On the distribution of a discrete sample path of a square-root diffusion

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. On the distribution of a discrete sample path of a square-root diffusion Michael B. Gordy 2012-12 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

On the distribution of a discrete sample path of a square-root diffusion∗ Michael B. Gordy Federal Reserve Board March 1, 2012 Abstract We derive the multivariate moment generating function for the stationary distribution of a discretesamplepathofnobservationsofasquare-rootdiffusion(CIR)process, X(t). Theform of the mgf establishes that the stationary joint distribution of (X(t ),...,X(t )) for any fixed 1 n vector of observation times t ,...,t is a Krishnamoorthy-Parthasarathy multivariate gamma 1 n distribution. As a corollary, we obtain the mgf for the increment X(t+δ)−X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws fromagammadistribution. Simpleclosed-formsolutionsforthemomentsoftheincrementsare given. Keywords: square-root diffusion; CIR process; multivariate gamma distribution; difference of gamma variates; Krishnamoorthy-Parthasarathy distribution; Kibble-Moran distribution; Bell polynomials. LetX followtheFeller(1951)square-rootdiffusionprocesswithstochasticdifferentialequation t (cid:112) dX = κ(θ−X )dt+σ X dW (1) t t t t where W is a Brownian motion. We assume that κ > 0, θ > 0 and σ > 0. This process is widely t usedineconomicsandfinance, especiallyinmodelinginterestratesandcorporatecreditrisk, where it is usually known as the CIR process after Cox, Ingersoll, and Ross (1985). In this paper, we derive the moment generating function for the stationary multivariate distribution of a discrete sample path of this process. Let X ≡ (X(t ),...,X(t )) be a discrete sample path for a given vector of ordered observation 1 n times t < t < ... < t . Let u denote the vector of auxilliary variables u ,...,u and let diag(u) 1 2 n 1 n be the diagonal matrix with diagonal entries u ,...,u . Let R be the symmetric n×n matrix with 1 n elements R[i,j] = exp(−(κ/2)|t −t |). I is the n×n identity matrix. Define the scale parameter i j n ν = σ2/(2κ). The central result of this paper is Theorem 1. The mgf of (X(t ),...,X(t )) under stationarity is 1 n M (u) = E[exp((cid:104)u,X(cid:105))] = det(I −νRdiag(u))−θ/ν X n ∗I thank Yacine A¨ıt-Sahalia, Luca Benzoni, Jens Christensen, Yang-Ho Park, Steven Shreve, Richard Sowers and DavidZelinskyforhelpfuldiscussion. BobakMoallemiprovidedexcellentresearchassistance. Theopinionsexpressed herearemyown,anddonotreflecttheviewsoftheBoardofGovernorsoritsstaff. Email: (cid:104)michael.gordy@frb.gov(cid:105). 1

The proof is set out in Section 1. ThedistributionofXisaspecialcaseofthebroaderclassofKrishnamoorthyandParthasarathy (1951)multivariategammadistributionswithmgfoftheformdet(I −Rdiag(u))−α fornonsingular n R and α > 0 (see also Kotz et al., 2000, §48.3.3). Series solutions for the density and cumulative distributionfunctionsaregivenbyRoyen(1994)forthecaseinwhichtheinverseofR istridiagonal (see also Kotz et al., 2000, §48.3.6), which applies for our matrix R. These series solutions are computationally practical only for low dimension n. The stationary square-root process has exponential decay in the autocorrelation function (Cont and Tankov, 2004, §15.1.2), so for pairs (i,j) in {1,...,n}2, the correlation corr(X(t ),X(t )) is i j given by ρ = exp(−κ|t −t |) = R[i,j]2. i,j i j From this relationship, the matrix R is known as the accompanying correlation matrix. In the bivariate case, the mgf has a simple form Corollary 1. The mgf of (X(t),X(t+δ)) under stationarity is M (u ,u ) = E[exp(u X(t)+u X(t+δ))] = ((1−νu )(1−νu )−ρu u )−θ/ν X 1 2 1 2 1 2 1 2 where ρ = exp(−κδ). ThisistheKibble-Moranbivariategammadistribution(seeKotzetal.,2000,§48.2.3). InSection2, we use this corollary to study the stationary distribution of the increment X(t+δ)−X(t) for fixed time-stepδ. Weshowthatthisincrementisequivalentindistributiontoascaleddifferencebetween two independent gamma variates, and provide a simple closed-form solution for the moments of this distribution. Applications are discussed in the concluding section. 1 Moment generating function It is well known that the transition distribution for X(t+δ) given X(t) is noncentral chi-squared.1 Letting M denote the conditional mgf for X(t+δ) given X(t), we have c M (u;δ,x) = E[exp(uX(t+δ))|X(t) = x] c (cid:16) (cid:16) (cid:17) (cid:17)−θ/ν (cid:18) e−κδu (cid:19) = 1−ν 1−e−κδ u exp x (2) 1−ν(1−e−κδ)u As the square-root diffusion is a Markov process, we have E[exp(u X(t ))|X(t ),X(t ),...,X(t )] n n n−1 n−2 1 = E[exp(u X(t ))|X(t )] = M (u ;t −t ,X(t )) n n n−1 c n n n−1 n−1 1See Alfonsi (2010) for a summary of basic properties of the square-root diffusion. 2

To exploit the conditional mgf, we write M in nested form: X M (u) = E[exp(u X(t )+...+u X(t ))M (u ;t −t ,X(t ))] X 1 1 n−1 n−1 c n n n−1 n−1 = (1−ν(1−ρ )u )−θ/ν· n−1,n n (cid:20) (cid:20) (cid:18) (cid:19)(cid:12) (cid:21)(cid:21) ρ n−1,n u n (cid:12) E exp(u 1 X(t 1 )+...+u n−2 X(t n−2 ))E exp(u n−1 X(t n−1 ))exp X(t n−1 ) (cid:12)X(t n−2 ) 1−ν(1−ρ )u (cid:12) n−1,n n = (1−ν(1−ρ )u )−θ/ν· n−1,n n E[exp(u X(t )+...+u X(t ))M (u˜ ;t −t ,X(t ))] (3) 1 1 n−2 n−2 c n−1 n−1 n−2 n−2 where ρ u n−1,n n u˜ = u + n−1 n−1 1−ν(1−ρ )u n−1,n n Repeating this process n−1 times in total, we get (cid:34) n (cid:35)−θ/ν (cid:89) M (u) = (1−ν(1−ρ )u˜ ) E[exp(u˜ X(t ))] X k−1,k k 1 1 k=2 where the modified auxilliary variables have the forward recursive relationship ρ u˜ k,k+1 k+1 u˜ = u + (4) k k 1−ν(1−ρ )u˜ k,k+1 k+1 for k = 1,...,n and where we fix u˜ = 0 (so u˜ = u ). The stationary distribution of X(t ) is n+1 n n 1 gamma with shape parameter θ/ν and scale parameter ν, which has mgf M (u) = (1−νu)−θ/ν (5) Γ so we arrive at (cid:34) n (cid:35)−θ/ν (cid:89) M (u) = (1−νu˜ ) (1−ν(1−ρ )u˜ ) (6) X 1 k−1,k k k=2 Equation (6) is computationally convenient but analytically cumbersome. Let Q(u) be the expression inside the brackets, so that M (u) = Q(u)−θ/ν. We now simplify Q(u) by writing it as X a finite series in powers of ν. Let S(n,k) be the set of subsequences of length k from the sequence 1,...,n, so that if s ∈ S(n,k), then 1 ≤ s(1) < s(2) < ... < s(k) ≤ n. For k = 1,...,n, define the functions  1 if k = 0,   f k (u) = (cid:80)n i=1 u i if k = 1, (7)    (cid:80) s∈S(n,k) (cid:16) (cid:81)k i= − 1 1(1−ρ s(i),s(i+1) ) (cid:17)(cid:16) (cid:81)k i=1 u s(i) (cid:17) otherwise. In Appendix A, we prove 3

Proposition 1. n (cid:88) Q(u) = (−ν)kf (u) k k=0 To prove Theorem 1, we need to prove that det(I −νRdiag(u)) has the same expansion as in n Proposition 1. Recall that the characteristic polynomial of a square n×n matrix A is defined as det(λI −A). For a subsequence s ∈ S(n,k), let A denote the kth order diagonal minor of A with n s elements A [i,j] = A[s(i),s(j)] and let Ω (A) for 1 ≤ k ≤ n be defined as s k (cid:88) Ω (A) = det(A ). k s s∈S(n,k) For notional convenience, we define Ω (A) = 1 = f (u). Then the characteristic polynomial of A 0 0 has the expansion (Gantmacher, 1959, §III.7) det(λI −A) = λn−Ω (A)λn−1+Ω (A)λn−2−...+(−1)nΩ (A) (8) n 1 2 n Substituting λ = 1 and A = νRdiag(u) in (8), we have det(I −νRdiag(u)) = 1−Ω (νRdiag(u))+Ω (νRdiag(u))−...+(−1)nΩ (νRdiag(u)) n 1 2 n n n (cid:88) (cid:88) = (−1)kΩ (νRdiag(u)) = (−ν)kΩ (Rdiag(u)) (9) k k k=0 k=0 Since diag(u) is a diagonal matrix, the diagonal minor (Rdiag(u)) is equal to the product s R diag(u ). Thus, we have s s (cid:88) (cid:88) Ω (Rdiag(u)) = det((Rdiag(u)) ) = det(R )det(diag(u )) (10) k s s s s∈S(n,k) s∈S(n,k) For the case of k = 1, det(R ) = 1 for all s ∈ S(n,1), so s n (cid:88) Ω (Rdiag(u)) = u = f (u). 1 i 1 i=1 For the case of 2 ≤ k ≤ n, we make use of this lemma:2 Lemma 1. Let A be an m×m matrix with elements A[i,j] = exp(−(c/2)|t −t |) for some constant i j c ≥ 0 and vector of nonnegative t ,...,t . Then det(A) = (cid:81)m−1(1−exp(−c|t −t |)). 1 m i=1 i+i i Proof. Let B be an m×m matrix with diagonal elements B[i,i] = 1/A[i,i+1] for i = 1,...,m−1 and B[m,m] = 1, B[i,i+1] = −1 on each element of the superdiagonal, and zero elsewhere. It is easily verified that the matrix C = BA is lower triangular with diagonal entries C[i,i] = 1/A[i,i+1]−A[i,i+1] = (1/A[i,i+1])· (cid:0) 1−A[i,i+1]2(cid:1) for i = 1,...,m−1 and C[m,m] = 1. Thus, det(C) (cid:81)m C[i,i] m (cid:89) −1 det(A) = = i=1 = (1−A[i,i+1]2) det(B) (cid:81)m B[i,i] i=1 i=1 2I thank David Zelinsky for suggesting the proof of this lemma. 4

The diagonal minor R takes on the same form as R, i.e., there is a vector t(cid:48) = (t ,...,t ) s s(1) s(k) such that R has elements R [i,j] = exp(−(κ/2)|t(cid:48) −t(cid:48)|). Applying Lemma 1, for any 2 ≤ k ≤ n s s i j and s ∈ S(n,k) we have k−1 k−1 (cid:89) (cid:89) det(R ) = (1−exp(−κ(t −t ))) = (1−ρ ) s s(i+1) s(i) s(i),s(i+1) i=1 i=1 Since det(diag(u )) = (cid:81)k u , we have s i=1 s(i) (cid:88) Ω (Rdiag(u)) = det(R )det(diag(u )) k s s s∈S(n,k) (cid:32)k−1 (cid:33)(cid:32) k (cid:33) (cid:88) (cid:89) (cid:89) = (1−ρ ) u = f (u) (11) s(i),s(i+1) s(i) k s∈S(n,k) i=1 i=1 We substitute equation (11) into equation (9) and arrive at the same expansion as in Proposition 1. This completes the proof of Theorem 1. 2 Moments of the increments d Under stationarity, X −X = X −X for all t, so without loss of generality we examine the t+δ t δ 0 stationary distribution of ∆ = X −X . From Corollary 1, δ δ 0 M (u;δ) = M (−u,u) = (1−ν2(1−ρ)u2)−θ/ν ∆ X (cid:16) (cid:112) (cid:112) (cid:17)−θ/ν (cid:16) (cid:112) (cid:17) (cid:16) (cid:112) (cid:17) = (1−ν 1−ρu)(1+ν 1−ρu) = M u 1−ρ ·M −u 1−ρ (12) Γ Γ where ρ = exp(−κδ) and M is the univariate mgf for X(t). An immediate implication of (12) is Γ that ∆ is equivalent in distribution to (1−ρ)1/2 times ∆ . Furthermore, ∆ is equivalent in δ ∞ ∞ distribution to the difference between two independent draws from the stationary distribution of X(t). This gives a very simple method for sampling from the stationary distribution of ∆ . δ Consider the general problem of the moments of the difference between two independent and iid identically distributed (iid) gamma variates. Let Z ,Z ∼ Ga(α,ν) for shape parameter α > 0 and 1 2 scale parameter ν > 0, and define Y = Z −Z . The nth cumulant of Y is 1 2 ψ = (1+(−1)n)(n−1)!ανn. n Central moments are obtained from the cumulants via the complete Bell polynomials, i.e., E[Yn] = B (ψ ,ψ ,...,ψ ). n 1 2 n For any sequence c ,c ,..., the Bell polynomials satisfy 1 2 B (νc ,ν2c ,...,νnc ) = νnB (c ,c ,...,c ) n 1 2 n n 1 2 n so E[Yn] = νnB (0,2α1!,0,2α3!,0,2α5!,...). (13) n Furthermore, since the distribution is symmetric around zero, we know that the odd moments E (cid:2) Y2n+1(cid:3) are zero. In Appendix B, we prove a general identity on the complete Bell polynomials: 5

Lemma 2. Let k be a positive integer and let ξ ,ξ ,... be the sequence of integers defined by k,1 k,2 (cid:40) k if j = 0 (mod k), ξ = k,j 0 otherwise. Then for any scalar α ∈ (cid:60)+, (kn)!Γ(α+n) B (ξ α0!,ξ α1!,...,ξ α(kn−1)!) = kn k,1 k,2 k,kn n! Γ(α) where Γ(·) is the Gamma function. For any positive integer m not divisible by k, B (ξ α0!,ξ α1!,...,ξ α(m−1)!) = 0. m k,1 k,2 k,m It follows immediately that the even central moments of Y are E (cid:2) Y2n(cid:3) = ν2nB (0,2α1!,0,2α3!,0,2α5!,...) = ν2n (2n)!Γ(α+n) (14) 2n n! Γ(α) and the odd central moments are zero. As kurtosis is often of particular interest, we note E (cid:2) Y4(cid:3) = 3(1+1/α). E[Y2]2 Application to the moments of ∆ is direct. We substitute α = θ/ν and get even moments δ E (cid:2) ∆2n(cid:3) = (1−exp(−κδ))nν2n (2n)!Γ(θ/ν +n) (15) δ n! Γ(θ/ν) The kurtosis of ∆ is 3(1+ν/θ), which is invariant with respect to the time increment δ. δ 3 Conclusion Our main result is a simple closed-form expression for the moment generating function of the stationary multivariate distribution of a discrete sample path of a square-root diffusion process. We establish that the distribution is within the Krishnamoorthy-Parthasarathy class, and thereby draw a connection between a stochastic process and a multivariate distribution that each first appeared in the literature in 1951. Our result has application to estimation of parameters of the continuous-time square-root processfromadiscretesample. Itgivesasimpleandcomputationallyefficientwaytogeneratemoment conditions for the generalized method of moments estimator of Chan et al. (1992). The empirical characteristic function approach of Jiang and Knight (2002) can also be easily implemented. Indeed, Jiang and Knight consider the example of a square-root diffusion, but their solution to the characteristic function corresponds roughly to our intermediate equation (6), rather than to the simple form in our Theorem 1. Threeofourauxilliaryresultsmayhaveapplicationelsewhere. First,Lemma1providesasimple solution to the determinant of the autocorrelation matrix for a discrete sample of any process with exponential decay in autocorrelation. This decay rate holds in a large class of stationary 6

Markov processes, including Gaussian and non-Gaussian Ornstein-Uhlenbeck processes as well as the square-root process (Cont and Tankov, 2004, §15.1.2, §15.3.1). Second, our Bell polynomial identity in Lemma 2 generalizes a known relationship between Bell polynomials and the Gamma function (i.e., for the case of k = 1 in our lemma). Finally, we provide a simple formula for the moments of the difference of two iid gamma variates. It complements existing results that allow the variates to differ in scale parameter (Johnson et al., 1994, §12.4.4), but which lead to more complicated expressions for the moments. A Proof of Proposition 1 Let us define Q˜ = 1−νu˜ and, for 2 ≤ m ≤ n, recursively define 1 1 Q˜ = (1−ν(1−ρ )u˜ )Q˜ m m−1,m m m−1 Since u˜ = 0, we have Q˜ = Q(u). We similarly generalize the f functions as n+1 n k  1 if k = 0,   f˜ m,k (w 1 ,...,w m ) = (cid:80)m i=1 w i if k = 1,    (cid:80) s∈S(m,k) (cid:16) (cid:81)k i= − 1 1(1−ρ s(i),s(i+1) ) (cid:17)(cid:16) (cid:81)k i=1 w s(i) (cid:17) otherwise. Observe that the set S(m,k) can be expressed as the union of two disjoint subsets S(m,k) = S(m−1,k)∪{(s,m)|s ∈ S(m−1,k−1)}. The latter set is equivalent to the subset of S(m,k) for which s(k) = m. This implies that the f˜ functions have the recurrence relation f˜ (w ,w ,...,w ) = f˜ (w ,w ,...,w ) m,k 1 2 m m−1,k 1 2 m−1 (cid:32)k−1 (cid:33)(cid:32) k (cid:33) (cid:88) (cid:89) (cid:89) + 1 (1−ρ ) w (16) {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 We now demonstrate m (cid:88) Q˜ = (−ν)kf˜ (u ,u ,...,u ,u˜ ) (17) m m,k 1 2 m−1 m k=0 by induction. For the case m = 1, Q˜ = 1−νu˜ = f˜ (u˜ )−νf˜ (u˜ ) 1 1 1,0 1 1,1 1 whichsatisfiesequation(17). For2 ≤ m ≤ n,letusassumethat(17)issatisfiedforQ˜ ,Q˜ ,...,Q˜ . 1 2 m−1 Then Q˜ = (1−ν(1−ρ )u˜ )Q˜ m m−1,m m m−1 m−1 (cid:88) = (1−ν(1−ρ )u˜ ) (−ν)kf˜ (u ,u ,...,u ,u˜ ). (18) m−1,m m m−1,k 1 2 m−2 m−1 k=0 7

Since f˜ is linear in each argument, m−1,k f˜ (u ,u ,...,u ,u˜ ) = f˜ (u ,u ,...,u ,u ) m−1,k 1 2 m−2 m−1 m−1,k 1 2 m−2 m−1 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) ρ m−1,m u˜ m + 1 (1−ρ ) u {s(k)=m−1} s(i),s(i+1) s(i) 1−ν(1−ρ )u˜ m−1,m m s∈S(m−1,k) i=1 i=1 Substituting into (18), we get m−1 (cid:34) (cid:88) Q˜ = (−ν)k (1−ν(1−ρ )u˜ )f˜ (u ,u ,...,u ,u ) m m−1,m m m−1,k 1 2 m−2 m−1 k=0 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33)(cid:35) (cid:88) (cid:89) (cid:89) +ρ u˜ 1 (1−ρ ) u (19) m−1,m m {s(k)=m−1} s(i),s(i+1) s(i) s∈S(m−1,k) i=1 i=1 Collecting terms on (−ν)k, we can write m (cid:88) Q˜ = (−ν)kg˜ m m,k k=0 where g˜ = f˜ (u ,...,u ) = 1 = f˜ (u ,...,u ,u˜ ), m,0 m−1,0 1 m−1 m,0 1 m−1 m and g˜ = (1−ρ )u˜ f˜ (u ,...,u ) = f˜ (u ,...,u ,u˜ ) m,m m−1,m m m−1,m−1 1 m−1 m,m 1 m−1 m and, for 1 ≤ k < m, g˜ = f˜ (u ,...,u )+(1−ρ )u˜ f˜ (u ,...,u ) m,k m−1,k 1 m−1 m−1,m m m−1,k−1 1 m−1 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) +ρ u˜ 1 (1−ρ ) u (20) m−1,m m {s(k)=m−1} s(i),s(i+1) s(i) s∈S(m−1,k) i=1 i=1 8

The last term is (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) ρ u˜ 1 (1−ρ ) u m−1,m m {s(k)=m−1} s(i),s(i+1) s(i) s∈S(m−1,k) i=1 i=1 (cid:32)k−2 (cid:33) (cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) = ρ u˜ 1 (1−ρ ) (1−ρ ) u m−1,m m {s(k)=m−1} s(i),s(i+1) s(k−1),m−1 s(i) s∈S(m−1,k) i=1 i=1 (cid:32)k−2 (cid:33) (cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) = u˜ 1 (1−ρ ) (1−ρ ) u m {s(k)=m−1} s(i),s(i+1) s(k−1),m s(i) s∈S(m−1,k) i=1 i=1 (cid:32)k−2 (cid:33) (cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) −u˜ 1 (1−ρ ) (1−ρ ) u m {s(k)=m−1} s(i),s(i+1) m−1,m s(i) s∈S(m−1,k) i=1 i=1 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) = u˜ 1 1 (1−ρ ) u m {s(k−1)<m−1} {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 −(1−ρ )u˜ f˜ (u ,...,u ) m−1,m m m−2,k−1 1 m−2 so (cid:16) (cid:17) g˜ = f˜ (u ,...,u )+(1−ρ )u˜ f˜ (u ,...,u )−f˜ (u ,...,u ) m,k m−1,k 1 m−1 m−1,m m m−1,k−1 1 m−1 m−2,k−1 1 m−2 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) +u˜ 1 1 (1−ρ ) u m {s(k−1)<m−1} {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 By equation (16), (cid:16) (cid:17) (1−ρ )u˜ f˜ (u ,...,u )−f˜ (u ,...,u ) m−1,m m m−1,k−1 1 m−1 m−2,k−1 1 m−2 (cid:32)k−2 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) = u˜ 1 (1−ρ ) u (1−ρ ) m {s(k−1)=m−1} s(i),s(i+1) s(i) m−1,m s∈S(m−1,k−1) i=1 i=1 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) = u˜ 1 1 (1−ρ ) u m {s(k−1)=m−1} {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 so (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) g˜ = f˜ (u ,...,u )+u˜ 1 1 (1−ρ ) u m,k m−1,k 1 m−1 m {s(k−1)=m−1} {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) +u˜ 1 1 (1−ρ ) u m {s(k−1)<m−1} {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 (cid:32)k−1 (cid:33)(cid:32)k−1 (cid:33) (cid:88) (cid:89) (cid:89) = f˜ (u ,...,u )+u˜ 1 (1−ρ ) u m−1,k 1 m−1 m {s(k)=m} s(i),s(i+1) s(i) s∈S(m,k) i=1 i=1 = f˜ (u ,...,u ,u˜ ) m,k 1 m−1 m 9

where the last equality is by equation (16). Thus, g˜ = f˜ (u ,...,u ,u˜ ) for k = 0,...,m, m,k m,k 1 m−1 m so equation 17 is proved. Substituting m = n and u˜ = u , we arrive at Proposition 1. n n B Proof of the Bell polynomial identity For any sequence of scalars c ,c ,..., the generating function of the complete Bell polynomials is 1 2 (cid:32) (cid:88) ∞ xn (cid:33) (cid:88) ∞ xn exp c = B (c ,c ,...,c ) (21) n n 1 2 n n! n! n=1 n=0 where we fix B = 1. When c = ξ α(j −1)!, we have 0 j k,j (cid:32) (cid:88) ∞ xn (cid:33) (cid:32) (cid:88) ∞ xkn (cid:33) (cid:32) (cid:88) ∞ yn (cid:33) (cid:88) ∞ yn exp c = exp kα = exp α = B (α0!,α1!,...,α(n−1)!) n n n! kn n n! n=1 n=1 n=1 n=0 where we introduce the change of variable y = xk. Using identities from Comtet (1974, pp. 135, 136) and DLMF (2010, §26.8.7), we have n (cid:88) Γ(α+n) B (α0!,α1!,...α(n−1)!) = |s(n,k)|αk = n Γ(α) k=1 where s(n,k) denotes the Stirling number of the first kind. Restoring the original variable x, we have (cid:32) (cid:88) ∞ xn (cid:33) (cid:88) ∞ Γ(α+n)yn (cid:88) ∞ Γ(α+n)(kn)! xkn exp c = = (22) n n! Γ(α) n! Γ(α) n! (kn)! n=1 n=0 n=0 Matching terms to the right hand side of (21) with the same power of x, we have (kn)!Γ(α+n) B (ξ α0!,ξ α1!,...,ξ α(kn−1)!) = kn k,1 k,2 k,kn n! Γ(α) Whenever m is not a multiple of k, the coefficient on xm in the right hand side of (22) is zero, so B (ξ α0!,ξ α1!,...,ξ α(m−1)!) = 0. m k,1 k,2 k,m 10

References Aur´elien Alfonsi. Cox-Ingersoll-Ross (CIR) model. In Rama Cont, editor, Encyclopedia of Quantitative Finance. John Wiley & Sons, 2010. K. C. Chan, G. Andrew Karolyi, Francis A. Longstaff, and Anthony B. Sanders. An empirical comparison of alternative models of short-term interest rate. Journal of Finance, 47(3):1209– 1227, July 1992. Louis Comtet. Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Company, Dordrecht, 1974. Rama Cont and Peter Tankov. Financial Modelling with Jump Processes. Chapman & Hall, Boca Raton, 2004. John C. Cox, Jonathan E. Ingersoll, Jr., and Stephen A. Ross. A theory of the term structure of interest rates. Econometrica, 53(2):385–407, March 1985. DLMF. Digital Library of Mathematical Functions, 7 May 2010. URL http://dlmf.nist.gov/. William Feller. Two singular diffusion problems. Annals of Mathematics, Second Series, 54(1): 173–182, July 1951. F.R. Gantmacher. Matrix Theory, volume I. Chelsea, New York, 1959. George J. Jiang and John L. Knight. Estimation of continuous-time processes via the empirical characteristicfunction. Journal of Business and Economics Statistics, 20(2):198–212, April2002. Norman L. Johnson, Samuel Kotz, and N. Balakrishnan. Continuous Univariate Distributions, volume 1. John Wiley & Sons, New York, second edition, 1994. Samuel Kotz, N. Balakrishnan, and Norman L. Johnson. Continuous Multivariate Distributions, volume 1: Models and Applications. John Wiley & Sons, New York, second edition, 2000. A.S. Krishnamoorthy and M. Parthasarathy. A multivariate gamma-type distribution. Annals of Mathematical Statistics, 22:549–557, 1951. Correction (1960), 31, 229. Thomas Royen. On some multivariate gamma-distributions connected with spanning trees. Annals of the Institute of Statistical Mathematics, 46(2):361–371, 1994. 11