v = P(t, T ) = exp

f(t, s) ds
Since the outcome of future interest rates is not known in advance it is reasonable to model instantaneous forward rates {f(t, s)}0 s t as stochastic processes. In this context we may interpret f(t, s) as the overnight interest rate at (future) time t as seen from time s. The case f(t, t) = : r(t) is simply the overnight rate, or short rate. The literature is replete with examples on interest rates. A small sample of papers, not otherwise cited in the text, is this (Va ek 1977; Rendleman and Bartter 1980; Cox, Ingersoll, c Jr., and Ross 1985; Lee and Ho 1986; Black, Derman, and Toy 1990; Ritchken and Sankarasubramanian 1995; Musiela 1995; Chen 1996a; Chen 1996b; Bjrk, Christensen, and Gombani 1998; Bjrk and Gombani 1999; Vargiolu 1999; Filipovi and Zabczyk 2002; Aihara and c Bagchi 2005; Filipovi and Tappe 2008). All address stochastic interest rates in nancial c modeling. Of interest within are these references including co-author Marek Musiela: (Brace and Musiela 1994; Brace, Gatarek, and Musiela 1997; Musiela and Rutkowski 1997; Goldys, Musiela, and Sondermann 2000). As mentioned above the classical duration is based on the assumption that interest rates are at or piecewise at. This assumption is quite unrealistic and only applies to sensitivity measurements with respect to parallel shifts of interest rates. The latter is especially unsatisfying for a trader who manages a complex portfolio of interest-rate-sensitive securities (as, e.g., caps, swaps, bond options, etc.) In this case it would be desirable to measure the interest rate risk of the portfolio with respect to the stochastic uctuations of the whole term structure or even the yield surface, that is (1.6) where Y(t, T ) is the yield given by 1 log P(t, T ) T t Here x in Equation (1.6) stands for the time-to-maturity. Using the relation of Equation (1.5) we can represent the yield surface Yt (x) : = Y(t, t + x) as Y(t, T ) =

(t, x) Y(t, t + x),

1 Yt (x) = ft (s) ds, x
where ft (s) : = f(t, t + s). Because of the linear correspondence of Equation (1.7) between the yield curves Yt () and the forward curves ft () we can and will refer to (1.8) (t, x) ft (x)

Here we require that (, T ) be a deterministic Borel-measurable function and (, T ) a predictable process for all T wrt the P-completed ltration Ft generated by a one-dimensional Brownian motion Bt , t 0. Now, let us reparametrize the forward rates by the time-to-maturity x = T t; that is, let us consider the forward curves ft (x) : = f(t, t + x) An application of the generalized It formula shows that under certain conditions on (, T ), (, T ) the forward curves ft (x) satisfy the rst order SPDE (2.4) dft (x) = d ft (x) + t (x) dt + t (x) dBt , dx
as in (Kunita 1997, Theorem 3.3.1). Here we use the notation t (x) : = (t, t + x), t (x) : = (t, t + x). Note that Equation (2.4) is referred to as the Musiela equation in the literature. See, e.g., (Carmona and Tehranchi 2006). See also (Da Prato and Zabczyk 1992) and the references therein for more information about SPDEs. A deciency of the model of Equation (2.4) is that it does not capture the feature of maturity-specic risk. A model with such a property would enable hedging of bond options with unique portfolio strategies. On the other hand, it would meet the intuitive requirement that maturities of the bonds underlying the bond option are used in the hedging portfolio. A more realistic model than that of Equation (2.4), which takes into account maturityspecic risk, would consequently have the form (2.5) dft (x) = d ft (x) + t (x) dt + t (x) dBt (x), dx
where each noise Bt (x) stands for the risk arising from the the time-to-maturity x. Here we may think of Bt (x) as a Brownian sheet in t and x. So Equation (2.5) can be recast as (2.6)

dft (x) =

d ft (x) + t (x) dt + dx

t (x) dBt (x),

k 1 (k)
where (), k 1, are deterministic measurable functions and Bt , k 1, are independent one-dimensional Brownian motions. In what follows we want to assume that the forward curves are modeled by functions of a Hilbert space H. This space should exhibit the natural feature that evaluation functionals on it are continuous; that is, (2.7) x : H R f f(x)
d is continuous on H for all x. Further, it is desirable that the generator A : = dx in Equation (2.6) admits a strongly continuous semigroup St on H. The semigroup St is the left shift operator given by

(St f)(x) = f(t + x)

The following family {Hw } of Hilbert spaces of Sobolev type introduced by (Filipovi 2001) c fullls the above-mentioned conditions: Let w : [0, ) (0, ) be a non-decreasing function

such that

1 dx < w(x)

Then Hw is dened as

f : [0, ) R f is absolutely continuous and

f (x) w(x) dx < ,

and is equipped with the scalar product
f, g In the sequel we require that

= f(0)g(0) +

f (x)g (x)w(x) dx

t (), t () H, a.e.,

Consider the special case that t (x) = (x)ft (x) for a deterministic function (x). Then, t t using integrating factors we observe that the mild solution of the SDE of Equation (2.6) is explicitly given by the Gaussian random eld

ft (x) = exp (2.9) +

(s, t + x) ds f(0, t + x)
(u, t + x) du (k) (s, t + x) dBt (x)
Now, let Wt be a Q-Wiener process, where Q is a symmetric non-negative operator on a separable Hilbert space U with Trace(Q) <. Set U0 = Q1/2 (U), which is a Hilbert space with norm h 0 : = Q1/2 (h) , u U0 Denote by L2 (U, H) the space of HilbertSchmidt operators from U to H with the operator norm L2. Further, let uk , k 1, be an orthonormal basis of U, and suppose that there exists a Borel-measurable map : [0, T ] L(U0 , H) such that t Q1/2 (uk ) = t () and t Q1/2 L2 (U, H)
for all (t, k) in Equation (2.6), where represents the composition of operators. Then we can (k) view Bt 0 t T , k 1, in Equation (2.6) as a Wiener process Bt cylindrically dened on U, and rewrite Equation (2.6) as (2.10) dft = Aft + t dt + t dWt
In the sequel we assume that there exists a predictable unique strong solution t ft () C([0, T ]; H) to Equation (2.10). Remark 2.1. Suppose that t = b(t, ft ) in Equation (2.10), where b : [0, T ] H H is a Borelmeasurable map. Then the following set of conditions provide sufcient criteria for the existence of a unique strong solution of Equation (2.10). (i) ft is a unique mild solution of Equation (2.10). (ii) f0 Dom(A); Sts b(s, x) Dom(A); Sts s u Dom(A), u U0 , t (iii) ASts b(s, x) (iv) ASts s See, e.g., (Kai 2006). Assume that t is invertible for all 0 (2.11)

t[0,T ] H H

q(t s) x
for some q L1 ([0, T ]; R+ ).
= g(t s), for some g L2 ([0, T ]; R+ ).
T a.e. and that the integrability condition

sup E exp 1 [Aft + t ] t

holds for some > 0. Then Girsanovs Theorem [see, e.g., (Bensoussan 1971)] applied to Equation (2.10) entails that (2.12) where

dft = t dWt ,

Wt = Wt (s) ds
is a Q-Wiener process under the change of measure P given by

T T 0 ds

P(A) = E 1A exp

(s), dWs

with (2.13) (t) : = 1 [Aft + t ] t
Consequently ft is a Gaussian Ft -martingale with respect to P. Dene

converges in L2 () as 0 for k K, then D F L2 (; K) exists and the above limit equals (D F, k)K. Since the measure P in Equation (2.3) is equivalent to P we see that the convergence of Expression (2.17) to D F, k K also holds in probability with respect to the image measure of the forward curves under the original measure P. Therefore, if F = T is the terminal value of a bond portfolio, we may interpret the Malliavin derivative D F as a sensitivity measure of the uctuations of the whole yield surface in this portfolio. The latter observation gives rise to introduce an expanded concept of duration as follows. Denition 2.1 (Stochastic duration). Let F be a square integrable functional of the forward curve f wrt P. Assume that F is Malliavin differentiable wrt f. Then the stochastic duration of
F is stochastic process D F L2 (, F, P; K) Remark 2.2. We shall mention that we also could have introduced our concept of stochastic duration wrt mild solutions ft of Equation (2.10). In this case one can replace Condition (2.11) by assuming that 2 sup E exp 1 [t ] 0 < t

t[0,T ]

for some > 0. Compared to mild solutions, strong solutions are rare. However, from the viewpoint of applications we have in mind it is technically more convenient to deal with strong solutions. See Section 3. We want to illustrate this concept by calculating the stochastic duration of certain interest rate claims. For this purpose we need the following auxiliary results. The rst Lemma gives a chain rule for the Malliavin derivative D. Lemma 2.2 (Chain Rule). Let F be Malliavin differentiable with respect f, i.e., F D1,2. Further, suppose that g : R R is continuously differentiable with bounded derivative. Then g(F) D1,2 and Du g(F) = g (F)Du F for each u K. Here g stands for the derivative of g. Proof. The proof follows from arguments in the Brownian motion case. See (Di Nunno, ksendal, and Proske 2008, Theorem 3.5) or (Nualart 1995, Proposition 1.2.2). The next Lemma pertains to the closability of the Malliavin derivative. Lemma 2.3 (Closability). Let F L2 (P) and (Fk )k

D1,2 such that

Fk F in L2 (P) and D Fk converges in L2 (P; K) Then F D1,2 and D Fk D F in L2 (P; K)
Proof. See the arguments in (Di Nunno, ksendal, and Proske 2008, Theorem 3.3). Example 2.1 (Zero Coupon Bond). As before let P(t, T ) be the price at time t of a zero coupon bond, which pays $1 at maturity T. Then using the instantaneous forward rates f(t, s), 0 t s, we have that

ft (x) dx

We nd that

T t T t

Dr,y ft (x) dx

1[0,t] (r) dx

= (T t)1[0,t] (r), where 1[0,t] is the indicator function of [0, t]. Then the chain rule of Lemma 2.2 (in connection with Lemma 2.3) shows that the stochastic duration D P(t, T ) of P(t, T ) in the HJM model is given by Dr,y P(t, T ) = (T t)P(t, T ) 0 , if 0 r t , otherwise
So Dr,y P(t, T )/P(t, T ), 0 r t, has the form of the classical duration in Section 1. The latter expression seems to suggest that we should rather use D F/F as a generalized duration than D F. However, a general interest rate claim F may be zero for a positive probability. Therefore it is reasonable to introduce D F as an expanded concept of duration. Note that our denition does not generalize Macaulays duration in the sense that D F gives the classical duration if the interest rate claim F is deterministic, that is, a functional of a deterministic (piecewise at) yield surface. The explanation for this is that the duration concepts are based on different interest rate models. The classical duration presumes yield surfaces which are at or piecewise at. Such a model is fundamentally different from a stochastic interest rate model. For example, under our conditions yield surfaces in our [risk-neutral] HJM model only assume a certain constant value with probability zero. In view of this we may therefore consider the stochastic duration as a concept which is analogous to the classical one in the HJM setting. Example 2.2 (Interest Rate Cap). Consider a cap of the form F = R(t, T ) K
where K is the cap rate and R(t, T ) the average interest rate given by

1 R(t, T ) = r(s) ds T t

Here r(t) = f(t, t) is the overnight interest rate, also known as the short rate. We observe that

1 r(s) ds T t

1 = Dr,y r(s) ds T t

1 Dr,y fs (0) ds T t

= 1[0,t] (r)
Now let us approximate the (x) : = (x K)+ by functions {n } with n (x) = (x) for |x K| and 0 n (x) 1 for all x 1 n
Then it follows from Lemma 2.2 and Lemma 2.3 that Dr,y F = 1[K,) R(t, T ) 1[0,t] (r) Example 2.3 (Asian Option). Let us also have a look at the following Asian type of option dened as 1 F= (x2 x1 )(T2 T1 ) Then 1 Dr,y F = (x2 x1 )(T2 T1 ) = 1[0,t] (r) 3. E STIMATION OF S TOCHASTIC D URATION C ONSTRUCTION OF I MMUNIZATION S TRATEGIES

x 2 T2 x 2 T2

ft (x) dt dx

x 1 T1

1[0,t] (r) dt dx

AND THE

In the previous section we introduced the concept of stochastic duration Dt,y F and gave examples of interest rate derivatives F whose stochastic duration can be computed explicitly. In general, the stochastic duration of an interest claim or a complex bond portfolio cannot be determined explicitly. The latter is also due to the fact that, e.g., a dynamically hedged bond portfolio is a stochastically weighted sum of interest rate claims. The weights of the portfolio or hedging strategy at any time point are usually complicated functionals of the stochastic forward curve. In order to overcome this deciency we aim at resorting to an estimate of Dt,y F. A reasonable estimate of Dt,y F could be the expected stochastic duration of F given the observed forward curves fs , 0 s t. This estimate naturally appears in the ClarkOcone formula or as a solution of a backward stochastic differential equation [BSDE]. Using the fact that the set exp I1 (h)
is total in L2 (, F, P) one nds in connection with Relation (2.15) the ClarkOcone formula wrt the forward curves ft takes the following form. See also(Di Nunno, ksendal, and Proske 2008).
F = EP [F] + E[D (F) | Fs ] dfs , s
where the B([0, T ]) F, B(H )-measurable map D (F) : [0, T ] H can be linearly isometrically identied with the Malliavin derivative, i.e., stochastic duration, D F in Denition 2.1. Further, F L2 (, F, P) is in the domain of D and Ft is the P-completed ltration generated by fs , 0 s t. The H -valued conditional expectation E[D (F) | Ft ], t 0 t T
can be regarded as an estimation of D F. Now let us us have a look at the BSDE

Yt = YT Zs dfs ,

where YT = F. Then we observe that Zt = E[D (F) | Ft ] t P a.e.
for 0 t T , a.e. We wish to recast the dynamics of the solution (Yt , Zt ) in Equation (3.1) wrt the original measure P. Since t is invertible t-a.e. we see that the natural ltration of Wt coincides with the ltration Ft. Assume that there exists a unique strong solution f of the SPDE t
= 1 [Af + s (s, )] ds + Wt , s s
where Wt is the Q-cylindrical Wiener process in Equation (2.12). See, e.g., (Prvt and Rckner 2007) for criteria about the existence and uniqueness of solutions of non-linear SPDEs. Remark 3.1. Let t = b(t, ft ) in Equation (3.2) for a Borel measurable map b : [0, T ] H H. Impose on A the rather strong condition to be a bounded operator on H. Further assume that the drift coefcient F(t, x) : = 1 [Ax + b(t, x)] satises a linear growth and Lipschitz condition wrt x, t uniformly in t. Then the Picard iteration gives a unique strong solution of Equation (3.2). The Assumption of Equation (3.2) entails that the natural ltration of Wt is given by Ft. Then it follows from Equation (2.12) that the solution (Yt , Zt ) in Equation (3.1) has the following BSDE dynamics under P.

T T Yt = YT + Zs [Afs + s (s, )] ds Zs dWs t t
YT = F, where W is the square integrable H-valued martingale given by

= s dWs

So we see that the estimate Zt of the stochastic duration of F satises the forward-backward stochastic partial differential equation [FBSPDE] dft = Aft + t dt + t dWt

YT = F,

where F is a measurable functional of the solution of the forward SPDE, i.e., of the forward curves ft. For more information about linear forward-backward S(P)DEs the reader may consult (Ma and Yong 1999). See also (ksendal, Proske, and Zhang 2005). Remark 3.2. In view of nancial applications it would be desirable to develop a numerical approximation scheme for solutions (Yt , Zt ) of FBSPDEs of the type of Equation (3.3). In general, this is a challenging task. A possible ansatz to this problem (in some special cases) would be to employ the results in (Zhang 2004) or in (Nakayama 2002) in connection with Galerkin approximation. Another approach could be based on nite element or nite difference schemes in a backward stochastic partial differential equation [BSPDE] setting. In the framework of the linear Gaussian model, as in Equation (2.9), for the forward curves one can simplify further the numerical analysis by using dimension reduction techniques as, e.g., principal component analysis of interest rate data. See (Carmona and Tehranchi 2006). Remark 3.3. Using stochastic distribution theory the concept of stochastic duration for interest rate claims F D1,2 can be extended to the case of claims contained in a space of generalized random variables which comprises the space of square integrable functionals of the forward curves wrt P. See, e.g., (stnel 1995) or (Da Prato and Zabczyk 1992). As a consequence we may still interpret Zt in Equation (3.3) as an estimate of the stochastic duration of a claim F, when F L2 (P) L2 (P). Finally, we want to discuss an extension of the concept of delta hedge of interest rate sensitive securities developed by (Hull and White 1994) to a stochastic setting, which involves the uctuations of the whole yield surface. The purpose of delta hedge is to immunize portfolios of interest-rate-sensitive securities under Hos interest rate scenario (Ho 1992). In other words, the idea devised by (Hull and White 1994) is to neutralize given nancial positions in interest-rate derivatives against parallel shifts of i-years spot rates (or key rates). We want to propose a mathematical framework which facilitates the construction of immunization strategies of interest-rate-sensitive portfolios in the sense of (Hull and White 1994) wrt stochastic uctuations of the yield surface. In fact, we aim at minimizing the exposure of given nancial positions to interest rate risk by going short in bonds of a generalized bond portfolio, that is, of self-nancing portfolios composed of innitely many bonds of any maturity. To this end we need some notions and conditions. Suppose that the generalized HJMmodel [see Equation (2.10)] for the forward curves ft fullls the HJM no-arbitrage condition

t (x) =

(k) t (x)

(k) Ix t 0

t (u) du + t
where the processes t , k
1, are the Fourier coefcients of a predictable H-valued process t =
Here {ek } is an optimal normal basis of H. Further t , k and Ix is a linear functional in H dened by
1, is given as in Equation (2.6)

Ix (f) = f(u) du

We remark that the processes t , k 1, admit the interpretation of market prices of risk wrt different bond maturities. Now let us consider the discounted bond price curve Pt () given by
Pt (x) = exp We require that the conditions

fs (0) ds fs (x) ds

s , dWs

t s 1/2

ds <
hold for all t 0. Then, using Its Formula and Girsanovs Theorem one nds that

(3.4) where

P(t, T ) = P(0, T ) P(s, T )IT s s dWs ,

Wt = Wt + s ds

is a Q-Wiener process under a local martingale measure P. Dene (3.5) t (, x) = Pt (x)Ix t Let G be a separable Hilbert space in C [0, ) such that evaluation functionals x on G are continuous and the semigroup St of left shift operators is strongly continuous on G. See Equations (2.7) and (2.8). From now forward we assume that t in Equation (3.5) is a
predictable L(U0 , G)-valued process such that
ds < a.e. The latter implies that
the bond price curves Pt are G-valued and satisfy dPt = APt dt t dWt or dPt = (APt t [t ]) dt t dWt in the mild sense. Now let us consider generalized bond portfolios. See (Bjrk, Masi, Kabanov, and Runggaldier 1997). That is, the wealth process Vt of such portfolios is given by Vt = Vt () : = t [Pt ()], G -valued t 0, where t is a predictable process. The process t can be regarded as the trading strategy of an investor who manages a portfolio with innitely many bonds of any maturity. For example, the strategy t = T t stands for buying and holding a zero-coupon bond with maturity T , since t [Pt ()] = P(0, T ). Assume that
ds < 0, is self-nancing if there is a
for all t 0. Then we shall say that a trading strategy t , t V0 R such that

(3.6) for all t

Vt () s s dWs = V0
0 a.e. where Vt () is the discounted wealth process given by Vt () = t [Pt ()]
See, e.g., (Bjrk, Masi, Kabanov, and Runggaldier 1997). We denote the set of all self-nancing strategies by A. Remark 3.4. In the innite-dimensional HJM-framework the existence of a unique martingale measure does not imply in general that the bond market given by Equation (3.4) is complete. The latter is a deciency not shared by nite-rank models. However, since the kernels of t , as in Equation (3.5), are zero t-a.e. our bond market is approximately complete in the following sense. For all > 0 there exists a strategy A

EP [h] + s s dWs h

< ,
where h a discounted contingent claim. See, e.g., (Bjrk, Masi, Kabanov, and Runggaldier 1997). Suppose that a trader is long in interest rate securities at time t 0 whose price process is Lt. In order to neutralize the risk coming from the uctuations of the yield surface the trader wishes to go short in the generalized bond portfolio, as in Equation (3.6), for a self-nancing strategy A such that minimizes at any time point the worst-scenario interest rate
sensitivity of the resulting portfolio. More precisely, the trader tries to nd a A such that

D Lt Vt ()

2 K dt

D Lt Vt ( )

where K is the RKHS of the forward curves. Note that sup

D F, k

for an interest claim F D1,2. So Equation (2.17) admits the interpretation that D F K is the worst-scenario sensitivity with respect to all directional interest changes k K. Using the estimate Z = Z (F) for the stochastic duration D (F) in the FBSPDE of Equation (3.3) for F = Lt Vt () [see Remark 3.3 and Relation 2.15] the optimization problem of Equation (3.7) then takes the form

Zu Lt Vt () u

Zu Lt Vt ( ) u

du dt <

for A. We see that the construction of an immunized bond portfolio reduces to an optimal control problem of the FBSPDE of Equation (3.3) or the FBSPDE

Vt () = V0 () s s dWs

0 T T Yt = YT + Zs [Afs + s (s, )] ds Zs dWs t t
YT = F, where F = Lt Vt () for each t, if Lt is a measurable functional of V (). An approach to tackle this problem could be based on a stochastic maximum principle for FBSPDEs. See (Haadem, Kettler, Mandrekar, Proske, and ksendal 2011). From a practical point of view it would be important to nd numerical approximation schemes for a delta hedge A. Remark 3.5. (1) It is conceivable that the concept of g-expectation by (Peng 1997) for BSDEs can be generalized to FBSPDEs of the type of Equation (3.3). The latter would enable the construction of risk measures of functionals of forward curves. Such a construction would reveal the role of the stochastic duration as a building block for general interest rate risk measures. (2) We point out that our framework also allows for the denition of stochastic convexity, that is, a measure of curvature wrt the uctuations of the yield surface. It makes sense to dene

the stochastic duration of a twice Malliavin differentiable interest rate claim F as D D (F) L2 (, F, P; K K) A CKNOWLEDGMENT The authors wish to thank Vidyadhar S. Mandrekar for his valuable comments on this work.
A PPENDIX A. M ACAULAY DURATION AND PORTFOLIO IMMUNIZATION A.1. Macaulay duration. Consider the discrete and continuous cases separately. A.1.1. Discrete case. In Macaulays original concept duration was the weighted average by present value of the number of periods to maturity for a series of cash ows, typically those of interest and principal payments for a bond, normalized by the total present value (Macaulay 1938). For notation, let V be the present value (or price) of the bond, r > 0 be the [constant] rate of interest, and n be the number of periods to maturity. The expression 1 (1 + r)n r is the closed form for the present value of an annuity in arrears for n periods at rate r, reecting the typical payment scheme of a bond, e.g., a United States Treasury bond. Therefore the Macaulay duration dMac has the following denition for equally spaced cash ows of size C and return of principal P. A(r, n) =

C dMac : = C or

k(1 + r)k + nP(1 + r)n , (1 + r)k + P(1 + r)n

k=1 n k=1

log C A(r, n) + P(1 + r)n r In the simple case of a single cash ow a zero coupon bond Macaulay duration reduces to the number of periods n to that payment, justifying the name. Soon, however, practitioners began preferring a version of duration as the simple negative of the derivative of V with respect to r, dropping the factor (1 + r). This version became known as the modied duration dmod , with this denition. (A.1) dMac = (1 + r) (A.2) dmod : = log C A(r, n) + P(1 + r)n r
Such redenition provides the relationship dMac = (1 + r)dmod , so that the modied duration of a zero coupon bond is (1 + r)n. In ordinary parlance, either form of duration is stated as a positive number, e.g., The duration of this bond is ten years, as indicated. A rationale exists, however, for stating the duration as a negative number, reecting the inverse relationship between changes in the level of interest and changes in price. Such versions, inverting the minus signs of Equations (A.1) and (A.2), more typically appear in Taylor series expansions of bond price, and in more developed mathematical expositions. The latter approach is assumed in this paper. A.1.2. Continuous case. The continuous case is a straightforward extension of the discrete case. Let C, as previously, be the cash ow assigned to a single period, but consider it divided equally into j parts owing at the ends of j equally spaced sub-periods. As well, consider the interest rate r as that assigned to the entire period, but let it be divided by j providing a sub-rate for compounding across the sub-periods.

The term C A(r, n) of Equation (A.1) then becomes C A(r, n) : = lim C 1 (1 + r/j)jn j j r/j rn 1e =C r
So, if 1 ern , r then Equations (A.1) and (A.2), respectively, become A(r, n) : = dMac = and dmod = log C A(r, n) + Pern , r log C A(r, n) + Pern r
in the latter case because limj (1 + r/j) = 1. So (A.3) dMac = dmod ,
justifying the use of the combined name continuous duration for both versions. As in the case of discrete Macaulay duration, in the simple case of a zero coupon bond continuous duration reduces to the number of periods n to that payment. An alternative description of this result is that the modied duration is a continuous approximation to the Macaulay duration, or conversely, the Macaulay duration is a discrete approximation to the modied duration. As n with rn constant the two denitions merge. It is stated without proof that the other common form of annuity timing, payments in advance, i.e., at the beginnings of the compounding periods rather than at the ends, results in the same continuous forms of Equation (A.3). A.2. Portfolio immunization. An active part of portfolio management is the targeting of a specic duration. For example, a pension fund manager may wish to have a value certain at some future time t = T , starting at t = 0 now. Consider two portfolios A and B, with respective durations dA and dB , and present values (prices) of vA and vB. If these portfolios are combined, then the new portfolio A + B has duration vB vA dA+B = dA + dB vA + vB vA + vB If A be the portfolio to be immunized to desired duration dA+B , then one can solve for vB knowing all other quantities. Specically, vB = dA+B dA vA , dB dA+B
which may be positive or negative. If negative one can interpret the result as an amount proportioned to portfolio B to be sold from portfolio A to achieve the objective, or alternatively, the amount to sell short of portfolio B.
Bond immunization is a very big business. In recent years Japanese banking interests have been heavy buyers of 30-year United States Treasury Bond strips having a duration of 30 years in order to extend the durations of portfolios. The activity has been so signicant as to keep the longest-term yields below those of somewhat shorter-term yields for extended periods of time, even in strongly positive yield curve environments otherwise.

References

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(Paul C. Kettler) C ENTRE OF M ATHEMATICS FOR A PPLICATIONS D EPARTMENT OF M ATHEMATICS U NIVERSITY OF O SLO P.O. B OX 1053, B LINDERN N0316 O SLO N ORWAY E-mail address: paulck@math.uio.no URL: http://www.math.uio.no/paulck/ (Frank Proske) C ENTRE OF M ATHEMATICS FOR A PPLICATIONS D EPARTMENT OF M ATHEMATICS U NIVERSITY OF O SLO P.O. B OX 1053, B LINDERN N0316 O SLO N ORWAY E-mail address: proske@math.uio.no URL: http://www.math.uio.no/ (Mark Rubtsov) C ENTRE OF M ATHEMATICS FOR A PPLICATIONS D EPARTMENT OF M ATHEMATICS U NIVERSITY OF O SLO P.O. B OX 1053, B LINDERN N0316 O SLO N ORWAY E-mail address: mark.rubtsov@cma.uio.no URL: http://www.math.uio.no/

LVY-COPULA-DRIVEN FINANCIAL PROCESSES
PAUL C. KETTLER Abstract. This paper proposes a general non-Gaussian Ornstein-Uhlenbeck model for a joint nancial process based on marginal Lvy measures joined by a Lvy copula. Simulated processes then result from choices of marginal measures and Lvy copulas, with resulting statistics and inferences. Selected for analysis are the 3/2-stable and Gamma marginal Lvy measures, along with Clayton, Gumbel, and Complementary Gumbel Lvy versions of ordinary [probability] copulas, with the last two being here introduced. A relationship between the original coupled subordinated processes and the terminal dependency relationship between the simulated variables is observed and calibrated. Normal inverse Gaussian and tempered stable measures are also noted, as are additional Lvy copulas constructed from the Gumbel and Frank ordinary copulas, with some analysis and suggestion for using them in future research.
1. Introduction A recent work of Fred Espen Benth with the author (Benth and Kettler 2006) investigated the relationship between electricity and gas prices by estimating marginal distributions and a theoretical copula joining them. That study simulated the model process, concluding with option prices for the spark spread, the dierence of these two prices. This paper proposes a general non-Gaussian Ornstein-Uhlenbeck subordinated model for a joint nancial process. The model is founded not on process laws and corresponding marginal distributions with an ordinary [probability] copula, but rather on marginal Lvy measures joined by a Lvy copula. Simulated processes then result from choices for these measures and copula. Statistical analysis produces summary results, and a section on theory probes the relationship between an originating subordinator and the terminal relationship of the simulated variables. The principal inferences and conclusions of this study are that the choice of Lvy copula is not material in dierentiating the character or statistics of the price series, and that the terminal ordinary copula of the logarithmic price relatives is nearly the independent copula, regardless of choice of subordinating Lvy processes. These ndings imply that nancial processes modeled in this fashion are robust across functional forms and parameter settings. As well, the resulting logarithmic price relatives exhibit marked departure from normal distributions, an anticipated result, given the character of the marginal driving Lvy measures. The
Date: 12 January 2009. 2000 Mathematics Subject Classication. Primary: 91B24, 91B70. Secondary: 62M10, 62M20. 1991 Journal of Economic Literature Subject Classication. C51, G13. Key words and phrases. Lvy copula, nance, stochastic processes, model construction, simulation, time series. The author wishes to thank Fred Espen Benth and Frank Proske for valuable insights. The R Foundation for Statistical Computing made available the statistical packages for this study (R Development Core Team 2005; Wrtz et al. 2005; Genz, Bretz, and Hothorn (R port) 2005).

calibrations are interesting, as evidenced in various summary statistics such as the AndersonDarling test for normality. Appendices A and B provide background information respectively on Lvy copulas and Ornstein-Uhlenbeck processes.
2. A general subordinated model The paper is a report on research into the joint behavior of stock prices when they are dened in a geometric process with dependence on subordinated pure jump Ornstein-Uhlenbeck Lvy process. Within the subordinated process one joins marginal Lvy measures by a specied Lvy copula to produce stochastic variables then introduced into the geometric process. This structure of subordination is a Background Driving Lvy Process (BDLP) in the manner of Barndor-Nielsen and Shephard. See, e.g., (Barndor-Nielsen and Shephard 2001, Section 1.1, pp. 167169). Here is the setup, beginning with the coupled Ornstein-Uhlenbeck process in the two dimensional case. (2.1) dYt1 = 1 Yt1 dt + dL1 , Y01 = 0 t dYt2 = 2 Yt2 dt + dL2 , Y02 = 0, t
where L1 and L2 are the subordinators. The variables (Yt1 , Yt2 ) then enter the geometric t t equations as follows.
1 d log St = (1 + 1 Yt1 ) dt + 2 d log St = (2 + 2 Yt2 ) dt + Yt1 dBt , log S0 = 2 Yt2 dBt , log S0 = 0,
where Bt Bt are Brownian motions. The experimental design then calls for simulation of the joint Ornstein-Uhlenbeck process of Equations (2.1) with a Lvy copula, followed by simulation of the joint geometric process of Equations (2.2). The study begins by examining the relationship of the subordinators through a Lvy copula, by example, and continues through analysis of the simulated joint stock price series, with accompanying tables and charts.
3. Random selection from a Lvy copula Recall that a Lvy copula is like an ordinary copula in that it is a function which retains all of the dependence information of a Lvy measure, while leaving all of the remaining information in the marginal Lvy measures. Let ( dx dy) be such a bivariate Lvy measure. Tail integrals of this measure, which are the analogues of distribution functions, are dened as follows. First 2 for the joint measure, in this study supported on R+ ,

U (a, b) : =

( dx dy)
and for the marginal measures,

U1 (a) : =

U2 (b) : =
The Lvy copula CL (u, v), then, dened on the same domain, is this.

CL (u, v) : = U U1 (u), U2 (v) ,
or equivalently, CL U1 (a), U2 (b) : = U (a, b), assuming all inverses are dened in the generalized sense. Remark. One may think of a Lvy copula as itself a joint Lvy measure with uniform margins, much in the same sense that an ordinary copula is a joint probability distribution with uniform margins. In the case of the Lvy copula the uniform margins extend from [0, ], whereas with the ordinary copula they extend from [0, 1]. In the sense of information, the Lvy copula provides only the dependency, and nothing of the margins, again in the same sense as an ordinary copula with its margins. Together, the Lvy copula and its margins provide the entire information set of the joint Lvy measure. For illustration consider a Clayton-Lvy copula subordinator model with common -stable marginal Lvy measures. This is one of the six pairwise choices of copula and marginal measure for the later simulations. At the heart of selecting a jump pair is the choice of point in the copular domain. A presentation on this process appears here (Tankov 2003, Example 5.1, p. 20), and follows this plan. Let C(u, v) be a Clayton-Lvy copula as such. C(u, v) : = u + v

, (u, v) [0, )2 , > 0

To simulate a joint -stable subordinator on the chosen unit time interval one generates processes Xs and Ys given the common marginal tail integral of (3.1) for which the inverse is (3.2) U 1 (y) = y

U (x) = x ,

The -stable subordinator has nite activity if < 1 because |x| integrates the measure of the small jumps. In the simulations, however, the choice is = 3/2 to be more representative of what is observed in the nancial markets. Specically in the present context, applying the Fundamental Theorem of the Calculus to U (x),

xU (x) =

>0 1
Call i the ith jump time of a Poisson process with intensity , and select a pair (Wi,1 , Wi,2 ) of independent uniform variates on [0, 1]. Then,

(1) Xs

U 1 (i )1{[0,s]} (Wi,1 ) U 1 F 1 (Wi,2 |i ) 1{[0,s]} (Wi,1 ),

(2) Xs =

Further, given the conditional distribution on the copula as (3.5) it follows that (3.6) F
u F (u, v) = 1+ F (v|u) = u v

1 (1+ )

(Wi,2 |u) = u

Wi,21+

Equations (3.4) appear also in (Cont and Tankov 2004, Chapter 6, Section 5, p. 195). The Clayton-Lvy copula is the only one of the copulas chosen for simulation which admits a closed-form expression for the inverse of the conditional copula distribution. The others require numeric inversion procedures for their conditional distributions. For the present modeling purposes one wishes to simulate the BDLP by selecting jump times {i }, 1 i N (T ), from a standard Poisson process over a revised time interval [0, T ], (1) (2) and then to calculate paired jumps xi , xi at these times. As the distribution of a waiting time i : = i i1 , with 0 = 0, conventionally, is (i ) = 1 exp(i ) = Wi,1 , so i = 1 (Wi,1 ) One then constructs the {i } iteratively as i = i1 + i , continuing until determining N (T ). Next, with a view to the discrete simulation, construct an inventory of N (T ) paired jumps in the manner of Equations (3.4), of which this is the ith. (3.7) xi xi

= U 1 (i ) = U 1 F 1 (Wi,2 |i )
These jumps shall appear in order of the {Wi,1 }, as well, in harmony with Equations (3.4).
Remark. This diagram shows the sequence of calculations to produce the ith pair of jump (1) (2) components xi , xi. i = u v 1 1

U F 1 (Wi,2 |)

xi 4. Models
One may address models other than the Clayton-Lvy subordinator model with -stable margins by allowing either other copulas or other margins, or both. Further, one may consider n bidirectional copulas and margins, meaning those non-trivially supported on R \ {0}, with or without subordination. Among the marginal choices are the Gamma, normal inverse Gaussian (NIG), and the tempered stable processes (including as a limit the variance gamma,) and the bidirectional -stable process, among others. Copula choices include the Gumbel-Lvy, herein dened, and Complementary Gumbel-Lvy, called complementary because its generator is the inverse of the Gumbel-Lvy generator.1 The study now proceeds to examine some combinations of these seriatim. For the -stable and Gamma processes tail integrals and their inverses exist in closed form. For the -stable processes one has Equations (3.1) and (3.2). For the Gamma processes one has these. (4.1) for which the inverse is (4.2) UG

U G (x) = e x ,

y 1 (y) = max 0, log
See (Barndor-Nielsen and Shephard 2001, Section 2.3.4, p. 175). The Lvy measure NIG (x) on R \ {0} of the NIG(, , , ) process is this, with notation of (Barndor-Nielsen 1998, p. 47, Equation 2.9). K1 () is the modied Bessel function of third order and index 1. As well, > 0 and 0 || . (4.3) NIG (x) = K1 ( |x|)ex |x|
The Lvy measure TS (x) on R \ {0} of the tempered stable processes is this, with notation of (Cont and Tankov 2004, Chapter 4, Section 5, p. 119, Equation 4.26). As well, c , c+ , , and + are positive coecients, and > 0. c |x| c+ (4.4) TS (x) = 1{x<0} + 1+ e+ x 1{x>0} 1+ e x |x| The limiting case for = 0, and c : = c = c+ is the Lvy measure of the variance gamma process. See (Tankov 2006, p. 3, Equation 2.4).
1Your author has chosen these names in honor of the late Professor Emil Julius Gumbel, founder of extreme

value theory and Nazi antagonist. As there are many ways to chose Lvy copulas inspired by ordinary copulas, these are only two such choices.
The inverse tail integrals of the NIG and tempered stable processes are only known by numerical approximation. Though these processes are of interest to nancial economists and mathematicians, these ideas are left for future study. For pertinent reading on the relationship between process probability and Lvy densities, including that of the Gamma distribution, see (Barndor-Nielsen 2000). Both the NIG and tempered stable processes have innite activity, for the measures do not integrate |x| near {0}, cf. Equation(3.3). Following are the functional representations of the named Lvy copulas, including a bidirectional Clayton-Lvy version (Tankov 2006, p. 6, Equation 3.1), adapted from ordinary copulas of the same names. See (Cherubini, Luciano, and Vecchiato 2004, p. 124) for a presentation on ordinary copulas. Included for comparison are the Product-Lvy (Independent) and FrchetLvy upper limit copula C (u, v); no analogous Lvy version exists for the Frchet-Lvy lower limit copula. Clayton-Lvy: (4.5) C(u, v) = u + v

, >0

Clayton-Lvy, bidirectional: (4.6) CB (u, v) = |u|
1{uv0} (1 )1{uv<0} , > 0
Gumbel-Lvy: (4.7) CG (u, v) = exp log(u + 1)

+ log(v + 1)

1, > 0
Complementary Gumbel-Lvy: (4.8) CG (u, v) = log exp u + exp v 1
Product-Lvy (Independent) for marginal Lvy measures 1 , 2 [0, ]: u v C (u, v) = u + v 0 : (u, v) [0, 1 ] [2 ] : (u, v) [0, 2 ] [1 ] : (u, v) = (1 , 2 ) : elsewhere
Frchet-Lvy Upper: (4.10) C (u, v) = min(u, v)
The following functions generate, respectively, the Clayton-Lvy, Gumbel-Lvy, and Complementary Gumbel-Lvy copulas in Archimedean analogy to their corresponding ordinary
copulas. In each case : [0, ] [0, ]. C (x) : = x G (x) : = [log(x + 1)] G (x) : = exp x 1 For a discussion of Lvy copula generation see (Kallsen and Tankov 2006, Section 6, pp. 21 23) and (Tankov 2003, Proposition 4.5, pp. 1516). Note that G () and G () are inverses of each other (after re-parameterizing to 1/ in either formulation.) Figures 1 and 2 display a Clayton-Lvy copula and its level curves; Figures 3 and 4 display a Gumbel-Lvy copula and its level curves; Figures 5 and 6 display a Complementary GumbelLvy copula and its level curves. In each case = 1. Observe the vertical scale of these. C(20, 20) = 10.0000 for the Clayton-Lvy copula; CG (20, 20) = 3.5826 for the Gumbel-Lvy copula; CG (20, 20) = 10.2439 for the Complementary Gumbel-Lvy copula. Compare these values with C(20, ) = C(, 20) = 20, as for the other (and all) Lvy copulas. Alternative generation of Lvy copulas comes from reference to an ordinary copula by way of a generator : [0, 1] [0, ]. Such procedures extend the possibilities for creating useful copulas in empirical research. For instance, one can begin with ordinary copulas such as the Gumbel and Frank, respectively. CG (u, v) = exp ( log u) + ( log v)

[1, ), with Product copula for = 1 exp(u) 1 exp(v) CF (u, v) = log 1 + exp() 1 (, ) \ {0}, with Product copula for = 0 For a discussion of Lvy copula generation in this form also see (Kallsen and Tankov 2006, loc. cit.) and (Tankov 2003, loc. cit.). An example of such a generator, as proered in (Tankov 2004, Theorem 5.1, pp. 167169) is (x) = x/(1 x); another is (x) = log(1 x). 5. Simulation The simulation proceeds in two phases, the rst to develop the subordinated process, as displayed in Equations (2.1), the second to develop the geometric process, as displayed in Equations (2.2). Six models are selected, taking 3/2-stable subordinators or the Gamma subordinators, and coupling them by a Clayton-Lvy, Gumbel-Lvy, or Complementary GumbelLvy copula, with chosen parameters. The calculations include charts in the Clayton-Lvy copula choice to illustrate the ndings. 5.1. The subordinated process. The way is clear now to devise an algorithm for generating sequences of jumps joined by a Lvy copula. This algorithm generalizes mutatis mutandis to marginal processes other than the -stable and to copulas other than the Clayton-Lvy, as this paper explores in the sequel. Consider now that U () is the tail integral of an arbitrary Lvy measure.
(1) Select and T , then create a series of jump times {i }, 1 i N (T ), by exponential delay. Note that if : = U 1 (T ), then jumps smaller than , dened now as small (1) jumps, in the xi series will be eliminated, for x1 x2 . xN (T ) , owing to the monotonicity of U (). (1) (2) (2) Calculate an inventory of incremental jump component pairs xi , xi. (3) Calculate Yt1 , Yt2 iteratively as the accumulation of these jumps following interjump exponential declines. Select the jumps for inclusion at time j on the order of the {Wi,1 }, now indexing the BDLP by the jump times, as follows.

N (T ) (1) (1) (1)

Yj1 (5.1)

i=1 N (T )

xi 1{j1 /T <Wi,1 j /T } , Y01 = 0 xi 1{j1 /T <Wi,1 j /T } , Y02 = 0

i=1 (2)

1 Yj2 = e2 j Yj1 +
The rst terms on the right of Equations (5.1) represent the inter-jump exponential declines of the Ornstein-Uhlenbeck process, whereas the second terms represent the accumulated jumps occurring between times j1 and j. The jumps are indicated (literally) for inclusion by the {Wi,1 }, but actually occur when the next jump time j appears. By this means the subordinator remains stationary in that the expected size of the accumulated jumps at a jump time is proportional to the waiting time. Remark. Jumps catalogued by this algorithm in the xi complement the small jumps. Observe that is such that (5.2)
series are dened large jumps, to

U () = U U 1 (T ) = T

Thus the Lvy measure of the large jumps, and therefore the intensity of the compound Poisson process they represent, is T , independent of U (). The small jumps, and a method for including them in the study, is the subject of Section 5.4. 5.2. Finite sample bias. In selecting pairs of jumps, the rst coordinate jump, computed as in the rst of Equations (3.7), is limited to a lower bound of , as reported in Subsection 5.1. The second coordinate jump, computed as in the second of Equations (3.7), is not so limited. In consequence, a bias exists in jump selection leading to expected lower values in the second jump. The phenomenon is most pronounced for the Clayton-Lvy copula, so the correction proposed is only implemented in that case. To counteract the observed bias the simulations also restrict the second coordinate jump to a lower bound of. This selection arrives in a direct manner by choosing the uniform random variable Wi,2 not on the interval [0, 1], but rather on the interval [0, ri ], with ri = F (T |i ) chosen by the following reasoning. The revised requirement is that (5.3) So U xi

= U 1 F 1 (Wi,2 |i )

= F 1 (Wi,2 |i ) U () = T
by Equation (5.2) and because U () is monotone decreasing. Therefore, (5.4) F U xi
i = Wi,2 F (U ()|i ) = F (T |i ) = : r,
independent of U (), as F (|i ) is monotone increasing. An alternative plan would be to require E xi
This scheme, while better in some ways, would make ri dependent on U (), as revealed by Equation (5.3). 5.3. The geometric process. Herein one simply takes the Yj1 , Yj2 terms developed by simulating the subordinated Ornstein-Uhlenbeck process, inserting them into the discrete time version of the geometric process, cf. Equations (2.2), as so. This is implementation of Eulers Method (rst order) on the deterministic part. (5.5)
1 log Sj = log Sj1 + (1 + 1 Yj1 )j + 2 log Sj = log Sj1 + (2 + 2 Yj1 )j + 1 Yj1 Bj , log S0 = Yj1 Bj , log S0 = 0, log Sj , log Sj
where Bj Bj are Brownian motions. Exponentiating the the recovery of the Sj , Sj

series allows

series.
5.4. Amussen-Rosiski modication. The processes articulated in Section 4 are necessarily approximate in that small jumps, those below the threshold of such as those computed in the -stable and Gamma processes in Equations (3.2) and (4.2), are ignored. One can improve on this methodology by employing a method articulated by Amussen and Rosiski to approximate the small jumps by a Brownian motion. The primary reference is (Amussen and Rosiski 2001), with additional presentations in (Rosiski 2006; Prause 1999; Rosiski 1991). The essence of the argument, with results incorporated in the simulations of this study, is that one can approximate the small jumps of a Lvy process of innite measure frequently, but not always, by a Brownian motion with drift. Therein, the authors provide a necessary and sucient condition that the normal approximation, as this capability is called, does not hold for any process with nite Lvy measure, such as the compound Poisson process, nor for the Gamma process, but does hold for the -stable process for the entire admissible set { 0 < 2}. See Equation (5.8) below. For the NIG process see (Amussen and Rosiski 2001, Theorem 2.1 and Proposition 2.1, and Examples 2.12.5, pp. 484486). 1 For the simulations using Gamma Lvy margins, is set to T so that G : = U G (T ) = 0, G (0) = < reecting the state of the Gamma process as having no small jumps. Insofar as U , the Gamma process has nite variation, and thus is a compound Poisson process. Figure 7 displays conditional copula distribution functions in the manner of Equation (3.5), which appears for the Clayton-Lvy copula along with similar formulations for the GumbelLvy and Complementary Gumbel-Lvy copulas. In each case the point of conditional evaluation is u = 2. The rank of vertical scaling described for the copulas is evident in these measures also for evaluations at (u, v) = (2, 5), at the right hand boundary of this chart. Figure 8 displays the marginal Lvy measure for the 3/2-stable subordinate process with parameter = 1. For the Gamma subordinate process (not shown) the parameter choices

Gumbel-Lvy copula,) the independent limit of the Clayton family. Further, the projected views of the empirical copulas, as appearing in Figures 9 and 10 for the ClaytonLvy copula with 3/2-stable margins show only patterns which are attributable to the accumulation of computational errors; specically they exhibit low amplitude wave patterns typical of truncation errors in evaluating transcendental functions by series methods. (2) The logarithmic price relative series are distinctly not normal, exhibiting signicant skewness and kurtosis, as revealed by all the Anderson-Darling and related statistics appearing in Table 2. This is an expected result, given the nature of the driving 3/2-stable and Gamma Lvy marginal subordinating processes. (3) The choice of copula is not important in determining the quality of the inferences in the two items above. 6. Conclusions This study established that the proposed model provides a computationally reasonable scheme for generating nancial processes. The model incorporates the freedom to describe the dependency relationship between variables with the generality of a Lvy copula, while also permitting exible jump processes as often required. Financial process modeling of the fashion proposed by this study appears to be robust across choices of marginal Lvy measures and Lvy copulas. Subtle distinctions are evident, but in general all of the developed processes are remarkably similar. Planned future research includes delving into the theory of Lvy-copula-driven nancial processes by establishing a set of rst principles, thus enabling informed prediction of terminal processes and copulas from the subordinators, ex ante.
Appendices A. Lvy copulas A Lvy copula is much like an ordinary [probability] copula in that it distills the dependence information from a joint measure in this instance a joint Lvy measure leaving the information from the marginal measures distinct. The main dierence to an ordinary copula is that one integrates the Lvy measures in the upper tails rather than in the lower tails. This convention obtains for frequently the Lvy measures of interest are supported on the positive real line, as with subordinators, and Lvy measures become uniformly unbounded at the origin. For background reading on ordinary copulas see (Nelsen 1998), for Lvy copulas see (Cont and Tankov 2004, Sections 5.55.7, pp. 145165). 2 For illustration, let be a Lvy measure on R+. If A : = [x, ] R+ B : = R+ [y, ] then the tail integral of the measure (analogous to a probability distribution function) is L(x, y) =

and the marginal tail integrals are L1 (x) =

and L2 (y) =

The marginal measures are independent if and only if they are supported respectively on the axes. 2 The Lvy copula K(u, v) : R+ R+. Specically, K(u, v) = L L1 (u), L1 (v) , or equivalently K L1 (x), L2 (y) = L(x, y) Some properties of the Lvy copula are easy to establish. First, the copula is grounded with uniform margins. K(0, v) = K(u, 0) = 0 and K1 (u) = K(u, ) = u K2 (v) = K(, v) = v
Further, if has a density l(x, y), with marginal densities l1 (x) and l2 (y), then they relate to the copular density k(u, v) as follows. k(u, v) = K(u, v) = L(x, y) uv xy L1 L2 x y = l(x, y) l1 (x)l2 (y)
Two special Lvy copulas are worthy of note, the independent copula K (u, v) and the completely dependent copula K (u, v), as follows. K (u, v) = u 1{v=} + v 1{u=} K (u, v) = min(u, v) B. Ornstein-Uhlenbeck processes B.1. Introduction. This appendix sets forth principles for the analysis of dependency among specied kinds of Lvy processes. These are ones of the Ornstein-Uhlenbeck description, with subordination. That is, each has a component of a pure jump process with non-negative jumps included with the usual Brownian motion. The study of these processes dates to a seminal paper by George Eugene Uhlenbeck and Leonard Salomon Ornstein, two highly respected Dutch mathematical physicists (Uhlenbeck and Ornstein 1930). Some say Laplace gave the process its start in 1810 (Jacobsen 1996), but almost all who observe say the subject is correctly named. These two researchers were contemporaries and collaborators with some of the great names of physics, including Neils Bohr, Satyendra Bose, Paul Dirac, Paul Ehrenfest, Albert Einstein, Josiah Gibbs, Samuel Goudsmit, Werner Heisenberg, Oskar Klein, Wolfgang Pauli, Dirk Jan Struik, and others. Among the discoveries of these persons are phenomena of high level current interest. Dirac invented the word boson for a particle conforming to the BoseEinstein statistics, and today physicists are searching for the elusive Higgs boson (after Peter Higgs of the University of Edinburgh) to extend the Standard Model. This is not even to mention the Ornstein-Uhlenbeck process itself, still and increasingly, a subject of intense modern investigation. That a physical process was at the beginning of this concept is evident simply by reading the cited paper. This was a study of the behavior of a molecule of a gas responding to a systematic force friction and to a random inuence uctuation. This was not the study of the mathematical formulation we now call the Wiener process (of course, after Norbert Wiener, another luminary of the time.) Nonetheless, Ornsteins and Uhlenbecks formulation was almost exactly what one sees today in modern mathematical terminology with drift coecients and Brownian dierentials. This is remarkable prescience. The present study oers a contribution in the realm of dependency analysis. Motivation is manifold. Jump processes drive some of the most important events in business and commerce, economics and nance, in the natural and physical sciences, in biological systems and the occurrence of natural disasters, in subatomic physics and theories of the universe, in human behavior itself. Evident among these phenomena is the suusion of dependency relationships. Subatomic events have consequences in space and time; earthquakes occur not in isolation; galaxies inuence one another, and, of course, people interact in many ways. On closer examination these dependencies rarely present themselves neatly in the form, say, of multinormal covariance matrices. Research questions abound, with signicant positive consequences for understanding and explaining. This appendix examines the O-U process, as all aectionately

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