ONLINE SUPPLEMENT TO: Asymptotic Properties of Monte Carlo Estimators of Derivatives published in Management Science
J´erˆome B. Detemple School of Management, Boston University, 595 Commonwealth Avenue, Boston, Massachusetts 02215, and CIRANO, 2020, University St., 25th Floor, Montr´eal, Qu´ebec, H3A 2A5, CANADA,
[email protected]
Ren´e Garcia Economics Department, Universit´e de Mont´eal and CIREQ, C.P. 6128, Succursale Centre-ville, Montr´eal H3C 3J7, CANADA, and CIRANO, 2020, University St., 25th Floor Montr´eal, Qu´ebec, H3A 2A5, CANADA,
[email protected] Marcel Rindisbacher J. L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, M5S 3E6, CANADA, and CIRANO, 2020, University St., 25th Floor, Montr´eal, Qu´ebec, H3A 2A5, CANADA,
[email protected]
Appendix Appendix A: Expected Approximation Errors The proofs of our results, in Appendix B, rely on a theorem of Detemple, Garcia and Rindisbacher (2004) that characterizes the asymptotic expected approximation error. To state this result we need to introduce the following notation. Recall that ∂A, ∂Bj are d × d matrices of Jacobians. The tangent process, in this multivariate setup, solves the linear equation d X d∇t,x Xs = ∂A(Xs )ds + ∂Bj (Xs )dWsj ∇t,x Xs with ∇t,x Xt = In . j=1
With this notation we have,
(A-1)
Theorem 5: Let g ∈ C 3 (Rd ) be such that the uniform integrability condition h i lim lim sup Et 1{kN (g(X N )−g(XT ))k>r} N |g(XTN ) − g(XT )| = 0 r→∞
T
N
(A-2)
holds ( P-a.s.). Then, as N → ∞, 1 1 N Et g(XTN ) − g(XT ) → Kt,T (x) ≡ Et [∂g(XT )V1 (t, T ) + V2 (t, T )] 2 2
(A-3)
where the random variables V1 (t, T ) and V2 (t, T ) are Z T d X −1 (∇t,x Xs ) V1 (t, T ) = −∇t,x XT ∂A(Xs )dXs + [∂Bj A](Xs )dWsj t
−
d X
j=1
[(∂Bj )(∂Bj )Bi ](Xs )dWsi
i,j=1
Z
T
+∇t,x XT
d d X X [∂k (∂l ABl,j )Bk,j ] (Xs )ds (∇t,x Xs )−1 [∂Bj ∂Bj A] −
t
Z +∇t,x XT
j=1 T
(∇t,x Xs )−1
t
d X
j,k,l=1
([∂[∂Bj ∂Bj Bi ]Bi − ∂Bi ∂Bj ∂Bj Bi ](Xs )) ds
i,j=1
Z
d X
T
V2 (t, T ) = − t
νi,j (s, T )ds.
i,j=1
where νi,j (s, T ) ≡ [hi,j (∇t,x X· )−1 [(∂Bj )Bi ](X), W i ]s with hi,j t ≡ Et [Djt (∂g(XT )∇t,x XT ei )]. An explicit expression for νi,j (s, T ) is in Detemple, Garcia and Rindisbacher (2004). In order to derive the asymptotic expected approximation error, on the right hand side of (A-3), an expansion, based on the mean value theorem, of the error g(XTN ) − g(XT ) is used. This gives a linear SDE with four autonomous components. Three of these autonomous terms converge at the √ rate 1/N . The last one converges at the rate 1/ N , but has zero expectation. This difference in the convergence behavior of the autonomous terms in the error expansion explains the presence of two parts in the second order bias Kt,T (x). The limits of the first three terms give rise to V1 (t, T ). The fourth term leads to V2 (t, T ). The uniform integrability assumption in the theorem enables us to find the asymptotic expected approximation error by taking the expectation of the weak limit of the error expansion.
Appendix B: Proofs In what follows we set t = 0, without loss of generality, and write E0 for E0,x , P0 for P0,x , ∇x for ∇0,x , and KT (x) for K0,T (x). 2
M,N,τ
Proof of Theorem 2: Recall that f (0, x) ≡ E0 [g(XT )] and that the MCC estimator ∂x^ f (0, x) M,N,τ is given in (26). The error of the MCC estimator, e ε(0, x; M, N, τ ) ≡ ∂x^ f (0, x) − ∂x f (0, x), can be expanded as follows ∆τ W0 e ε(0, x; M, N, τ ) = E0 g(XT ) C(x) − ∂x f (0, x) τ M ∆ Wi 1 X τ 0 + g(XTi,N ) − g(XTi ) C(x) M τ i=1 M i X ∆τ W0 1 i ∆τ W0 g(XT ) C(x) − E0 g(XT ) + M τ τ i=1
where ∆τ W0 = Wτ − W0 and C is the generalized inverse of B defined in (13). In this expansion, XTi,N is an independent replication of the Euler discretization of the diffusion given in (4), XTi (resp. WTi ) is an independent replication of the terminal value of the diffusion (resp. of the Brownian motion driving the SDE) and ∆τ W0i is an independent replication of the increment in the Brownian motion. Our next three lemmas give the asymptotic errors for the three terms in this expansion. Lemma 1: Under the assumption of Theorem 2 we have 1 ∆τ W 0 1 E0 g(XT ) C(x) − ∂x f (0, x) → (∂v E0 [∂x f (v, Xv )B(Xv )])|v=0 C(x) τ τ 2 as 1/τ → ∞. RT Proof of Lemma 1: The Clark-Ocone formula g(XT ) = E0 [g(XT )] + 0 Ev [Dv g(XT )]dWv gives Z τ 1 ∆τ W0 1 E0 g(XT ) C(x) − ∂x f (0, x) = 2 β(v)dv τ τ τ 0 with β(v) ≡ E0 [Ev [Dv g(XT )]C(x) − ∂x f (0, x)]. Given that β(0) = 0, a second order Taylor apRτ proximation of β(v) yields lim1/τ →∞ τ12 0 β(v)dv = 21 ∂v β(v)|v=0 . But ∂v β(v)|v=0 = (∂v E0 [Ev [Dv g(XT )]C(x) − ∂x f (0, x)])|v=0 = (∂v E0 [∂x f (v, Xv )B(Xv )])|v=0 C(x) where the last equality uses Ev [Dv g(XT )] = ∂x f (v, Xv )B(Xv ). Substituting the expression for ∂v β(v)|v=0 in the limit above establishes the result in the lemma.
3
Lemma 2: Let KT (x) be given by equation (A-3) in Theorem 5. Under the assumption of Theorem 2 we have, as M → ∞, M ∆ Wi 1 1 X τM 0 → ∂KT (x)B(x) NM g(XTi,NM ) − g(XTi ) M τM 2 i=1
in probability, when NM , 1/τM → ∞ as M → ∞. Proof of Lemma 2: We proceed in two steps. In the first step we find a sequential limit. In the second step we establish that joint limits and sequential limits are the same. Define, eM,N,τ T
M ∆ Wi 1 1 X τ 0 N g(XTi,NM ) − g(XTi ) − ∂KT (x)B(x), ≡ M τ 2 i=1
N and note that eM,N,τ = eM,N,τ + eN,τ T 1,T 2,T + e3,T , with
eM,N,τ 1,T
M ∆ Wi ∆τ W0 1 X τ 0 i,N i N N g(XT ) − g(XT ) − E0 N g(XT ) − g(XT ) ≡ M τ τ i=1
eN,τ 2,T eN 3,T
Z τ 1 N ≡ E0 [Dv − D0 ]N g(XT ) − g(XT ) dv τ 0 1 ≡ D0 N E0 g(XTN ) − g(XT ) − ∂KT (x)B(x). 2
The expression for eN,τ 2,T is obtained by using the Clark-Ocone formula N
g(XTN )
− g(XT ) = E0 N g(XTN ) − g(XT ) +
Z
T
Ev Dv N g(XTN ) − g(XT ) dWv
0
to conclude Z τ ∆τ W0 1 N N Ev Dv N g(XT ) − g(XT ) dv . = E0 E0 N g(XT ) − g(XT ) τ τ 0 By the weak law of large numbers for i.i.d random sequences the first term vanishes in probability for fixed N and τ , i.e. P−limM →∞ eM,N,τ = 0 for all N, τ . By the continuous differentiability of the 1,T N Riemann integral limτ →∞ eN,τ 2,T = 0 for fixed N . Finally, by Theorem 5 that shows N E0 [g(XT ) −
g(XT )] → 21 KT (x) and by the chain rule of Malliavin that gives D0 KT (x) = ∂KT (x)B(x) we obtain M,N,τ limN →∞ eN → 0 in probability, if sequentially M, 1/τ, N → ∞. 3,T = 0. We conclude that eT
Next, we show that this sequential limit corresponds to the joint limit (i.e. when M, N, 1/τ → ∞ jointly) and thus to the limit along any “diagonal”. Invoking arguments in the proof of Theorem 4.2 in Billingsley (1968) (see also Lemma 6 in Phillips and Moon (1996)), it is necessary and sufficient to show that
> =0 lim sup P0 eM,N,τ
1,T
M,N,1/τ →∞
4
(B-4)
lim sup eN,τ 2,T = 0.
(B-5)
N,1/τ →∞
Condition (B-4) is proved by applying the Markov-type inequality,1 h i
M
! N,τ
E kH k1 N,τ X 0 γ T 1 {kHT k>M }
P0 + M γ−1 P0 kHTN,τ k ≤ M γ HTi,N,τ > ≤
M M
(B-6)
i=1
for > 0 and γ ∈]0, 1[ to the random variable HTi,N,τ
≡N
g(XTi,N )
−
∆ Wi ∆τ W0 τ 0 N . − E0 N g(XT ) − g(XT ) τ τ
g(XTi )
Let M, N, 1/τ → ∞. The uniform integrability condition (27) with r = M γ establishes that the first term on the right hand side of (B-6) converges to zero for all > 0. The second term also converges to zero as M γ−1 → 0 when M → ∞. We conclude that the left hand side converges to zero as M, N, 1/τ → ∞, thereby establishing (B-4). For condition (B-5) note the inequality sup N ≥N0 ,1/τ ≥1/τ0
keN,τ 2,T k
≤
0 ,τ0 e¯N 2,T
" #
N N ≡ sup E0 sup Dv ZT − D0 ZT ,
N ≥N0 v∈(0,τ0 ]
where supv∈(0,τ0 ] Dv ZTN − D0 ZTN is non-decreasing in τ0 and null for τ0 = 0. Taking the infimum 0 ,τ0 over N0 > 0, 1/τ0 > 0 on both sides gives (B-5) as inf N0 >0,1/τ0 >0 e¯N = 0. 2,T
Lemma 3: Under the assumption of Theorem 2 we have √
M i 1 X i ∆τM W0 i ∆τM W0 − E0 g(XT ) ⇒ OT (x) τM √ g(XT ) τM τM M i=1
(B-7)
where 1/τM → ∞ as M → ∞ and where OT (x) ∼ N 0, E0 [g(XT )2 ] . Proof of Lemma 3: First note that Ito’s lemma implies Z t+τ Z t+τ (∆τ Wt ) (∆τ Wt )0 = dWs (Ws − Wt )0 + (Ws − Wt ) (dWs )0 + τ Id , t
t 1
i
ˆ ˜ This inequality holds for a sequence of i.i.d. random vectors Z and any > 0, as E kZk1kZk>M γ = E [kZk] − ˜ E kZk1kZk≤M γ , by the triangle inequality and the Markov inequality (Kallenberg (1997), Lemma 3.1 page 40), ‚# "‚ M ‚ ‚ ˆ ˜ ˆ ˜ ‚ 1 X i‚ γ E kZk1{kZk>M γ } ≥ E [kZk] − M E 1{kZk≤M γ } ≥ E ‚ Z ‚ − M γ P (kZk ≤ M γ ) ‚M ‚ i=1 ‚ ‚ ! M ‚ ‚ ‚ 1 X i‚ ≥ P ‚ Z ‚ > M − M γ P (kZk ≤ M γ ) . ‚M ‚ ˆ
i=1
5
while the Clark-Ocone formula gives 2
2
g(XT ) = E0 [g(XT ) ] +
g(XT ) = E0 [g(XT )] +
d Z X
T
Ev [Djv g(XT )2 ]dWvj
j=1
0
d Z X
T
j=1
Ev [Djv g(XT )]dWvj .
0
It follows that mτ2T
≡ E0 g(XT )2 (∆τ W0 ) (∆τ W0 )0 Z τ Z τ 0 2 (Ws − W0 ) (dWs ) Ev Dv g(XT ) dWv = E0 0 0 Z τ Z τ 0 2 + τ E0 g(XT )2 Id dWs (Ws − W0 ) Ev Dv g(XT ) dWv +E0 0 Z τ0 2 = E0 (Wv − W0 )Ev Dv g(XT ) dv 0 Z τ 2 0 0 +E0 Ev Dv g(XT ) (Wv − W0 ) dv + τ E0 g(XT )2 Id , 0
and that
mτ1T
≡ E0 [g(XT )∆τ W0 ] = E0
R τ 0
Ev [Dv g(XT )] dv . From these expressions we obtain
the conditional variance at t = 0, ∆τ W0 1 τ 1 τ 1 τ 0 VAR0 g(XT ) √ = m −τ m m τ 2T τ 1T τ 1T τ Z τ 1 2 E0 = (Wv − W0 )Ev Dv g(XT ) dv τ 0Z τ 1 2 0 0 + E0 Ev Dv g(XT ) (Wv − W0 ) dv + E0 g(XT )2 Id τ 0 0 Z τ Z τ 1 1 Ev [Dv g(XT )] dv Ev [Dv g(XT )] dv . E0 E0 −τ τ τ 0 0 Given that terms one, two and four converge to zero as 1/τ → ∞ the limit ∆τ W0 lim VAR0 g(XT ) √ = E0 g(XT )2 Id τ 1/τ →∞ holds. The uniform integrability condition (28), which is sufficient for the Lindeberg Central Limit theorem for independent random variables to hold, can now be invoked to conclude that the weak limit is Gaussian with variance E0 g(XT )2 Id . Proof of Theorem 2 (continued): Combining Lemmas 1, 2 and 3, shows that2 2
The symbol OP (x) stands for “at most of order x in probability”. A sequence of random variables Z N is OP (N k ),
if for every > 0, there exists a real number r such that P(N −k |Z N | > r) < for all N . If N −k |Z N | ≤ r for all N pointwise, the sequence Z N is said to be at most of order N k . In this case we write O(N k ). In contrast, a sequence Z N is of smaller order than N k , denoted by o(N k ), if N −k |Z N | → 0 as N → ∞.
6
M,NM ,τM
∂x^ f (0, x)
− ∂x f (0, x) = OP
1 1 +√ + NM M τM
1 τM
−1 ! .
With the selection 1/τM = M δ /εc1 and NM = M γ /εc2 for some εc1 , εc2 , δ, γ > 0, it is seen that the efficient scheme is obtained for (γ, δ) = inf (arg maxγ,δ min{δ, (1 − δ)/2, γ}) = (1/3, 1/3). This proves the theorem. M,N,τ
j Proof of Theorem 3: The error b εj (0, x; M, N, τj , αj ) ≡ ∂x\ −∂xj f (0, x) of the MCFD j f (0, x) M,N,τj estimators ∂x\ given in (30), can be expanded for arbitrary αj ∈ R, as: j f (0, x)
b εj (0, x; M, N, τj , αj ) = R1j (0, x; αj , τj ) + R2j (0, x; M, N, τj , αj ) + R3j (0, x; M, τj , αj ) with R1j (0, x; τj , αj ) ≡
E0 [g (XT (x + αj τj ej )) − g (XT (x − (1 − αj )τj ej ))] − ∂xj f (0, x) τj
(B-8)
i,N i,N M (x − (1 − α )τ e ) (x + α τ e ) − g X g X X j j j j j j T T 1 R2j (0, x; M, N, τj , αj ) ≡ M τj i=1 ! M 1 X g XTi (x + αj τj ej ) − g XTi (x − (1 − αj )τj ej ) − (B-9) M τj i=1
! M 1 X g XTi (x + αj τj ej ) − g XTi (x − (1 − αj )τj ej ) (B-10) R3j (0, x; M, τj , αj ) ≡ M τj i=1 g (XT (x + αj τj ej )) − g (XT (x − (1 − αj )τj ej )) −E0 . τj In this expansion, ej = [0, ..., 1, ..., 0]0 is the j th unit vector of dimension d, XTi,N (x) is an independent replication of the Euler discretization of the SDE based on N points started at X0i = x, and XTi (x) is an independent replication of the continuous diffusion with initial value X0i = x. Our next three lemmas provide the asymptotic errors for R1j , R2j and R3j . Lemma 4: Let R1j (0, x; τj , αj ) be defined by (B-8) and suppose that f ∈ C 3 (Rd ). Then lim1/τj →∞
1 R τj2 1j
(0, x; τj , αj ) =
1 3 24 ∂xj f (0, x)
if αj = 1/2
lim 1/τj →∞
1 τj R1j
(0, x; τj , αj ) =
2αj −1 2 2 ∂xj f (0, x)
if αj 6= 1/2.
7
Proof of Lemma 4: Given the differentiability assumption on f we can use Taylor series expansions for f (0, x + αj τj ej ) and f (0, x − (1 − αj )τj ej ) around f (0, x) to write (see footnote 2 for a definition of o(·)) 1 1 2 αj − (1 − αj )2 ∂x2j f (0, x)τj + αj3 + (1 − αj )3 ∂x3j f (0, x)τj2 +o R1j (0, x; τj , αj ) = 2 6
1 τj
−2 ! .
If αj = 1/2 the first term vanishes and R1j (0, x; τj , αj ) is of order (1/τj )−2 . If αj 6= 1/2 the second term in the expansion is asymptotically negligible and R1j (0, x; τj , αj ) is of order (1/τj )−1 . The limits announced are obtained from this Taylor expansion. Lemma 5: Let KT (x) be defined by (A-3), R2j (0, x; M, N, τj , αj ) by (B-9) and select NM , τj,M such that NM , 1/τj,M → ∞ when M → ∞. Suppose that the assumptions of Theorem 3 hold. Then, for all αj ∈ [0, 1] we have NM R2j (0, x; M, NM , τj,M , αj ) → 12 ∂KT (x) in probability as M → ∞. Proof of Lemma 5: We proceed in two steps as in the proof of Lemma 2. Under the assumptions of Theorem 3 we can apply the Clark-Ocone formula to find R2j (0, x; M, N, τj ) = R21j (0, x; M, N, τj ) − R22j (0, x; M, N, τj ) with M g X i,N (x + αj τj ej ) − g X i (x + αj τj ej ) X T T 1 R21j (0, x; M, N, τj , αj ) ≡ M τj i=1 M g X i,N (x − (1 − αj )τj ej ) − g X i (x − (1 − αj )τj ej ) X T T 1 R22j (0, x; M, N, τj , τj ) ≡ . M τj i=1
The law of large numbers and the arguments in the proof of Theorem 5 in Detemple, Garcia and Rindisbacher (2004) imply N R21j (0, x; M, N, τj , αj ) → N R22j (0, x; M, N, τj , αj ) →
1 KT (x + αj τj ej ) , 2 τj 1 KT (x − (1 − αj )τj ej ) 2 τj
in probability, as M, N → ∞ sequentially. If next 1/τj → ∞, we obtain N R2j (0, x; M, N, τj , αj ) → 1 2 ∂j KT (x).
This establishes that N R2j (0, x; M, N, τj , αj ) → 21 ∂j KT (x) in probability as M, N, 1/τj
→ ∞ sequentially. The remainder of the proof parallels the proof of Lemma 2. To show that the same limit holds if M, N, 1/τ j → ∞ jointly, and therefore along any “diagonal”, it is sufficient to show that ! M 1 X τj ,αj τ ,α lim sup P0 N ∇xj g XTi,N (x) − E0 N ∇xjj j g XTN (x) > = 0 (B-11) M M,N,1/τj →∞ i=1
8
for all > 0 and τ ,α lim sup E0 N ∇xjj j g XTN (x) − ∂xj N E0 g(XTN (x)) − g(XT (x)) = 0.
(B-12)
N,1/τj →∞
As in the proof of Lemma 2, (B-11) follows from the Markov-type inequality (B-6) and the uniform integrability condition (31). Similarly (B-12) is satisfied, because ∂xj N E0 g(XTN (x)) − g(XT (x)) = N E0 ∇xj (g(XTN (x)) − g(XT (x))) and under the assumptions of Theorem 5, τ ,α lim sup E0 ∇xjj j N (g(XTN (x)) − g(XT (x))) = lim sup E0 ∇xj N (g(XTN (x)) − g(XT (x))) . N →∞
1/τj ,N →∞
We conclude that the joint limit corresponds to the sequential limit. Lemma 6: Let R3j (0, x; M, τj ) be defined by (B-10) and select τj,M such that 1/τj,M → ∞ when M → ∞. Suppose that the assumptions of Theorem 3 hold. For j = 1, . . . , d and for all √ M R3j (0, x; M, τj,M ) ⇒ QT as M → ∞, where QT ∼ αj ∈ [0, 1] we have the limit, j=1,...,d RT N 0, E0 [ 0 Lv L0v dv] with L0v = Ev [Dv (∂g(XT )Dv XT C(Xv ))]. Proof of Lemma 6: Let us first find the asymptotic variance of τ
HTj ≡
g (XT (x + αj τj ej )) − g (XT (x − (1 − αj )τj ej )) . τj
As g(XT (x)) ∈ D1,2 , we can apply the Clark-Ocone formula to obtain Z T Z T τ τj τj τj (Lvj )0 dWv Ev [Dv HT ]dWv ≡ HT − E0 [HT ] = 0
0
τ
and therefore, VAR0 [HTj ] = E0 [[H τj , H τj ]T ] =
RT 0
τ
τ
E0 [(Lvj )0 Lvj ]dv.
Define f (v, Xv (x)) ≡ Ev,Xv (x) [g(XT (Xv (x)))]. Commutativity of the Malliavin derivative and the conditional expectation gives τ (Lvj )0
= =
τ Dv Ev [HTj ]
= Dv
f (v, Xv (x + αj ej τj )) − f (v, Xv (x − (1 − αj )ej τj )) τj
1 ∂x f (v, Xv (x + αj ej τj ))B(Xv (x + αj ej τj )) τj 1 − ∂x f (v, Xv (x − (1 − αj )ej τj ))B(Xv (x − (1 − αj )ej τj )). τj
Using X(x − (1 − αj )ej τj ) − X(x + αj ej τj ) → 0 P0 -a.s., as τj → 0 we obtain ∂x f (v, Xv (x + αj ej τj )) − ∂x f (v, Xv (x − (1 − αj )ej τj )) τj 0 (Lv ) − B(Xv ) → 0 τj 9
τ
2 f (v, X )B(X ) → 0 P -a.s., as τ → 0. From P0 -a.s., as τj → 0, or, equivalently, (Lvj )0 − ∂x,x v v 0 j j τ
2 f (v, X )B(X ) = D ∂ f (v, X ), we conclude (L j )0 → D ∂ f (v, X ), P -a.s. as τ → 0, and ∂x,x v v v v x v v xj v 0 j j RT 0 j τ τ 0 j j hence [H , H ]T → 0 Ljv Ljv dv P0 -a.s., with Ljv = Dv ∂xj f (v, Xv ) = Ev Dv ∂g(XT )∇xj XT . τ
By assumption (32), Lvj is uniformly integrable so that almost sure convergence implies convergence of the mean. We conclude τ lim VAR0 HTj = E0
T
Z
τj →0
L0jv Ljv ds
.
0
This analysis holds for all j = 1, ..., d. With Lv = [L0jv ]j=1,...,d we obtain the asymptotic variance hR i τ T VAR0 HTj j=1,...,d = E0 0 Lv L0v dv . The asymptotic distribution follows from the Lindeberg Central Limit theorem for independent random variables. The uniform integrability condition (32) ensures that the Lindeberg condition for application of this theorem is satisfied. Proof of Theorem 3 (continued): Given the results of Lemmas 4, 5 and 6, we see that
M,NM ,τj,M
∂x\ f (0, x)
O P N1M + − ∂x f (0, x) = OP N1M +
√1 M √1 M
+ +
1
−2
τj,M
1 τj,M
−1
if αj = 1/2
if αj 6= 1/2.
Choose 1/τj,M = M δj /εκj1 and NM = M γ /εκ2 for some εκj1 , εκ2 , δj , γ > 0 and κ ∈ {f cd, f d}. If αj = 1/2 the efficient scheme is obtained for (γ, δj ) = inf arg maxγ,δj min{2δj , 1/2, γ} = (1/2, 1/4). For αj 6= 1/2, (γ, δj ) = inf arg maxγ,δj min{δj , 1/2, γ} = (1/2, 1/2) provides the efficient scheme. This proves the theorem. Proof of Theorem 4: The proof of Theorem 4 parallels the proof of Theorem 3. For the disconP 1,2 tinuous functions under consideration, ∞ j=1 γj 1Bj (x) ∈ D , Lemma 6 is replaced by: Lemma 7: Let R3j (0, x; M, τj ) be defined by (B-10) and select τj,M such that 1/τj,M → ∞ when M → ∞. Suppose that the assumptions of Theorem 4 hold. For j = 1, . . . , d and for all αj ∈ [0, 1] p we have the limit, M τj,M R3j (0, x; M, τj,M ) ⇒ QjT as M → ∞, where QjT ∼ N (0, V0 (x))) with V0 (x) =
∞ X
γk2 (2αj − 1) ∂xj P0 ({XT (x) ∈ Bk })
k=1
−2αj
∞ X
γk γl ∂xj P0 {XT (x) ∈ Bk } ∩ XT (x0 ) ∈ Bl |x0 =x
k,l=1
+2(1 − αj )
∞ X
γk γl ∂x0j P0 {XT (x) ∈ Bk } ∩ XT (x0 ) ∈ Bl |x0 =x ,
k,l=1
10
and where QjT and QkT are mutually independent. Proof of Lemma 7: We first calculate the asymptotic variance of g(x) = g c (x) + g d (x) where P τj c,τj d,τ g d (x) = ∞ + HT j with j=1 γj 1Bj (XT ). Let HT ≡ HT c,τj
≡
HT
g c (XT (x + αj τj ej )) − g c (XT (x − (1 − αj )τj ej )) √ τj
g d (XT (x + αj τj ej )) − g d (XT (x − (1 − αj )τj ej )) . √ τj h i τ c,τ d,τ Using g c (x)g d (x) = 0, we obtain VAR0 HTj = VAR0 HT j + VAR0 HT j . c,τ From the assumption g c (XT ) ∈ D1,2 and Lemma 6, we get VAR0 HT j = O((1/τj )−1/2 ) when d,τj
≡
HT
1/τj → ∞, where O(·) is defined in i h footnote 2 d,τ d,τ Next, we show that limτi ,τj →0 COV0 [HTd,τi , HT j ] − E0 HTd,τi HT j = 0 for all i, j. As d,τ COV0 [HTd,τi , HT j ]
=
h
d,τ E0 HTd,τi HT j
i
−
d,τj c,τj d,τi √1 E0 [H T ] → ∂xj f (0, x) we have E0 [HT ]E0 [HT ] τj d,τ E0 [HTd,τi ]E0 [HT j ] → 0, P0 −a.s., as 1/τi , 1/τj → ∞.
and
h
E0 HTd,τi
i
h
d,τ E0 HT j
i
,
= O ((1/τi )(1/τj ))−1/2 , and therefore
Hence,
d,τ VAR0 [HT j ]
− E0
d,τ 2 HT j
→ 0, P0 − a.s., as
1 →∞ τj
i h 1 1 d,τ d,τ COV0 [HTd,τi , HT j ] − E0 HTd,τi HT j → 0, P0 − a.s., as , → ∞. τi τj
(B-13) (B-14)
To derive the asymptotic limits of the variance and the covariance we therefore need the limits of i h d,τ d,τ 2 and E HTd,τi HT j . E HT j i h d,τj 2 d,τ To find the limits of E0 HT and E0 HTd,τi HT j , note that d,τj
HTd,τi HT
=
1 g d (XT (x + αi τi ei )) g d (XT (x + αj τj ej )) τi τj 1 +√ g d (XT (x − (1 − αi )τi ei )) g d (XT (x − (1 − αj )τj ej )) τi τj 1 −√ g d (XT (x + αi τi ei )) g d (XT (x − (1 − αj )τj ej )) τi τj 1 −√ g d (XT (x − (1 − αi )τi ei )) g d (XT (x + αj )τj ej )) τi τj √
where g d (XT (x)) g d (XT (x0 )) ≡ P0 (XT (x) ∈ Bk ) and qkl
(x, x0 )
P∞
≡
0 k,l=1 γk γl 1Bk (XT (x))1Bl (XT (x )). With P0 ({XT (x) ∈ Bk } ∩ {XT (x0 ) ∈ Bl }) and
11
(B-15)
(B-16)
the definitions pk (x) ≡ using pk (x) = qkk (x, x)
and qkl (x, x) = 0 for k = 6 l, we obtain ∞ X pk (x − (1 − αj )ej τj ) − pk (x) d,τj 2 2 pk (x + αj ej τj ) − pk (x) + E0 HT = γk τj τj k=1 ∞ X qkk (x + αj ej τj , x − (1 − αj )ej τj ) − qkk (x, x) −2 γk2 τj k=1 ∞ X qkl (x + αj ej τj , x − (1 − αj )ej τj ) − qkl (x, x) −2 γk γl τj k,l=1
k6=l τ
d,τ
and therefore with V0 j (x) ≡ VAR0 [HT j ], τ
V0 j (x) →
∞ X
γk2 (2αj − 1) ∂xj pk (x) − 2
k=1
∞ X
γk γl αj ∂xj qkl (x, x0 )|x0 =x − (1 − αj )∂x0j qkl (x, x0 )|x0 =x ,
k,l=1
P0 -a.s., as 1/τj → ∞. d,τ
Next, we show that E0 [HTd,τi HT j ] → 0 P0 -a.s. when i 6= j and 1/τi , 1/τj → ∞. Using (B-15) we obtain d,τ E0 [HTd,τi HT j ]
=
∞ X
γk γl hkl (x, α, τi , τj )
k,l=1
where hkl (x, α, τi , αj , τj ) ≡
qkl (x + αi ei τi , x + αj ej τj ) − qkl (x, x) √ τi τj +
qkl (x − (1 − αi )ei τi , x − (1 − αj )ej τj ) − qkl (x, x) √ τi τj
−
qkl (x + αi ei τi , x − (1 − αj )ej τj ) − qkl (x, x) √ τi τj
−
qkl (x − (1 − αi )ei τi , x + αj ej τj ) − qkl (x, x) . √ τi τj
It follows that r r τj τi 0 0 hkl (x, α, τi , αj , τj ) = αi ∂xi qkl (x, x) + αj ∂xj qkl (x, x )|x0 =x τj τi r r τj τi −(1 − αi )∂xi qkl (x, x) − (1 − αj )∂x0j qkl (x, x0 )|x0 =x τj τi r r τj τi −αi ∂xi qkl (x, x) + (1 − αj )∂x0j qkl (x, x0 )|x0 =x τj τi r r τj τi +(1 − αi )∂xi qkl (x, x) − αj ∂x0j qkl (x, x0 )|x0 =x + o(1) τj τi = o(1). 12
d,τj
Hence hkl (x, α, τi , τj ) → 0, P0 −a.s. as 1/τi , 1/τj → ∞. This shows that HTd,τi and HT
are
asymptotically uncorrelated, for i 6= j. The asymptotic distribution follows from the Lindeberg Central Limit theorem for independent random variables: the uniform integrability condition (32) ensures that Lindeberg’s condition is satisfied. Proof of Theorem 4 (continued): The results of Lemmas 4, 5 and 7 give
M,NM ,τj,M
∂x\ f (0, x)
−2 1 1 1 + τj,M OP NM + √M τ j,M − ∂x f (0, x) = −1 1 1 1 √ + τj,M OP NM + M τj,M
if αj = 1/2 if αj 6= 1/2.
Choose 1/τj,M = M δj /εκj1 and NM = M γ /εκ2 for some εκj1 , εκ2 , δj , γ > 0 and κ ∈ {f cd, f d}. For αj = 1/2, the efficient scheme is attained for (γ, δj ) = inf arg maxδj ,γ min{2δj , (1 − δj )/2, γ} = (2/5, 1/5). For αj 6= 1/2, (γ, δj ) = inf arg maxγ,δj min{δj , (1 − δj )/2, γ} = (1/3, 1/3) provides the efficient scheme. This proves the theorem.
Appendix C: Abstract Integration by Parts This Appendix shows how to use an integration by parts argument to handle a discontinuous payoff function when the transition density is unknown.3 Consider a diffusion with volatility coefficient B ∈ C 1 (Rn × Rd ) such that rank(B(x)) = n for all x ∈ Rn and define the adapted shift on the Wiener space, Ω = C 0 ([0, T ]; Rd ), λ
Z
θ (ω)v = ωv +
v
νt,s (ω)λds. t
This perturbation of the state space depends on a parameter λ ∈ Rn and a progressively measurable Rn × Rd -valued matrix process νt,s . To obtain our integration by parts formula we pass to a new Rv measure Pλt,x under which Wv + t νt,s λds, v ≥ t, is a standard Brownian motion process. The existence of this probability measure requires, by Girsanov’s theorem, Z v 0 Et,x E − (νt,s λ) dWs = 1, for all v ≥ t. t
Under this condition we can define 3
dPλ t,x dPt,x
R T ≡ E − t (νt,s λ)0 dWs .
The presentation is based on Bismut’s approach to Malliavin calculus (see Bichteler, Gravereaux and Jacod (1987)
for a comparison of this approach with Malliavin’s original approach).
13
Taking derivatives with respect to the parameter λ on both sides of the equality " # dPλt,x λ h(XT (θ )) = Et,x [h(XT )] Et,x dPt,x and evaluating the resulting expressions at λ = 0, gives, for h ∈ C 1 (Rd ) " ! # λ dPλt,x dP t,x ∂λ Et,x h(XT (θλ )) + ∂h(XT (θλ ))∂λ XT (θλ ) dPt,x dPt,x dPλ
Using ∂λ dPt,x = t,x
dPλ t,x dPt,x ∂λ
R
T 0 t (−νt,s λ) dWs
−
1 2
RT t
kνt,s λk2 ds and
dPλt,x =− dPt,x |λ=0
Z
T
(C-17)
= 0. |λ=0
dPλ t,x dPt,x |λ=0
= 1 gives
dWs0 νt,s .
t
Substituting in (C-17) yields the abstract integration by parts formula, Z T Et,x (dWs )0 νt,s h(XT (θλ ))|λ=0 = Et,x ∂h(XT ) ∂λ XT (θλ ) t
|λ=0
.
(C-18)
To identify ∂λ XT (θλ ) |λ=0 differentiate the integral representation Z
λ
XT (θ ) = x +
T
d Z X A Xs (θ ) ds +
λ
0
j=1
T
Bj
Xs (θ ) dWsj + νt,s λds λ
0
with respect to λ, to obtain Z T d X ∂A(Xs )ds + ∂λ XT (θλ ) = ∂Bj (Xs )dWsj ∂λ Xs (θλ ) |λ=0
0
j=1
The solution of this linear SDE is Z λ ∂λ XT (θ ) = ∇t,x XT (Xt ) |λ=0
where ∇t,x XT (Xt ) is the solution of
T
Z |λ=0
+
T
B(Xs )νt,s ds. 0
(∇t,x Xs (Xt ))−1 B(Xs )νt,s ds
t
(A-1).4
The process ∇t,x XT (Xt ) is the tangent process. (α)
For C defined in (13), selecting the process νt,s ≡ C(Xs )∇t,x Xs (Xt )αt,s for some progressively RT measurable process αt,· such that t αt,s ds = In , and substituting in (C-18) gives (18), that is Z T Et,x (dWs )0 C(Xs )∇t,x Xs (Xt )αt,s h(XT ) = Et,x [∂h(XT )∇t,x XT (Xt )] . t
To establish the same result when h ∈ L2 is not continuous, we use the fact that any square integrable function can be approximated by a sequence of infinitely differentiable functions with compact support. This enables us to proceed by first using the approximating sequence and then passing to the limit. See Fourni´e et. al. (1999) for detailed arguments. 4
` ´ Note that ∂λ XT (θλ ) |λ=0 exists whenever the stochastic flow is differentiable with respect to the initial condition.
A sufficient condition for this is XT ∈ D1,2 , which is satisfied when the diffusion is in the domain of the Malliavin derivative operator. See Bichteler, Gravereaux, Jacod (1987) for more on the differentiability of perturbed diffusions.
14
References Bichteler, K., J. B. Gravereaux, J. Jacod. 1987. Malliavin Calculus for Processes with Jumps. Gordon Breach Publishing Group, New York. Billlingsley, P. 1968. Convergence of Probability Measures. Wiley, New York. Detemple, J. B., R. Garcia, M. Rindisbacher. 2004. Asymptotic Properties of Monte Carlo Estimators of Diffusion Processes. Working Paper, CIRANO, forthcoming J. Econometrics. Fourni´e, E., J-M. Lasry, J. Lebuchoux, P-L. Lions, N. Touzi. 1999. Applications of Malliavin Calculus to Monte Carlo Methods in Finance. Finance Stochastics 3(4) 391-412. Kallenberg, O. 1997. Foundations of Modern Probability Theory. Springer-Verlag, New York. Phillips, P. C. B., H. R. Moon. 1999. Linear Regression Limit Theory for Nonstationary Panel Data. Econometrica 67(5) 1057-1111..
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