SSRN Finance

7/30/2019 SSRN Finance

1/20Electronic copy available at: http://ssrn.com/abstract=1404905

Review of Statistical Arbitrage, Cointegration,and Multivariate Ornstein-Uhlenbeck

Attilio Meucci1

[email protected]

this version: December 04, 2010latest version available at http://ssrn.com/abstract=1404905

Abstract

We introduce the multivariate Ornstein-Uhlenbeck process, solve it analytically,and discuss how it generalizes a vast class of continuous-time and discrete-time multivariate processes. Relying on the simple geometrical interpretationof the dynamics of the Ornstein-Uhlenbeck process we introduce cointegrationand its relationship to statistical arbitrage. We illustrate an application toswap contract strategies. Fully documented code illustrating the theory and theapplications is available for download.

JEL Classification: C1, G11

Keywords: "alpha", z-score, signal, half-life, vector-autoregression (VAR),moving average (MA), VARMA, stationary, unit-root, mean-reversion, Levy

processes

1 The author is grateful to Gianluca Fusai, Ali Nezamoddini and Roberto Violi for theirhelpful feedback

1



The multivariate Ornstein-Uhlenbeck process is arguably the model mostutilized by academics and practitioners alike to describe the multivariate dy-

namics offinancial variables. Indeed, the Ornstein-Uhlenbeck process is parsi-monious, and yet general enough to cover a broad range of processes. Therefore,by studying the multivariate Ornstein-Uhlenbeck process we gain insight intothe properties of the main multivariate features used daily by econometricians.

continuous time

discrete

Levy

BrownianOU

OU

unit root

(1)VAR ( )VAR n

Brownian

direct coverage

indirect coverage

randomwalk

motion

time

stationary

Levyprocesses

stat-arbcointegration

Figure 1: Multivariate processes and coverage by OU

Indeed, the following relationships hold, refer to Figure 1 and see below fora proof. The Ornstein-Uhlenbeck is a continuous time process. When sampledin discrete time, the Ornstein-Uhlenbeck process gives rise to a vector autore-gression of order one, commonly denoted by VAR(1). More general VAR(n)processes can be represented in VAR(1) format, and therefore they are also cov-ered by the Ornstein-Uhlenbeck process. VAR(n) processes include unit rootprocesses, which in turn include the random walk, the discrete-time counterpartof Levy processes. VAR(n) processes also include cointegrated dynamics, whichare the foundation of statistical arbitrage. Finally, stationary processes are aspecial case of cointegration.

In Section 1 we derive the analytical solution of the Ornstein-Uhlenbeckprocess. In Section 2 we discuss the geometrical interpretation of the solution.

Building on the solution and its geometrical interpretation, in Section 3 weintroduce naturally the concept of cointegration and we study its properties.In Section 4 we discuss a simple model-independent estimation technique forcointegration and we apply this technique to the detection of mean-revertingtrades, which is the foundation of statistical arbitrage. Fully documented code

2



that illustrates the theory and the empirical aspects of the models in this articleis available at www.symmys.com Teaching MATLAB.

1 The multivariate Ornstein-Uhlenbeck process

The multivariate Ornstein-Uhlenbeck process is defined by the following sto-chastic differential equation

dXt = (Xt ) dt + SdBt. (1)

In this expression is the transition matrix, namely a fully generic squarematrix that steers the deterministic portion of the evolution of the process; isa fully generic vector, which represents the unconditional expectation when thisis defined, see below; S is the scatter generator, namely a fully generic square

matrix that drives the dispersion of the process;Bt is typically assumed to be avector of independent Brownian motions, although more in general it is a vector

of independent Levy processes, see Barndorff-Nielsen and Shephard (2001) andrefer to Figure 1.

To integrate the process (1) we introduce the integrator

Yt et (Xt ) . (2)

Using Itos lemma we obtain

dYt = etSdBt. (3)

Integrating both sides and substituting the definition (2) we obtain

Xt+ = I e + eXt + t,, (4)where the invariants are mixed integrals of the Brownian motion and are thusnormally distributed

t,

Zt+t

e(u)SdBu N (0,) . (5)

The solution (4) is a vector autoregressive process of order one VAR(1), whichreads

Xt+ = c+CXt + t,, (6)

for a conformable vector and matrix c and C respectively. A comparison be-tween the integral solution (4) and the generic VAR(1) formulation (6) provides

the relationship between the continuous-time coefficients and their discrete-timecounterparts.

The conditional distribution of the Ornstein-Uhlenbeck process (4) is normalat all times

Xt+ N (xt+,) . (7)

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The deterministic drift reads

xt+ I e + ext (8)and the covariance can be expressed as in Van der Werf (2007) in terms of thestack operator vec and the Kronecker sum as

vec() ()1I e()

vec() , (9)

where SS0, see the proof in Appendix A.2. Formulas (8)-(9) describe thepropagation law of risk associated with the Ornstein-Uhlenbeck process: thelocation-dispersion ellipsoid defined by these parameters provides an indicationof the uncertainly in the realization of the next step in the process.

Notice that (8) and (9) are defined for any specification of the input para-meters , , and S in (1). For small values of , a Taylor expansion of these

formulas shows thatXt+ Xt + t,, (10)

where t, is a normal invariant:

t, N (, ) . (11)

In other words, for small values of the time step the Ornstein-Uhlenbeckprocess is indistinguishable from a Brownian motion, where the risk of the in-variants (11), as represented by the standard deviation of any linear combinationof its entries, displays the classical "square-root of " propagation law.

As the step horizon grows to infinity, so do the expectation (8) and thecovariance (9), unless all the eigenvalues of have positive real part. In that

case the distribution of the process stabilizes to a normal whose unconditionalexpectation and covariance read

x = (12)

vec() = ()1 vec() (13)

To illustrate, we consider the bivariate case of the two-year and the ten-yearpar swap rates. The benchmark assumption among buy-side practitioners isthat par swap rates evolve as the random walk (10), see Figure 3.5 and relateddiscussion in Meucci (2005).

However, rates cannot diffuse indefinitely. Therefore, they cannot evolve as

a random walk for any size of the time step : for steps of the order of a monthor larger, mean-reverting effects must become apparent.

The Ornstein-Uhlenbeck process is suited to model this behavior. We fit thisprocess for different values of the time step and we display in Figure 2 thelocation-dispersion ellipsoid defined by the expectation (8) and the covariance

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2y swap rate

=10y

= 2y

=1y

=6m

=1m

=1d

=1w

5yswap

rate daily rates

observations

Figure 2: Propagation law of risk for OU process fitted to swap rates

(9), refer to www.symmys.com Teaching MATLAB for the fully documentedcode.

For values of of the order of a few days, the drift is linear in the step andthe size increases as the square root of the step, as in (11). As the step increasesand mean-reversion kicks in, the ellipsoid stabilizes to its unconditional values

(12)-(13).

2 The geometry of the Ornstein-Uhlenbeck dy-namics

The integral (4) contains all the information on the joint dynamics of theOrnstein-Uhlenbeck process (1). However, that solution does not provide anyintuition on the dynamics of this process. In order to understand this dynamicswe need to observe the Ornstein-Uhlenbeck process in a different set of coordi-nates.

Consider the eigenvalues of the transition matrix : since this matrix has

real entries, its eigenvalues are either real or complex conjugate: we denote themrespectively by (1, . . . , K) and (1 i1) , . . . , (J iJ), where K + 2J =N. Now consider the matrix B whose columns are the respective, possiblycomplex, eigenvectors and define the real matrix A Re (B)Im (B). Then thetransition matrix can be decomposed in terms of eigenvalues and eigenvectors

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as follows AA

1, (14)

where is a block-diagonal matrix

diag (1, . . . , K,1, . . . ,J) , (15)

and the generic j-th matrix j is defined as

j

j j

j j

. (16)

With the eigenvector matrix A we can introduce a new set of coordinates

z A1 (x ) . (17)

The original Ornstein-Uhlenbeck process (1) in these coordinates follows from

Itos lemma and readsdZt = Ztdt +VdBt, (18)

where V A1S. Since this is another Ornstein-Uhlenbeck process, its solutionis normal

Zt N (zt,t) , (19)

for a suitable deterministic drift zt and covariance t. The deterministic driftzt is the solution of the ordinary differential equation

dzt = ztdt, (20)

Given the block-diagonal structure of (15), the deterministic drift splits intoseparate sub-problems. Indeed, let us partition the N-dimensional vector ztinto K entries which correspond to the real eigenvalues in (15), and J pairs ofentries which correspond to the complex-conjugate eigenvalues summarized by(16):

zt

z1,t, . . . , zK,t, z

(1)1,t , z

(2)1,t , . . . , z

(1)J,t , z

(2)J,t

0. (21)

For the variables corresponding to the real eigenvalues, (20) simplifies to:

dzk,t = kzk,tdt. (22)

For each real eigenvalue indexed by k = 1, . . . , K the solution reads

zk;t ektzk,0. (23)

This is an exponential shrinkage at the rate k. Note that (23) is properly de-fined also for negative values of

k, in which case the trajectory is an exponential

explosion. If k > 0 we can compute the half-life of the deterministic trend,namely the time required for the trajectory (23) to progress half way towardthe long term expectation, which is zero:

et ln 2k

. (24)

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5

10

15

20

-1-0.5

0

0.5

1

-0.5

0

0.5

1

0 >

0 0. In Figure3 we display the ensuing dynamics for the deterministic drift (21), which in

this context reads

zt, z(1)t , z

(2)t

: the motion (23) escapes toward infinity at an

exponential rate, whereas the motion (28)-(29) whirls toward zero, shrinking atan exponential rate. The animation that generates Figure 3 can be downloadedfrom www.symmys.com Teaching MATLAB.

Once the deterministic drift in the special coordinates zt is known, the de-terministic drift of the original Ornstein-Uhlenbeck process (4) is obtained byinverting (17). This is an affine transformation, which maps lines in lines and

planes in planesxt +Azt (30)

Therefore the qualitative behavior of the solutions (23) and (28)-(29) sketchedIn Figure 3 is preserved.

Although the deterministic part of the process in diagonal coordinates (18)splits into separate dynamics within each eigenspace, these dynamics are notindependent. In Appendix A.3 we derive the quite lengthy explicit formulas forthe evolution of all the entries of the covariance t in (19). For instance, thecovariances between entries relative to two real eigenvalues reads:

k,hk;t

=k,hk

k + hk

1 e(k+hk)t

, (31)

where VV0. More in general, the following observations apply. First, astime evolves, the relative volatilities change: this is due both to the differentspeed of divergence/shrinkage induced by the real parts s and s of the eigen-values, and to different speed of rotation, induced by the imaginary parts sof the eigenvalues. Second, the correlations only vary if rotations occur: if theimaginary parts s are null, the correlations remain unvaried.

Once the covariance t in the diagonal coordinates is known, the covarianceof the original Ornstein-Uhlenbeck process (4) is obtained by inverting (17) andusing the affine equivariance of the covariance, which leads to t = AtA0.

3 Cointegration

The solution (4) of the Ornstein-Uhlenbeck dynamics (1) holds for any choiceof the input parameters , and S. However, from the formulas for thecovariances (31), and similar formulas in Appendix A.3, we verify that if somek 0, or some j 0, i.e. if any eigenvalues of the transition matrix are null or negative, then the overall covariance of the Ornstein-Uhlenbeck Xt

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does not converge: therefore Xt is stationary only if the real parts of all theeigenvalues of are strictly positive.

Nevertheless, as long as some eigenvalues have strictly positive real part, thecovariances of the respective entries in the transformed process (18) stabilizesin the long run. Therefore these processes and any linear combination thereofare stationary. Such combinations are called cointegrated: since from (17) theyspan a hyperplane, that hyperplane is called the cointegrated space, see Figure3.

To better discuss cointegration, we write the Ornstein-Uhlenbeck process (4)as

Xt = (Xt ) + t,, (32)

where is the difference operator Xt Xt+Xt and is the transitionmatrix

e

I. (33)

If some eigenvalues in are null, the matrix e has unit eigenvalues: thisfollows from

e = AeA1, (34)

which in turn follows from (14). Processes with this characteristic are known asunit-rootprocesses. A very special case arises when all the entries in are null.In this circumstance, e is the identity matrix, the transition matrix isnull and the Ornstein-Uhlenbeck process becomes a multivariate random walk.

More in general, suppose that L eigenvalues are null. Then has rankN L and therefore it can be expressed as

0, (35)

where both matrices and are full-rank and have dimension (N L)N.The representation (35), known as the error correction representation of the

process (32), is not unique: indeed any pair e P0 and e P1 givesrise to the same transition matrix for fully arbitrary invertible matrices P.

The L-dimensional hyperplane spanned by the rows of does not dependon the horizon . This follows from (33) and (34) and the fact that since generates rotations (imaginary part of the eigenvalues) and/or contractions(real part of the eigenvalues), the matrix e maps the eigenspaces of intothemselves. In particular, any eigenspace of relative to a null eigenvalue ismapped into itself at any horizon.

Assuming that the non-null eigenvalues of have positive real part2 , therows of, or of any alternative representation, span the contraction hyper-planes and the process Xt is stationary and would converge exponentially

fast to the the unconditional expectation , if it were not for the shocks t,in (32).

2 One can take differences in the time series ofXt until the fitted transition matrix doesnot display negative eigenvalues. However, the case of eigenvalues with pure imaginary partis not accounted for.

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4 Statistical arbitrage

Cointegration, along with its geometric interpretation, was introduced above,building on the multivariate Ornstein-Uhlenbeck dynamics. However, cointegra-tion is a model-independent concept. Consider a generic multivariate processXt.

4.1 Model-independent cointegration

This process is cointegrated if there exists a linear combination of its entrieswhich is stationary. Let us denote such combination as follows

Ywt X0tw, (36)

where w is normalized to have unit length for future convenience.Ifw belongs to the cointegration space, i.e. Yw

tis cointegrated, its variance

stabilizes as t . Otherwise, its variance diverges to infinity. Therefore,the combination that minimizes the conditional variance among all the possiblecombinations is the best candidate for cointegration:

ew argminkwk=1

[Var {Yw|x0}] . (37)

Based on this intuition, we consider formally the conditional covariance ofthe process

Cov{X|x0} , (38)

although we understand that this might not be defined. Then we consider theformal principal component factorization of the covariance

EE, (39)

where E is the orthogonal matrix whose columns are the eigenvectors

E

e(1), . . . ,e(N)

; (40)

and is the diagonal matrix of the respective eigenvalues, sorted in decreasingorder

diag

(1), . . . , (N)

. (41)

Note that some eigenvalues might be infinite.The formal solution to (37) is ew e(N), the eigenvector relative to the

smallest eigenvalue (N). Ife(N) gives rise to cointegration, the process Ye(N)

t

is stationary and therefore the eigenvalue (N) is not infinite, but rather it

represents the unconditional variance of Ye(N)

t .If cointegration is found with e(N), the next natural candidate for another

possible cointegrated relationship is e(N1). Again, ife(N1) gives rise to coin-

tegration, the eigenvalue (N1)t converges to the unconditional variance of

Ye(N1)

t .

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In other words, the PCA decomposition (39) partitions the space into twoportions: the directions of infinite variance, namely the eigenvectors relative to

the infinite eigenvalues, which are not cointegrated, and the directions of finitevariance, namely the eigenvectors relative to the finite eigenvalues, which arecointegrated.

The above approach assumes knowledge of the true covariance (38), whichin reality is not known. However, the sample covariance of the process Xtalong the cointegrated directions approximates the true asymptotic covariance.Therefore, the above approach can be implemented in practice by replacing thetrue, unknown covariance (38) with its sample counterpart.

To summarize, the above rationale yields a practical routine to detect thecointegrated relationships in a vector autoregressive process (1). Without ana-lyzing the eigenvalues of the transition matrix fitted to an autoregressive dynam-ics, we consider the sample counterpart of the covariance (38); then we extractthe eigenvectors (40); finally we explore the stationarity of the combinations

Ye(n)

t for n = N , . . . , 1.

To illustrate, we consider a trading strategy with swap contracts. First, wenote that the p&l generated by a swap contract is faithfully approximated by thechange in the respective swap rate times a constant, known among practitionersas "dv01". Therefore, we analyze linear combinations of swap rates, which mapinto portfolios p&ls, hoping to detect cointegrated patterns.

In particular, we consider the time series of the 1y, 2y, 5y, 7y, 10y, 15y,and 30y rates. We compute the sample covariance and we perform its PCAdecomposition. In Figure 4 we plot the time series corresponding with the first,second, fourth and seventh eigenvectors, refer to www.symmys.com Teaching MATLAB for the fully documented code.

4.2 Z-score, alpha, horizon adjustments

To test for the stationarity of the potentially cointegrated series it is convenientto fit to each of them a AR(1) process, i.e. the univariate version of (4), to thecointegrated combinations:

Yt+

1 e

+ eYt + t,. (42)

In the univariate case the transition matrix becomes the scalar . Consistentlywith (23) cointegration corresponds to the condition that the mean-reversionparameter be larger than zero.

By specializing (12) and (13) to the one-dimensional case, we can compute

the expected long-term gain

t, |Yt E {Y}| = |Yt | ;

the z-score

zt, |Yt E {Y}|

Sd {Y}=

|Yt |p2/2

; (43)

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96 97 98 99 00 01 0 2 0 3 0 4 05-30

-25

-20

-15

-10

-5

0

5e i g e n d i re c t i o n n . 4 , t h e t a = 6. 0 7 1 5

y e a r

b

asis

p

o

in

ts

96 9 7 9 8 99 0 0 01 0 2 0 3 04 0 5-2000

-1500

-1000

-500

e i g e n d i re c t i o n n . 1 , t h e t a = 0 .2 7 1 1 4

y e a r

basis

p

oin

ts

9 6 9 7 98 9 9 00 01 0 2 03 04 05400

450

500

550

600

650

700

750

800e i g e n d i re c t i o n n . 2 , t h e t a = 1 . 4 0 5

y e a r

b

asis

po

in

ts

9 6 9 7 98 9 9 00 01 0 2 03 04 05-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0. 5

1

1. 5e i g e n d i re c t i o n n . 7 , t h e t a = 2 7 . 0 3 3 1

y e a r

b

asis

p

o

in

ts

eigendirection 1, = 0.27

basis

points

eigendirection 2, = 1.41

eigendirection 7, = 27.03eigendirection 4, = 6.07

basis

points

basis

points

basis

points

current value 1 z-score bands long- term expec tation

Figure 4: Cointegration search among swap rates

and the half-life (24) of the deterministic trend

e ln 2

. (44)

The expected gain (4.2) is also known as the "alpha" of the trade. The z-score

represents the ex-ante Sharpe ratio of the trade and can be used to generatesignals: when the z-score is large we enter the trade; when the z-score hasnarrowed we cash a profit; and if the z-score has widened further we take a loss.The half-life represents the order of magnitude of the time required before wecan hope to cash in any profit: the higher the mean-reversion , i.e. the morecointegrated the series, the lesser the wait.

In practice, traders cannot afford to wait until the half-life of the trade todeliver the expected (half) alpha. The true horizon is typically one day and thecomputations for the alpha and the z-score must be adjusted accordingly. Thehorizon-adjusted z-score is defined as

zt, |Yt E {Yt+}|

Sd{Yt+}= zt,

1 e

(1 e2

)

12

. (45)

In practice e is close to zero. Therefore we can write the adjustment aszt,zt,

rln 2

2

e. (46)12


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Notice that the relative adjustment for two cointegrated strategies rescales asthe square root of the ratio of their half-lives e(1) and e(2):

z(1)t,

z(1)t,

/z(2)t,

z(2)t,

se(1)e(2) . (47)To illustrate, for a horizon 1 day and a signal with a half-life e 2 weeksthe adjustment (46) is 2; for a short signal ej 1 day and a long signal ek 1 month the relative adjustment (47) is 5.

A caveat is due at this point: based on the above recipe, one would betempted to set up trades that react to signals from the most mean-revertingcombinations. However, such combinations face two major problems. First, in-sample cointegration does not necessarily correspond to out-of-sample results:as a matter of fact, the eigenseries relative to the smallest eigenvalues, i.e.those that allow to make quicker profits, are the least robust out-of-sample.Second, the "alpha" (4.2) of a trade has the same order of magnitude as itsvolatility. In the case of cointegrated eigenseries, the volatility is the squareroot of the respective eigenvalue (41): this implies that the most mean-revertingseries correspond to a much lesser potential return, which is easily offset by thetransaction costs.

In the example in Figure 4 the AR(1) fit (42) confirms that cointegrationincreases with the order of the eigenseries. In the first eigenseries the mean-reversion parameter 0.27 is close to zero: indeed, its pattern is very similarto a random walk. On the other hand, the last eigenseries displays a very highmean-reversion parameter 27.

The current signal on the second eigenseries appears quite strong: one wouldbe tempted to set up a dv01-weighted trade that mimics this series and buy it.However, the expected wait before cashing in on this trade is of the order ofe 1/1.41 0.7 years.

The current signal on the seventh eigenseries is not strong, but the mean-reversion is very high, therefore, soon enough the series should hit the 1-z-scorebands: if the series first hits the lower 1-z-score band one should buy the series,or sell it if the series first hits the upper 1-z-score band, hoping to cash inin e 252/27 9 days. However, the "alpha" (43) on this trade would beminuscule, of the order of the basis point: such "alpha" would not justify thetransaction costs incurred by setting up and unwinding a trade that involveslong-short positions in seven contracts.

The current signal on the fourth eigenseries appears strong enough to buyit and the expected wait before cashing in is of the order of

e 12/6.07 2

months. The "alpha" is of the order offi

ve basis points, too low for sevencontracts. However, the dv01-adjusted presence of the 15y contract is almostnull and the 5y, 7y, and 10y contracts appear with the same sign and can bereplicated with the 7y contract only without affecting the qualitative behaviorof the eigenseries. Consequently the trader might want to consider setting upthis trade.

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References

Barndorff-Nielsen, O.E., and N. Shephard, 2001, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (withdiscussion), J. Roy. Statist. Soc. Ser. B 63, 167241.

Meucci, A., 2005, Risk and Asset Allocation (Springer).

Van der Werf, K. W., 2007, Covariance of the Ornstein-Uhlenbeck process,Personal Communication.

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A Appendix

In this appendix we present proofs, results and details that can be skipped atfirst reading.

A.1 Deterministic linear dynamical system

Consider the dynamical systemz1z2

=

a bb a

z1z2

. (48)

To compute the solution, consider the auxiliary complex problem

z = z, (49)

where z z1 + iz2 and a + ib. By isolating the real and the imaginary partswe realize that (48) and (49) coincide.

The solution of (49) isz (t) = etz (0) . (50)

Isolating the real and the imaginary parts of this solution we obtain:

z1 (t) = Re

etz0

= Re

e(a+ib)t (z1 (0) + iz2 (0))

(51)

= Re

eat (cos bt + i sin bt) (z1 (0) + iz2 (0))

z2 (t) = Im

etz0

= Im

e(a+ib)t (z1 (0) + iz2 (0))

(52)

= Im eat (cos bt + i sin bt) z1 (0) + iz2 (0)or

z1 (t) = eat (z1 (0)cos bt z2 (0)sin bt) (53)

z2 (t) = eat (z1 (0)sin bt + z2 (0)cos bt) (54)

The trajectories (53)-(54) depart from (shrink toward) the origin at the expo-nential rate a and turn counterclockwise with frequency b.

A.2 Distribution of Ornstein-Uhlenbeck process

We recall that the Ornstein-Uhlenbeck process

dX

t =

(X

tm

) dt +S

dBt. (55)

integrates as follows

Xt = m+et (x0 m) +

Zt0

e(ut)SdBu. (56)

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The distribution ofXt conditioned on the initial value x0 is normal

Xt|x0 N (t,t) . (57)

The expectation follows from (56) and reads

t = m+et (x0 m) . (58)

To compute the covariance we use Itos isometry:

t Cov{Xt|x0} =

Zt0

e(ut)e0(ut)du, (59)

where SS0. We can simplify further this expression as in Van der Werf(2007). Using the identity

vec(ABC) (C0 A)vec(B) , (60)

where vec is the stack operator and is the Kronecker product. Then

vec

e(ut)e0(ut)

=

e(ut) e(ut)

vec() . (61)

Now we can use the identity

eAB = eA eB, (62)

where is the Kronecker sum

AMM BNN AMM INN + IMM BNN. (63)

Then we can rephrase the term in the integral (61) as

vec

e(ut)e(ut)

=

e()(ut)

vec () . (64)

Substituting this in (59) we obtain

vec(t) =

Zt0

e()(ut)

du

vec() (65)

= ()1 e()(ut)

t0

vec()

= ()1

I e()t

vec()

More in general, consider the process monitored at arbitrary times 0 t1 t2 . . .

Xt1,t2,...

Xt1

Xt2...

. (66)

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The distribution ofXt1,t2,... conditioned on the initial value x0 is normal

Xt1,t2,...|x0 N t1,t2,...,t1,t2,... . (67)The expectation follows from (56) and reads

t1,t2,...

m+et1 (x0 m)m+et2 (x0 m)

...

(68)

For the covariance, it suffices to consider two horizons. Applying Itos isometrywe obtain

t1,t2 Cov{Xt1 ,Xt2 |x0} =

Zt10

e(ut1)e0(ut2)du

= Zt10

e

(ut1)e

0

(ut1)

0

(t2t1)du = t1e

0

(t2t1)

Therefore

t1,t2,... =

t1 t1e

0(t2t1) t1e

0(t3t1)

e(t2t1)t1 t2 t2e

0(t3t2) ... e(t3t2)t2 t3

.... . .

, (69)

This expression is simplified heavily by applying (65).

A.3 OU (auto)covariance: explicit solution

For t z the autocovariance is

Cov{Zt,Zt+|Z0} = E

(Zt E {Zt|Z0}) (Zt+ E {Zt+|Z0})0 |Z0

(70)

= E

(Zt0

e(ut)VdBu

Zt+0

dBuV0e

0(ut)

0)e

0

= E

(Zt0

e(ut)VdBu

Zt0

dBuV0e

0(ut)

0)e

0

=

Zt0

e(ut)VV0e0(ut)du

e

0

= Zt

0

esVV0e0sdu e

0

=

Zt0

ese0sds

e

0,

where VV

0 = A1SSA01. (71)

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In particular, the covariance reads

(t) Cov{Zt|Z0} = Zt0

ese0sds. (72)

To simplify the notation, we introduce three auxiliary functions:

E (t) 1

et

(73)

D, (t)

2 + 2

et ( cos t + sin t)

2 + 2(74)

C, (t)

2 + 2

et (cos t sin t)

2 + 2. (75)

Using (73)-(73), the conditional covariances among the entries k = 1, . . . K

relative to the real eigenvalues read:

k,hk

(t) =

Zt0

ekk,hk

ehksds (76)

= k,hk

E

t; k + hk

.

Similarly, the conditional covariances (72) of the pairs of entries j = 1, . . . J relative to the complex eigenvalues with the entries k = 1, . . . K relative to thereal eigenvalues read:

(1)j,k (t)

(2)j,k

(t)

!=

Zt0

ejs

(1)j,k

(1)j,k

!eksds (77)

= Zt0

e(j+k)s (1)j,k cos(js) (2)j,k sin(js)(1)j,k sin(js) +

(2)j,k cos(js)

! ds,which implies

(1)j,k (t)

(1)j,k

Zt0

e(j+k)s cos(js) ds (78)

(2)j,k

Zt0

e(j+k)s sin(js) ds

= (1)j,kC

t; j + k, j

(2)j,kS

t; j + k, j

and

(2)j,k (t)

(1)j,kZt0

e(j+k)s sin(js) ds (79)

+(2)j,k

Zt0

e(j+k)s cos(js) ds

= (1)j,kS

t; j + k, j

+

(2)j,kC

t; j + k, j

.

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Finally, using again (73)-(73), the conditional covariances (72) among the pairsof entries j = 1, . . . J relative to the complex eigenvalues read:

(1,1)

j,hj(t)

(1,2)

j,hj(t)

(2,1)

j,hj(t)

(2,2)

j,hj(t)

!=

Zt0

ejs

(1,1)

j,hj(1,2)

j,hj

(2,1)

j,hj(2,2)

j,hj

!e

0

hjs

ds (80)

=

Zt0

e(j+hj)s

cos js sin jssin js cos js

(1,1)

j,hj(1,2)

j,hj

(2,1)

j,hj(2,2)

j,hj

!cos hjs sin hjs

sin hjs cos hjs

ds

Using the identity

ea ebec ed ! cos sin sin cos a bc d cos sin sin cos , (81)where

ea 12

(a + d)cos( ) +1

2(c b)sin( ) (82)

+1

2(a d)cos( + )

1

2(b + c)sin( + )

eb 12

(b c)cos( ) +1

2(a + d)sin( ) (83)

+1

2(b + c)cos( + ) +

1

2(a d)sin( + )

ec 12 (c b)cos( ) 12 (a + d)sin( ) (84)+

1

2(b + c)cos( + ) +

1

2(a d)sin( + )

ed 12

(a + d)cos( ) +1

2(c b)sin( ) (85)

+1

2(d a)cos( + ) +

1

2(b + c)sin( + )

we can simplify the matrix product in (80). Then, the conditional covariancesamong the pairs of entries j = 1, . . . J relative to the complex eigenvalues read

S(1,1)j,hj;t

=1

2

S(1,1)j,hj

+ S(2,2)j,hj

Cj+hj ,hjj (t) (86)

+ 12 S(2,1)j,hj S(1,2)j,hj Dj+hj,hjj (t)+

1

2

S(1,1)

j,hj S

(2,2)

j,hj

Cj+hj ,hj+j (t)

1

2

S(1,2)

j,hj+ S

(2,1)

j,hj

Dj+hj,hj+j (t) ;

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S(1,2)

j,hj;t

1

2 S(1,2)

j,hj S

(2,1)

j,hj Cj+hj ,hjj (t) (87)

+ 12S(1,1)

j,hj+ S(2,2)

j,hjDj+hj,hjj (t)

+1

2

S(1,2)

j,hj+ S

(2,1)

j,hj

Cj+hj ,hj+j (t)

+1

2

S(1,1)

j,hj S

(2,2)

j,hj

Dj+hj,hj+j (t) ;

S(2,1)j,hj;t

1

2

S(2,1)j,hj

S(1,2)

j,hj

Cj+hj ,hjj (t) (88)

1

2

S(1,1)

j,hj+ S

(2,2)

j,hj

Dj+hj,hjj (t)

+1

2 S(1,2)

j,hj+ S

(2,1)

j,hj Cj+hj ,hj+j (t)+

1

2

S(1,1)j,hj

S(2,2)

j,hj

Dj+hj,hj+j (t) ;

S(2,2)j,hj;t

1

2

S(1,1)j,hj

+ S(2,2)j,hj

Cj+hj ,hjj (t) (89)

+1

2

S(2,1)

j,hj S

(1,2)

j,hj

Dj+hj,hjj (t)

+1

2

S(2,2)j,hj

S(1,1)

j,hj

Cj+hj ,hj+j (t)

+1

2

S(1,2)j,hj

+ S(2,1)j,hj

Dj+hj,hj+j (t) .

Similar steps lead to the expression of the autocovariance.

20

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