P 1 [g, K 1 [g = k 1 (T, P[g. So, invertible Gaussian Volterra processes are prediction-invertible and the following corollary to Theorem 4.24 is obvious: Corollary. Let V be an invertible Gaussian Volterra process and let K[g i L 2 ([, T for all i = 1,..., N. Denote g(t := K[g(t k(t, t. Then the bridge V g admits the canonical representation (4.33 V g t where = V t s k(t, uk(t, u k 1 K 1 [ l g(u, (s du dv s, (T, s l g (u, v = g W (u g (u( g W 1 (u g W ij (t = t g i (s g j (s ds = g X ij (t. g(v g W (v,

21 Working Papers Example. An important example of an invertible Gaussian Volterra process is the fractional Brownian motion B with Hurst index H (, 1. It is a centered Gaussian process with B = and covariance function R(t, s = 1 ( t 2H + s 2H t s 2H. 2 It is well-known that the fractional Brownian motion is an invertible Gaussian Volterra process with K[f(s = c H s 1 2 H I H 1 2 T K 1 [f(s = 1 s 1 2 H I 1 2 H T c H Here I H 1 2 T and I 1 2 H T [, T of order H 1 2 and 1 2 H, respectively: I H 1 2 T [f(t = [( H 1 2 f (s, [( H 1 2 f (s. are the Riemann-Liouville fractional integrals over 1 T Γ(H 1 2 t t 1 d Γ( 3 2 H dt and c H is the normalizing constant ( 2HΓ(H + 1 c H = Here Γ(x = f(s (s t 3 2 H ds, for H > 1 2, f(s (s t H Γ( 3 2 H Γ(2 2H e t t x 1 dt ds, for H < 1 2, 1 2. is the Gamma function. (For the proofs of these facts and for more information on the fractional Brownian motion we refer to the monographs by Biagini et al. [4 and Mishura [ Application to Insider Trading We consider insider trading in the context of initial enlargement of filtrations. Our approach here is motivated by Imkeller [11. Consider an insider who has at time t = some insider information of the evolution of the price process on a financial asset S over a period [, T. We want to calculate the additional expected utility for the insider trader. To make the maximization of the utility of terminal wealth reasonable we have to assume that our model is arbitrage-free. In our Gaussian realm this boils down to assuming that the (discounted asset prices are governed by the equation ds t (5.1 = a t d M t + dm t, S t where S = 1, M is a continuous Gaussian martingale with strictly increasing M with M =, and the process a is F-adapted satisfying a2 t d M t < P-a.s. Assuming that the trading ends at time T ε, the insider knows some functionals of the return over the interval [, T. If ε = there is obviously

22 16 Working Papers arbitrage for the insider. The insider information will define a collection of functionals G i T (M = g i(t dm t, where g i L 2 ([, T, d M, i = 1,..., N, such that (5.2 g(t ds t S t = y = [y i N i=1, for some y R N. This is equivalent to the conditioning of the Gaussian martingale M on the linear functionals G T = [G i T N i=1 of the log-returns: [ N G T (M = g(t dm t = g i (t dm t. i=1 Indeed, the connection is where g(t dm t = y a, g =: y, [ a, g = [ a, g i N i=1 = N a t g i (t d M t. i=1 As the natural filtration F represents the information available to the ordinary trader, the insider trader s information flow is described by a larger filtration G = (G t t [,T given by G t = F t σ(g 1 T,..., G N T. Under the augmented filtration G, M is no longer a martingale. It is a Gaussian semimartingale with the semimartingale decomposition ( (5.3 dm t = d M t + l g (t, s dm s g (t g 1 (ty d M t, where M is a continuous G-martingale with bracket M, and which can be constructed through the formula (4.1. In this market, we consider the portfolio process π defined on [, T ε Ω as the fraction of the total wealth invested in the asset S. So the dynamics of the discounted value process associated to a self-financing strategy π is defined by V = v and dv t V t = π t ds t S t, or equivalently by ( (5.4 V t = v exp π s dm s + for t [, T ε, ( π s a s 1 2 π2 s d M s. Let us denote by, ε and ε the inner product and the norm on L 2 ([, T ε, d M. For the ordinary trader, the process π is assumed to be a non-negative F-progressively measurable process such that (i P[ π 2 ε < = 1. (ii P[ π, f ε < = 1, for all f L 2 ([, T ε, d M.

23 Working Papers 17 We denote this class of portfolios by Π(F. By analogy, the class of the portfolios disposable to the insider trader shall be denoted by Π(G, the class of non-negative G-progressively measurable processes that satisfy the conditions (i and (ii above. The aim of both investors is to maximize the expected utility of the terminal wealth V T ε, by finding an optimal portfolio π on [, T ε that solves the optimization problem max π E [U(V T ε. Here, the utility function U will the logarithmic utility function, and the utility of the process (5.4 valued at time T ε is ε ε ( log V T ε = log v + π s dm s + π s a s 1 (5.5 2 π2 s d M s = log v + = log v + ε ε π s dm s ε π s dm s π, 2a π ε π s (2a s π s d M s ( From the ordinary trader s point of view M is a martingale. So, T ε E π s dm s = for every π Π(F and, consequently, E [U(V T ε = log v E [ π, 2a π ε. Therefore, the ordinary trader, given Π(F, can solve the optimization problem max E [U(V T ε = log v + 1 π Π(F 2 max E [ π, 2a π ε π Π(F over the term π, 2a π ε = 2 π, a ε π 2 ε. Using the polarization identity gives π, 2a π ε = a 2 ε π a 2 ε a 2 ε, where the maximum is at π = a on [, T ε. This means that the optimal portfolio is π t = a t defined for all t in [, T ε. The corresponding maximal expected utility value is max E [U(V T ε = log v + 1 π Π(F 2 E [ a 2 ε. From the insider trader s point of view the process M is not a martingale under his information flow G. The insider can update his utility of terminal wealth (5.5 by considering (5.3, where M is a continuous G-martingale. This gives log V T ε = log v + + π, ε π s d M s π, 2a π ε l g (, t dm t g g 1 y ε.

24 18 Working Papers Now, the insider maximizes the expected utility over all π Π(G: max π Π(G E [log V T ε = log v + 1 [ ( 2 max E π, 2 a + π Π(G l g (, t dm t g g 1 y π ε. The optimal portfolio π for the insider trader can be computed in the same way as for the ordinary trader. We obtain π t = a t + l g (t, s dm s g (t g 1 (ty, t [, T ε. Since [ E a, l g (, s dm s g g 1 y ε =, we obtain that T ε = max E [U(V T ε max E [U(V T ε π Π(G π Π(F = 1 [ 2 E l g (, s dm s g g 1 y 2 ε. Now, let us use the short-hand notation G t := g(s dm s, g (t, s := g (t g (s, g 1 (t, s := g 1 (t g 1 (s. Then, by expanding the square 2 ε, we obtain [ 2 T ε = E l g (, s dm s g g 1 y 2 ε [ = E g g 1 ( y + G 2 ε = E +E [ ε [ ε y g 1 (tg(tg (t g 1 (ty d M t G t g 1 (tg(tg (t g 1 (tg t d M t.

25 Working Papers 19 Now the formula E[x Ax = Tr[ACovx + E[x AE[x yields 2 T ε = ε ε + y g 1 (tg(tg (t g 1 (ty d M t [ Tr g 1 (tg(tg (t g 1 (t g (, t d M t = y g 1 (T ε, y ε [ + Tr g 1 (tg(tg (t g 1 (t g ( d M t ε [ Tr g 1 (tg(tg (t d M t = (y g, a g 1 (T ε, (y g, a [ +Tr g 1 (T ε, g ( + log g (T ε. g ( We have proved the following proposition: 5.6. Proposition. The additional logarithmic utility in the model (5.1 for the insider with information (5.2 is T ε = max E [U(V T ε max E [U(V T ε π Π(G π Π(F = 1 ( 2 (y g, a g 1 (T ε g 1 ( (y g, a + 1 [( g 2 Tr 1 (T ε g 1 ( g ( g (T ε log. g ( 5.7. Example. Consider the classical Black and Scholes pricing model: ds t S t = µdt + σdw t, S = 1, where W = (W t t [,T is the standard Brownian motion. Assume that the insider trader knows at time t = that the total and the average return of the stock price over the period [, T are both zeros and that the trading ends at time T ε. So, the insider knows that where G 1 T = G 2 T = g 1 (t dw t = y 1 σ µ σ g 1, 1 T = µ σ g 1, 1 T g 2 (t dw t = y 2 σ µ σ g 2, 1 T = µ σ g 2, 1 T, g 1 (t = 1 T (t, g 2 (t = T t T.

26 2 Working Papers Then, by Proposition 5.6, T ε = 1 ( µ ( 2 g, 1T g 1 (T ε g ( 1 g, 1 T 2 σ + 1 [( g 2 Tr 1 (T ε g 1 ( g ( with g (T ε log, g ( g 1 (t = 6 T ( 4 T T T t ( T T t ( 2 6 T T T t ( T T t 2 12 T 3 for all t [, T ε. We obtain T ε = { 1 ( µ ( 2 T 3 ( T 2 ( T 3T 6T + 4T 2 σ ε ε ε ( T 3 ( T 2 ( ( T T log ε ε ε ε } T 1. Here it can be nicely seen that = (no trading at all and T = (the knowledge of the final values implies arbitrage. References [1 Alili, L. Canonical decompositions of certain generalized Brownian bridges. Electron. Comm. Probab. 7, 27 36, 22. [2 Baudoin, F. Conditioned stochastic differential equations: theory, examples and application to finance. Stochastic Processes and their Applications 1, no. 1, pp , 22. [3 Bender, C. and Elliott, R. On the Clark-Ocone theorem for fractional Brownian motions with Hurst parameter bigger than a half. Stochastic. Stoch. Rep. 75, no. 6, , 23. [4 Biagini, F., Hu, Y., Øksendal, B., and Zhang, T. Stochastic calculus for fractional Brownian motion and applications. Probability and its Applications (New York. Springer-Verlag London, Ltd., London, 28. [5 Campi, L., Çetin, U. and Danilova, A. Dynamic Markov bridges motivated by models of insider trading. Stochastic Process. Appl. 121, no. 3, , 211. [6 Chaumont, L. and Uribe Bravo, G. Markovian bridges: weak continuity and pathwise constructions. Ann. Probab. 39, no. 2, , 211. [7 Gasbarra, D., Sottinen, T., and Valkeila, E. Gaussian bridges. Stochastic analysis and applications, , Abel Symp. 2, Springer, Berlin, 27. [8 Gasbarra, D., Valkeila, E. and Vostrikova, L. Enlargement of filtration and additional information in pricing models: Bayesian approach. From stochastic calculus to mathematical finance, , Springer, Berlin, 26. [9 Hida, T., Hitsuda, M., Gaussian Processes. AMS Translations, [1 Hoyle, E., Hughston, L. P. and Macrina, A. Lévy random bridges and the modelling of financial information. Stochastic Process. Appl. 121, no. 4, , 211. [11 Imkeller, P. Malliavin s calculus in insider models: additional utility and free lunches. Conference on Applications of Malliavin Calculus in Finance (Rocquencourt, 21. Math. Finance 13, no. 1, , 23. [12 Janson, S. Gaussian Hilbert spaces. Cambridge Tracts in Mathematics 129. Cambridge University Press, Cambridge, 1997.

27 Working Papers 21 [13 Mishura, Yu. S. Stochastic calculus for fractional Brownian motion and related processes. Lecture Notes in Mathematics Springer-Verlag, Berlin, 28. [14 Pipiras, V. and Taqqu, M. Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7, no. 6, , 21. [15 Shiryaev, A. Probability. Second edition. Graduate Texts in Mathematics, 95. Springer-Verlag, New York, [16 Sottinen, T. On Gaussian processes equivalent in law to fractional Brownian motion. Journal of Theoretical Probability 17, no. 2, , 24. [17 Sottinen, T. and Tudor, C.A. On the equivalence of multiparameter Gaussian processes. Journal of Theoretical Probability 19, no. 2., , 26. Tommi Sottinen, Department of Mathematics and Statistics, University of Vaasa P.O.Box 7, FIN-6511 Vaasa, Finland address: Adil Yazigi, Department of Mathematics and Statistics, University of Vaasa P.O.Box 7, FIN-6511 Vaasa, Finland address: