Frédéric Lavancier
email: frederic.lavancier@univ-nantes.fr

Anne Philippe
email: anne.philippe@univ-nantes.fr

Donatas Surgailis
email: sdonatas@ktl.mii.lt

Covariance function of vector self-similar process

Keywords: operator self-similar process, vector fractional Brownian motion, crosscovariance function

may come

Covariance function of vector self-similar process

Introduction

A p-variate stochastic process X = {X(t) = (X 1 (t), • • • , X p (t)), t ∈ R} is said operator self-similar (os-s) if there exists a p × p matrix H (called the exponent of X) such that for any λ > 0, X(λt) = fdd λ H X(t),

(1.1) where = fdd means equality of finite-dimensional distributions, and the p × p matrix λ H is defined by the power series λ H = e H log λ = ∞ k=0 H k (log λ) k /k!. Os-s processes were studied in [START_REF] Laha | Operator self-similar stochastic processes in R d[END_REF], [START_REF] Hudson | Operator-self-similar processes in a finite-dimensional space[END_REF], Maejima and Hudson (1984), [START_REF] Sato | Self-similar processes with independent increments[END_REF] and other papers. A Gaussian os-s process with stationary increments ACCEPTED MANUSCRIPT is called operator fractional Brownian motion (ofBm). For p = 1 the class of ofBm coincides with fundamental class of fractional Brownian motions (fBm) (see e.g. [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF]). Recall that a fBm with exponent H ∈ (0, 1) can be alternatively defined as a stochastically continuous Gaussian process X = {X(t), t ∈ R} with zero mean and covariance

EX(s)X(t) = σ 2 2 (|s| 2H + |t| 2H -|t -s| 2H ), t, s ∈ R, (1.2)
where σ 2 = EX 2 (1). The form of covariance of general ofBm seems to be unknown and may be quite complicated. The structure of ofBm and stochastic integral representations were studied in [START_REF] Didier | Integral representations of operator fractional Brownian motion[END_REF].

A particular case of os-s processes corresponds to diagonal matrix

H = diag(H 1 , • • • , H p ). In this case, relation (1.1) becomes (X 1 (λt), • • • , X p (λt)) = fdd (λ H 1 X 1 (t), • • • , λ Hp X p (t)).
(1.3) Below, a p-variate process X satisfying (1.3) for any λ > 0 will be called vector self-similar (vs-s) and a stochastically continuous Gaussian vs-s process with stationary increments (si) will be called a vector fractional Brownian motion (vfBm).

Vs-s processes (in particularly, vfBm) seem to be most useful for applications and statistical analysis of multiple time series. They arise as limits of normalized partial sums of multivariate long memory processes with discrete time, see [START_REF] Marinucci | Weak convergence of multivariate fractional processes[END_REF], [START_REF] Davidson | The functional central limit theorem and weak convergence to stochastic integrals[END_REF], [START_REF] Chung | Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes[END_REF], [START_REF] Davidson | Alternative frequency and time domain versions of fractional Brownian motion[END_REF], [START_REF] Robinson | Multiple local Whittle estimation in stationary systems[END_REF], [START_REF] Lavancier | A two-sample test for comparison of long memory parameters[END_REF]. In particularly, vs-s processes appear in the OLS estimation in multiple linear regression model [START_REF] Chung | Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes[END_REF], the multiple local Whittle estimation [START_REF] Robinson | Multiple local Whittle estimation in stationary systems[END_REF] and the two-sample testing for comparison of long memory parameters [START_REF] Lavancier | A two-sample test for comparison of long memory parameters[END_REF].

Note from (1.3) that each component X i = {X i (t), t ∈ R}, i = 1, • • • , p of vs-s process is a (scalar) self-similar process, the fact which is not true for general os-s processes.

The present paper obtains the general form of the (cross-)covariance function of vs-s si process X with finite variance and exponent H = diag(H 1 , • • • , H p ), 0 < H i < 1. According to Theorem 2.1 below, under some regularity condition, for any i, j = 1, • • • , p, i = j with

H i + H j = 1, there exist c ij , c ji ∈ R such that for any s, t ∈ R cov(X i (s), X j (t)) = σ i σ j 2 c ij (s)|s| H i +H j + c ji (t)|t| H i +H j -c ji (t -s)|t -s| H i +H j , (1.4)
where σ 2 i := var(X i (1)) and

c ij (t) := c ij , t > 0, c ji , t < 0.
(1.5)

A similar expression (involving additional logarithmic terms) for the covariance cov(X i (s), X j (t)) is obtained in the case H i + H j = 1. We prove Theorem 2.1 by deriving from vss si property a functional equation for the cross-covariance of the type studied in Aczél ACCEPTED MANUSCRIPT (1966) and [START_REF] Aczél | Further uniqueness theorems for functional equations[END_REF] and using the result of the last paper (Theorem 2.2 below) about the uniqueness of this equation. Section 3 discusses the existence of vfBm with covariance as in (1.4). We start with a double sided stochastic integral representation similar to [START_REF] Didier | Integral representations of operator fractional Brownian motion[END_REF]:

X(t) = R (t -x) H-.5 + -(-x) H-.5 + A + + (t -x) H-.5 - -(-x) H-.5 - A -W (dx),
(1.6) where H -.5 := diag(H 1 -.5, • • • , H p -.5), x + := max(x, 0), x -:= max(-x, 0), A + , A - are real p × p matrices and

W (dx) = (W 1 (dx), • • • , W p (dx)
) is a Gaussian white noise with zero mean, independent components and covariance EW

i (dx)W j (dx) = δ ij dx. According to Proposition 3.1, if 0 < H i < 1, H i + H j = 1, i, j = 1, • • • , p then the cross-covariance of X in (1.6) has the form as in (1.4) with c ij = 2c ij φ ij /σ i σ j , φ ij := B(H i + .5, H j + .5)/ sin((H i + H j )π), (1.7)
where the matrix C = (c ij ) is given by

C := cos(Hπ)A + A * + + A -A * -cos(Hπ) (1.8) -sin(Hπ)A + A * -cos(Hπ) -cos(Hπ)A + A * -sin(Hπ).
Here and below, A * denotes the transposed matrix, sin(Hπ

) := diag(sin(H 1 π), • • • , sin(H p π)), cos(Hπ) := diag(cos(H 1 π), • • • , cos(H p π)).

The form of the covariance function of vs-s process

Recall that a random process X = {X(t), t ∈ R} has stationary increments (si

) if {X(t + T ) -X(T ), t ∈ R} = fdd {X(t) -X(0), t ∈ R} for any T ∈ R.
Theorem 2.1 Let X = {X(t), t ∈ R} be a 2nd order process with values in R p . Assume that X has stationary increments, zero mean, X(0) = 0, and that X is vector self-similar with exponent

H = diag(H 1 , • • • , H p ), 0 < H i < 1 (i = 1, • • • , p).
Moreover, assume also that for any

i, j = 1, • • • , p, the function t → EX i (t)X j (1) is continuously differentiable on (0, 1) ∪ (1, ∞). Let σ 2 i > 0 denote the variance of X i (1), i = 1, • • • , p. (i) If i = j, then for any (s, t) ∈ R 2 , we have EX i (s)X i (t) = σ 2 i 2 |s| 2H i + |t| 2H i -|t -s| 2H i . (2.1) (ii) If i = j and H i + H j = 1, then there exists c ij , c ji ∈ R such that for any (s, t) ∈ R 2 , (1.4) holds. (iii) If i = j and H i + H j = 1, then there exists d ij , f ij ∈ R such that for any (s, t) ∈ R 2 , we have EX i (s)X j (t) = σ i σ j 2 {d ij (|s| + |t| -|s -t|) + f ij (t log |t| -s log |s| -(t -s) log |t -s|)} . (2.2) (iv) The matrix R = (R ij ) i,j=1,•••,p is positive definite, where R ij :=    1, i = j, c ij + c ji , i = j, H i + H j = 1, d ij , i = j, H i + H j = 1.
Proof. (i) Follows from the well-known characterization of covariance of (scalar-valued) self-similar stationary increment process.

(ii)-(iii) From part (i), it follows that E(X i (t) -X i (s)) 2 = σ 2 i |t -s| 2H i and hence {X i (t)
, t ∈ R} is stochastically continuous on the real line, for any i = 1, • • • , p. Whence, it follows that EX i (s)X j (t) is jointly continuous on R 2 and vanishes for t = 0 or s = 0, for any

i, j = 1, • • • , p. Denote r(s, t) := EX i (s)X j (t), H := 1 2 (H i + H j ). Let R + := {u : u > 0}, R -:= {u : u < 0}. From vs-s and si properties we obtain r(λs, λt) = λ 2H r(s, t), (2.3) r(s, t) = r(s + T, t + T ) -r(s + T, T ) -r(T, t + T ) + r(T, T ) (2.4)
for any reals s, t, T and any λ > 0. Substituting

s = t = u, λ = 1/|u| into (2.3) one obtains r(u, u) = 2κ ± |u| 2H , u ∈ R ± , 2κ ± := r(±1, ±1).
(2.5)

Substituting s = t = -1, T = 1 into (2.4) and using r(0, 0) = r(1, 0) = r(0, 1) = 0 yields r(-1, -1) = r(1, 1). Next, substituting T = u, s = t = vu into (2.4) and using (2.5) with

κ := κ + = κ -one obtains r(u, v) + r(v, u) = r(v, v) + r(u, u) -r(v -u, v -u) = 2κ(|v| 2H + |u| 2H -|u -v| 2H ) (u, v ∈ R). (2.6) Next, let g ± (t) := r(±1, t) (t ∈ R). Then (2.3) implies r(s, t) = |s| 2H g ± (±t/s) (s ∈ R ± , t ∈ R).
(2.7) Equation (2.4) with s = 1 and (2.7) yield, for all t ∈ R

g + (t) = (T + 1) 2H g + T + t T + 1 -T 2H g + T + t T -(T + 1) 2H g + T T + 1 + T 2H g + (1) (T > 0), (2.8)

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and for all t ∈ R, equation (2.4) with s = -1 and (2.7) yield

g -(t) = (T -1) 2H g + T + t T -1 -T 2H g + T + t T -(T -1) 2H g + T T -1 + T 2H g + (1) (T > 1).
(2.9)

We claim that the general solution of functional equations (2.8)-(2.9) has the form:

• When H = 1/2, g + (t) =      c ′ + c ′ |t| 2H -c ′ |1 -t| 2H , t < 0, c ′ + c|t| 2H -c ′ |1 -t| 2H , 0 < t < 1, c ′ + c|t| 2H -c|1 -t| 2H , t > 1, , (2.10
)

g -(t) =      c + c ′ |t| 2H -c ′ |t + 1| 2H , t < -1, c + c ′ |t| 2H -c|t + 1| 2H , -1 < t < 0, c + c|t| 2H -c|t + 1| 2H , t > 0;
(2.11)

• When H = 1/2, g + (t) = f (t log |t| -(t -1) log |t -1|) , t < 0, d (1 ∧ t) + f (t log |t| -(t -1) log |t -1|) , t > 0, , (2.12
)

g -(t) = d (1 ∧ |t|) + f (t log |t| -(t + 1) log |t + 1|) , t < 0, f (t log |t| -(t + 1) log |t + 1|) , t > 0, (2.13)
with some c, c ′ , d, f ∈ R. It follows from (2.10)-(2.13) and (2.7) that the covariance r(s, t) = EX i (s)X j (t) for (s, t) ∈ R 2 has the form as in (1.4), (2.2), with c ij = 2c/σ i σ j , c ji = 2c ′ /σ i σ j in the case 2H = H i + H j = 1, and

d ij = 2d/σ i σ j , f ij = 2f /σ i σ j in the case 2H = H i + H j = 1.
To show the above claim, note, by direct verification, that (2.10)-(2.13) solve equations (2.8)-(2.9). Therefore it suffices to show that (2.10)-(2.13) is a unique solution of (2.8)-(2.9).

Let t > 1. Differentiating (2.8) with respect to t leads to

g ′ + (t) = (T + 1) 2H-1 g ′ + T + t T + 1 -T 2H-1 g ′ + T + t T .
(2.14)

Let x := t, y := (T + t)/T . Then (x, y) ∈ (1, ∞) 2 and the mapping (t, T ) → (x, y) :

(1, ∞) × (0, ∞) → (1, ∞) 2 is a bijection. Equation (2.14) can be rewritten as (2.16) Equation (2.15) belongs to the class of functional equations treated in [START_REF] Aczél | Lectures on functional equations and their applications[END_REF] and [START_REF] Aczél | Further uniqueness theorems for functional equations[END_REF]. For reader's convenience, we present the result from [START_REF] Aczél | Lectures on functional equations and their applications[END_REF] which will be used below.

g ′ + (F (x, y)) = y -1 x + y -1 2H-1 g ′ + (x) + x x + y -1 2H-1 g ′ + (y), ( 2

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Theorem 2.2 [START_REF] Aczél | Further uniqueness theorems for functional equations[END_REF]) There exists at most one continuous function f satisfying the functional equation

f (F (x, y)) = L(f (x), f (y), x, y)
(2.17) for all x, y ∈ A, B and the initial conditions

f (a 1 ) = b 1 , f (a 2 ) = b 2 (a 1 , a 2 ∈ A, B , a 1 = a 2 ) if F is continuous in A, B × A, B , L(u, v, x, y
) is strictly monotonic in u or v and F is intern (i.e., F (x, y) ∈ (x, y) for all x = y ∈ A, B ), where A, B is a closed, half-closed, open, finite or infinite interval.

Unfortunately, equation (2.15) does not satisfy the conditions of Theorem 2.2 (with f = g + and A, B = (1, ∞)), since F in (2.16) is not intern. Following [START_REF] Aczél | Further uniqueness theorems for functional equations[END_REF], we first apply some transformations of (2.15) so that Theorem 2.2 can be used. Note, taking T = t/(t -1) in (2.14) one obtains

g ′ + (t) = K(t)g ′ + t 2 2t -1 , (2.18)
where

K(t) := (2t -1) 2H-1 t 2H-1 + (t -1) 2H-1 . (2.19) Let F (x) := F (x, x) = x 2 2x-1 . Then F is strictly increasing from (1, ∞) onto (1, ∞). Let G(x, y) := F -1 (F (x, y)) = xy + x(x -1)y(y -1) x + y -1 , (2.20)
with F -1 (y) = y + y(y -1). Note F (x, y) > 1, G(x, y) > 1 for (x, y) ∈ (1, ∞) 2 . From (2.18) with t = G(x, y) one obtains

g ′ + (G(x, y)) = g ′ + (F (x, y))K(G(x, y)).
(2.21)

Combining (2.15) and (2.21) one gets

g ′ + (G(x, y)) = L(g ′ + (x), g ′ + (y), x, y), (2.22)
where

L(u, v, x, y) := y -1 x + y -1 2H-1 u + x x + y -1 2H-1 v K(G(x, y)).
(2.23)

The fact that G in (2.20) is intern follows from its definition and monotonicity in x and y, implying

x = G(x, x) ≤ G(x, y) ≤ G(y, y) = y

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for any x ≤ y, x, y ∈ (1, ∞). Since L is monotonic in u or v, so Theorem 2.2 applies to functional equation (2.22) and therefore this equation has a unique continuous solution g ′ + on the interval (1, ∞), given boundary conditions

g ′ + (a i ) = b i (i = 1, 2), 1 < a 2 < a 1 < ∞. Form of g + on [1, +∞) : • Assume H = 1/2. Let a 1 := 2, b 1 := cH(2 2H -2).
In view of (2.18), the other boundary condition can be defined by a 2 = a 2 1 /(2a 1 -1) = 4/3, b 2 := g ′ + (2)/K(2) and so equations (2.22) and (2.14) have a unique solution for single boundary condition Aczél and Hosszú (1965, p.51)

g ′ + (2) = cH(2 2H -2). (See also
.) Since g ′ + (t) = 2Hc(t 2H-1 -(t -1) 2H-1
) is a solution of (2.14) with this boundary condition, it follows that this solution is unique. Hence it also follows that

g + (t) = c ′ + ct 2H -c(t -1) 2H t ∈ [1, ∞), (2.24)
for some c ′ ∈ R.

• When H = 1/2. A particular solution of (2.14) when t > 1 is log(t) -log(t -1). For the same reason as above, the general solution of (2.14) is thus d ′ (log(t) -log(t -1)) where d ′ ∈ R. It follows that

g + (t) = d + d ′ (t log t -(t -1) log(t -1)) t ∈ [1, ∞), (2.25)
for some d, d ′ ∈ R.

Form of g + on (0, 1) : Putting t = 1 in (2.8) results in

(T + 1) 2H g + T T + 1 + T 2H g + T + 1 T = g + (1)[(T + 1) 2H + T 2H -1].
Whence, for s := T /(T + 1) ∈ (0, 1) and using (2.24) one obtains

g + (s) = -s 2H g + (1/s) + g + (1)[1 + s 2H -(1 -s) 2H ].
(2.26)

This gives, for s ∈ (0, 1),

• when

H = 1/2, g + (s) = c ′ + cs 2H -c ′ (1 -s) 2H ,
• when H = 1/2, g + (s) = ds + d ′ (s log s -(s -1) log(s -1)).

Therefore, relations (2.10) and (2.12) have been proved when t > 0.

Form of g + on (-∞, 0) : This case follows from (2.8) taking T = -t.

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Remark 3.2 Let H i + H j = 1, i, j = 1, • • • , p and let cij , φ ij be defined as in (1.7). ¿From Proposition 3.1 (3.1), (3.2) it follows that the matrix C = (c ij ) satisfies (1.8). In this context, a natural question arises to find easily verifiable conditions on the matrices C and H such that there exist matrices A + , A -satisfying the quadratic matrix equation in (1.8). In other words, for which C and H there exists a vfBm X with cross-covariance as in (1.4)? While the last question does not seem easy, it becomes much simpler if we restrict the class of vfBm's X in (1.6) to causal representations with A -= 0. In this case, equation (1.8) becomes C = cos(H)A + A * + . Clearly, the last factorization is possible if and only if the matrix cos(H) -1 C is symmetric and positive definite.

Proof. For all s, let

I + ik (s) := R (s -x) H i -.5 + -(-x) H i -.5 + W k (dx), I - ik (s) := R (s -x) H i -.5 - -(-x) H i -.5 - W k (dx).
Using the above notation,

X i (s) = p k=1 a + ik I + ik (s) + a - ik I - ik (s) and EX i (s)X j (t) = α ++ ij EI + i1 (s)I + j1 (t) + α +- ij EI + i1 (s)I - j1 (t) (3.5) + α -+ ij EI - i1 (s)I + j1 (t) + α -- ij EI - i1 (s)I - j1 (t).
Let H i + H j = 1. From Stoev and Taqqu (2006, Th. 4.1), taking there a + = 1, a -= 0, H(s) = H i and H(t) = H j , we obtain

EI + i1 (s)I + j1 (t) = ψ H cos (H j -H i ) π 2 - (H i + H j )π 2 sign(s) |s| H i +H j + cos (H j -H i ) π 2 + (H i + H j )π 2 sign(t) |t| H i +H j -cos (H j -H i ) π 2 - (H i + H j )π 2 sign(s -t) |s -t| H i +H j ,
where

ψ H := Γ(H i + .5)Γ(H j + .5)Γ(2 -H i -H j ) π(H i + H j )(1 -H i -H j ) = B(H i + .5, H j + .5) sin((H i + H j )π) . Therefore, EI + i1 (s)I + j1 (t) = B(H i + .5, H j + .5) sin((H i + H j )π) b ij (s)|s| H i +H j + b ji (t)|t| H i +H j -b ij (s -t)|s -t| H i +H j , where b ij (s) = cos(H i π), if s > 0 cos(H j π), if s < 0.
Similarly, taking a + = 0, a -= 1, H(s) = H i and H(t) = H j in Stoev and Taqqu (2006, Th. 4 Therefore, using (3.5), we obtain (3.3) and (3.4). Proposition 3.1 is proved.

  B(H i + .5, H j + .5) sin((H i + H j )π) b ji (s)|s| H i +H j + b ij (t)|t| H i +H jb ji (st)|s -t| H i +H j . x) H i -.5 (xt) H j -.5 dx -1 {s>0} s 0 (sx) H i -.5 x H j -.5 dx H i -.5 (xt) H j -.5 dx = B(H i + .5, H j + .5) (st)Substituting these formulas into (3.5) we obtain (3.2) and (3.1).Next, let H i + H j = 1. We get similarly from Theorem 4.1 of[START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Process[END_REF] Expressions (3.6) and (3.7) remain true when H i + H j = 1 and they can be rewritten as

	.1), we obtain
	EI -i1 (s)I -j1 (t) =
	Finally,
	EI + i1 (s)I -j1 (t) =	1 {s>t}	t	s	(s -1 {t<0} (-x) H i +H j 0 t + -s H i +H j + -t	H i +H j -	(3.6)
	and
	EI -i1 (s)I +				H i +H j +	-t	H i +H j +	-s	H i +H j -	.	(3.7)
	EI + i1 (s)I + j1 (t) = sin(H and 1 π B(H i + .5, H j + .5) π 2
	EI -i1 (s)I -j1 (t) = sin(H EI + 1 π B(H i + .5, H j + .5) π 2 i1 (s)I -j1 (t) = EI -i1 (s)I + j1 (t) = -1 2 B(H

j1 (t) = B(H i + .5, H j + .5) (ts) i π)(|s| + |t| -|s -t|) -cos(H i π)(s log |s|t log |t| -(st) log |s -t|) i π)(|s| + |t| -|s -t|) + cos(H i π)(s log |s|t log |t| -(st) log |s -t|) . i + .5, H j + .5)(|s| + |t| -|s -t|).

* The joint work was partially supported by the research project MATPYL of the Fédération de Mathématiques des Pays de Loire. † The author is partially supported by the Lithuanian State Science Foundation grant no. T-70/09.

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Form of g -: The relations (2.11) and (2.13) are deduced from (2.10) and (2.12) thanks to relation (2.9).

(iv) Follows from the fact that R is the covariance matrix of random vector (X 1 (1)/σ 1 ,

Theorem 2.1 is proved.

Stochastic integral representation of vfBm

In this section we derive the covariance function of vfBm X = {X(t), t ∈ R} given by double-sided stochastic integral representation in (1.6). Denote

where

6). Clearly,

Note, each of the processes

) is a well-defined fractional Brownian motion with index H i ∈ (0, 1); see e.g. [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF].

Proposition 3.1 The covariance of the process defined in (1.6) satisfies the following properties

(3.1) (ii) If H i + H j = 1 then for any s, t ∈ R, the cross-covariance EX i (s)X j (t) of the process in (1.6) is given by (1.4), with

) is given by (2.2), with

(3.4)


Anna Krasnosielska
email: akrasno@mini.pw.edu.pl

A version of the Elfving problem with random starting time

Keywords: stopping game, Poisson process, random starting time, random horizon

An optimal stopping time problem with a Poisson process, discount function and random starting time is considered. Generalizations to a problem with random horizon and to a multi-person stopping game with priorities are presented.

Introduction

An agent, who is waiting for some permit to run the business, is interested in some projects. He will obtain the permit at some random time M. Until M, he observes the market and learns which project should be accepted. The projects appear according to a Poisson process and have assigned values Y 1 , Y 2 , . . ., which are i.i.d. random variables. At each time only one project is presented. The agent is allowed to accept only one project at the time of its appearance and he is interested in accepting the most profitable one. The profit from the accepted project with value Y n at time τ n is g(Y n r(τ n )) if τ n ≥ M, and 0 otherwise, where r(•) and g(•) express the change of the value of the projects in time and the agent's subjective value of projects, respectively.

The results of the paper extend those obtained in the paper by [START_REF] Elfving | A persistency problem connected with a point process[END_REF] and Ferenstein andKrasnosielska (2009a, 2009b). [START_REF] Elfving | A persistency problem connected with a point process[END_REF] considered

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the problem of selling a commodity where offers Y 1 , Y 2 , . . . appear according to a Poisson process. The offers Y i are i.i.d. random variables independent of the jump times τ 1 , τ 2 , . . . of the Poisson process. The owner obtains a reward Y n r(τ n ) if he accepts the offer Y n at time τ n , where r(•) is a discount function. [START_REF] Siegmund | Some problems in the theory of optimal stopping[END_REF] removed several additional assumptions from Elfving solution. Ferenstein and Krasnosielska (2009a) generalized the Elfving problem to the case of a multi-person stopping game with priorities. A model with random horizon based on the Elfving problem was considered in Ferenstein and Krasnosielska (2009b). [START_REF] Stadje | An optimal k-stopping problem for the Poisson process[END_REF] generalized the Elfving problem to a multi stopping time problem. [START_REF] Kühne | Approximation of optimal stopping problems[END_REF] obtained the approximations for the stopping of i.i.d. sequences in the domain of max-stable laws. [START_REF] Cowan | An optimal selection problem associated with the Poisson process[END_REF] considered a continuous-time version of the secretary problem in which the offers appear at jump times of the Poisson process up to fixed time T . [START_REF] Bruss | On an optimal selection problem of Cowan and Zabczyk[END_REF] extended the model of [START_REF] Cowan | An optimal selection problem associated with the Poisson process[END_REF] to the case of a compound Poisson process. [START_REF] Szajowski | A game version of the Cowan-Zabczyk-Bruss' problem[END_REF] generalized the problem considered by [START_REF] Bruss | On an optimal selection problem of Cowan and Zabczyk[END_REF] to the case of a two-person game with random priority. A full-information best choice problem with random starting point was considered by [START_REF] Porosiński | Full-information best choice problem with random starting point[END_REF]. The starting point (an observation which cannot be chosen) is a positive random variable independent of observations which are i.i.d. random variables. The observations appear according to a renewal process. A decision maker obtains some imperfect information about the random starting point. The aim of the decision maker is to choose the largest observation. [START_REF] Sakaguchi | Best choice problems for randomly arriving offers during a random lifetime[END_REF] analyzed the full information secretary problem with Poisson stream of offers arriving in a time interval of random length. [START_REF] Samuel-Cahn | Optimal stopping with random horizon with application to the full-information best-choice problem with random freeze[END_REF] considered an optimal stopping time problem with independent random horizon M and a known number of offers. The problem has been translated to a problem with discounting. Multi-person games with offers having uniform distribution and appearing according to a Poisson process were considered in [START_REF] Dixon | Equilibrium points for three games based on the Poisson process[END_REF]. A two-person game with random priority and two Poisson streams of offers was considered in [START_REF] Sakaguchi | Best-choice problems with random priority on a two-Poisson stream[END_REF]. In this game player i has higher priority at the times when the observations are from stream i. General models of multi-person discretetime games with priorities were analyzed in [START_REF] Enns | On a multi-person time-sequential game with priorities[END_REF]. A multi-person stopping game with a known and finite number of observations was considered in [START_REF] Ramsey | N person stopping games with players given priority randomly[END_REF]. A game with a large number of offers and players was analyzed in [START_REF] Ramsey | A large population job search game with discrete time[END_REF].

In the paper the notations and the solution of the Elfving problem follow those of Chow et al. (1971, pp. 113-118).

Let (Ω, F , P) be a probability space on which all random variables are considered. Let Y 1 , Y 2 , . . . be independent, non-negative random variables with continuous distribution function F , 0 < τ 1 < τ 2 < • • • be the jump times of a homogeneous Poisson process with intensity 1, Y 0 = 0, τ 0 = 0. Let M be a nonnegative random variable with known distribution function F M . We assume that the sequences {Y n } and {τ n } are independent and they are independent of M. Let us introduce various σ-fields of events dependent on available observations:

F n = σ(Y 1 , . . . , Y n , τ 1 , . . . , τ n ), F * n = σ(F n , I(M ≤ τ 0 ), . . . , I(M ≤ τ n )), n = 1, 2, . . ., F 0 = {∅, Ω}, F * 0 = σ(F 0 , I(M ≤ τ 0 )),
X * n = g(Y n r(τ n ))I(M ≤ τ n ), (1)
X * ∞ = lim sup n→∞ X * n and r : R + 0 → R + 0 and g : R + 0 → R + 0 are Borel functions. We assume that r is bounded, continuous from the right, g is increasing, continuous, g(0) = 0, and there exist functions g 1 and g 2 such that |g(yr(x 1 ))-

g(yr(x 2 ))| ≤ |g 1 (y)g 2 (x 1 , x 2 )| for x 1 , x 2 ≥ 0 and y ∈ supp Y 1 , E|g 1 (Y 1 )| < ∞, g 1 (0) = 0, ∞ 0 |g 2 (x 1 + t, x 2 + t)|dt < c|x 1 -x 2 |
for some constant c ≥ 0. Moreover, we assume that there exist constants m and U such that U > m ≥ 0, F M (U) > 0, r(s) = 0 for s ∈ [0, m) ∪ [U, ∞) and r(s) > 0 for s ∈ [m, U). If m > 0, then U < ∞, and if m = 0, then U ≤ ∞. Let g -1 denote the inverse function for g. Let S be the set of all points of discontinuity of the functions r(•) and F M (•) and all points of non-differentability of the function

F M (•). Assume that S ∩ (m, U) = {s 1 , . . . , s k }, k < ∞, s 1 < • • • < s k , s 0 = m, s k+1 = U and ∞ 0 h(x)dx < ∞, where h(x) = E(g(Y 1 r(x))). Denote F (y) = 1 -F (y).
Note that the conditions concerning the functions r and g are satisfied if r is increasing at most on a finite number of intervals and either g ), and 0 elsewhere. Remark 2.1 From Equation (1), the fact that g(•) is nonnegative and the def-

(x) = ax p , p > 0, a > 0, E(Y p 1 ) < ∞, ∞ 0 (r(x)) p dx < ∞ or E(Y 1 ) < ∞ and g(x) is a Lipschitz function. For example g(x) = ln(x+1), x ≥ 0, r(x) = x for x ∈ [0, 8), r(x) = 1 for x ∈ [8, U
inition of h(•) we get E( ∞ n=1 X * n ) ≤ E( ∞ n=1 g(Y n r(τ n ))) = ∞ n=1 E(h(τ n )) < ∞ and consequently X * ∞ = lim n→∞ X * n = 0 and E(sup(X * n )) < ∞.
Note that m may be interpreted as the time up to which we plan to observe the market and learn, max{M, m} as the time up to which we will actually learn, r as the discount function if it is non-increasing and r(0) = 1, g as the utility function.

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To solve the above problem we need to find a solution of a generalization of the Elfving problem without random starting time.

The generalization of the Elfving problem

Let the above assumptions be satisfied and let our goal be to find sup t∈T EX t (u), where

X n (u) = g(Y n r(u + τ n )), u ≥ 0. We have E(sup n X n (u)) < ∞ and X ∞ (u) = 0. Note that {Z n,u } ∞ n=0 , Z n,u = (Y n , u+τ n ), is a homogeneous Markov chain with respect to {F n } ∞ n=0 , and X n (u) is the function of Z n,u only. More- over, since Y i are i.i.d. random variables, E(ess sup t∈T n+1 E(X t (u) | F n+1 ) | F n ) is a function of u + τ n only, say V (u + τ n ). Moreover, the Markov time σ(u) = inf{n ≥ 0 : X n (u) ≥ V (u + τ n )} is optimal in T , where inf ∅ = +∞. Since X 0 (u) = 0, it follows that σ(u) = inf{n ≥ 1 : X n (u) ≥ V (u + τ n )} and sup t∈T E(X t (u)) = E max{X 0 (u), V (u + τ 0 )} = V (u). From the inequality sup t∈T |E(X t (u 1 )) -E(X t (u 2 ))| ≥ | sup t∈T E(X t (u 1 )) -sup t∈T E(X t (u 2 ))
| and the properties of the functions g and r we deduce that V (•) is continuous (for more details see Appendix A). Set

y(s) = g -1 (V (s)) r(s) for s ∈ [m, U) (2)
and y(s) = 0 elsewhere. Define f u (v) = P (τ σ(u) > v). Note that f u (•) is continuous. Moreover, the assumptions on r(•), g(•) and the form of σ(u) Chow et al. (1971, pp. 114-115)). Note that f u (v) is differentiable with respect to v in each of the intervals (s i -u, s i+1 -u), i = j + 1, . . . , k, where u ∈ (s jm, s j+1m), j = 0, . . . , k.

ensure that f u (m) = 1. Hence, f u (v) = exp(-u+v u+m F (y(v ′ ))dv ′ ) for m ≤ v < U -u and 0 ≤ u ≤ U -m (see
The optimal mean reward V (u) may be expressed as follows:

V (u) = E(X σ(u) (u)) = E(E(g(Y σ(u) r(u + τ σ(u) )) | τ σ(u) , σ(u))). Note that for τ σ(u) ∈ [m, U -u) and u ∈ [0, U -m) the conditional distri- bution of g(Y σ(u) r(u + τ σ(u) )) given τ σ(u) , σ(u) is the same as the conditional distribution of g(Y σ(u) r(u + τ σ(u) )) given {Y ≥ y(u + τ σ(u)
)}, where Y, τ σ(u) are independent and Y has the distribution function F . Hence, using the formula for f u (v) we obtain for u ∈ (s jm, s j+1m), j = 0, . . . , k,

V (u) = U u+m ∞ y(v) g(xr(v))dF (x) exp - v u+m F (y(v ′ ))dv ′ dv.
(3)

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Equation ( 3) is differentiable with respect to u in each of the intervals (s jm, s j+1m), j = 0, . . . , k, hence for u ∈ (s jm, s j+1m) we have

d du V (u) = F (y(m + u))V (u) - ∞ y(m+u) g(xr(m + u))dF (x). ( 4
)
Remark 2.2 Equation (4) can be solved by recursion. Note that V (u) = 0 for u ∈ [Um, ∞). Hence, using the boundary condition V (Um) = 0 and the fact that V (u) is continuous, we obtain V (u) for u ∈ (U -2m, Um). In the same way, we can compute

V (u) for u ∈ (max{U -nm, 0}, U -nm + m), n = 3, 4, . . . . Hence, from continuity of V (u), we obtain V (u) for u ∈ [0, ∞).
The proof of the theorem below is based on Chow et al. (1971, pp. 115-118).

Theorem 2.1 (i) A continuous function Ṽ (•) satisfies (3) if and only if Ṽ (•) satisfies (4) and Ṽ (u) → 0 as u → U -m. (ii) If a continuous function Ṽ (•) satisfies (3) on [0, U -m) and Ṽ (•) = 0 elsewhere, then Ṽ (•) = V (•) and σ(u) = inf{n ≥ 1 : X n (u) ≥ Ṽ (u + τ n )}.
Proof. (i) The fact that Ṽ (•) satisfying (3) has to satisfy (4) has been shown above. Moreover, note that the right-hand side of ( 3) is non-negative, exp(-

v u+m F (ỹ(v ′ ))dv ′ ) ≤ 1 and 0 ≤ ∞ ỹ(v) g(xr(v))dF (x) ≤ E(g(Y 1 r(v))) = h(v)
, where ỹ(s) = g -1 ( Ṽ (s))/r(s) for s ∈ [m, U) and ỹ(s) = 0 elsewhere. Hence Ṽ (u) → 0 as u → Um. Now, suppose that a continuous function Ṽ (•) satisfies (4) and Ṽ (u 5) in each of the intervals (s jm, s j+1m), j = 0, . . . , k, we obtain

) → 0 as u → U -m. For u ∈ [0, U -m), define V 1 (•) by V 1 (u) = U u+m ∞ ỹ(v) g(xr(v))dF (x) exp - v u+m F (ỹ(v ′ ))dv ′ dv (5) and V 1 (u) = 0 for u ≥ U -m. Note that V 1 (u) is continuous for u ≥ 0 and V 1 (u) = Ṽ (u) = 0 for u ≥ U -m. We wish to show that V 1 (u) = Ṽ (u) for u ∈ [0, U -m]. Note that V 1 (u) → 0 as u → U -m. Moreover, differentiating (
d du V 1 (u) = F (ỹ(m + u))V 1 (u) - ∞ ỹ(m+u) g(xr(m + u))dF (x). ( 6
) Assume that V 1 (u 0 ) = Ṽ (u 0 ) at some point u 0 ∈ (s k -m, U -m). Let u 1 be the first point after u 0 such that u 1 ∈ (s k -m, U -m) and V 1 (u 1 ) = Ṽ (u 1
). If such a u 1 does not exist we set u 1 = Um. From ( 6) and the assumption that Ṽ (•) satisfies ( 4) we obtain

d(V 1 (u)-Ṽ (u))/du = F (ỹ(m+u))(V 1 (u)-Ṽ (u)). Hence,

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for u ∈ (u 0 , u 1 ) we have ln

|V 1 (u) -Ṽ (u)| = ln |V 1 (u 0 ) -Ṽ (u 0 )| + u u 0 F (ỹ(m + u ′ ))du ′ .
For u → u 1 the left-hand side of the above equation goes to -∞ while the right-hand side is finite. This contradiction showes that

V 1 (u) = Ṽ (u) for u ∈ (s k -m, U -m). From the continuity of the functions V 1 (u) and Ṽ (u) we obtain V 1 (u) = Ṽ (u) for u ∈ [s k -m, U -m]. Now by recursion we conclude that V 1 (u) = Ṽ (u) for u ≥ 0.
(ii) Let u ≥ 0 be fixed. Suppose that there exists a continuous function Ṽ (u+z) such that for u + z ∈ (s jm, s j+1m), z ≥ 0, j = 0, . . . , k,

Ṽ (u + z) = U u+z+m ∞ ỹ(v) g(xr(v))dF (x) exp - v u+z+m F (ỹ(v ′ ))dv ′ dv, (7)
where ỹ(s) = g -1 ( Ṽ (s))/r(s) for s ∈ [m, U) and ỹ(s) = 0 elsewhere, and

Ṽ (u + z) = 0 for u + z ≥ U. Define σ1 (u + z) = inf{n ≥ 1 : Y n ≥ ỹ(u + z + τ n )}.
Considerations similar to those above show that E(X σ1 (u+z) (u + z)) is equal to the right-hand side of ( 7). Let

H v = σ(N(v ′ ), v ′ ≤ v, Y 1 , . . . , Y N (v) ), where N(v), v ≥ 0, is a homogeneous Poisson process with intensity 1. Then H τn = F n for n ≥ 1, where H τn = {A ∈ F : A ∩ {τ n ≤ v} ∈ H v for all v ≥ 0}.
From the properties of the exponential distribution we see that the conditional joint distribution of (Y

N (v)+1 , τ N (v)+1 ), (Y N (v)+2 , τ N (v)+2 ), . . . given H v is the same as the unconditional joint distribution of (Y 1 , v + τ 1 ), (Y 2 , v + τ 2 ), . . . . Hence, P (X σ(u+z) (u + z) ≤ t, σ(u + z) > N(v) | H v ) = I(σ(u + z) > N(v))P (X σ(u+z+v) (u + z + v) ≤ t).
Therefore, for all u

+ z ∈ [0, U -m), v ∈ [0, U -u -z), z ≥ 0, we have I(σ(u + z) > N(v)) Ṽ (u + z + v) = I(σ(u + z) > N(v))E(X σ(u+z) (u + z) | H v ). Hence, Ṽ (u + z) = E(X σ(u+z) (u + z)) = E(E(X 1 (u + z)I(X 1 (u + z) ≥ Ṽ (u + z + τ 1 )) +X σ(u+z) (u + z)I(X 1 (u + z) < Ṽ (u + z + τ 1 )) | F 1 )) = E(max{X 1 (u + z), Ṽ (u + z + τ 1 )}). Define Γ(y, u + z) = max{g(yr(u + z)), Ṽ (u + z)} for y ∈ [0, ∞), u + z ∈ [0, U -m), z ≥ 0. Then E( Γ(Y 1 , u + z + τ 1 )) = Ṽ (u + z).
Hence, using the facts that the sequences {Y n } and {τ n } are independent and Y i are i.i.d. random variables, we see that

E( Γ(Y n+1 , u + τ n+1 ) | F n ) = Ṽ (u + τ n ). Defining ACCEPTED MANUSCRIPT γn (u) = Γ(Y n , u + τ n ) for n ≥ 1, we get E(γ n+1 (u) | F n ) = Ṽ (u + τ n ). Hence, γn (u) = max{X n (u), E(γ n+1 (u) | F n )}. (8) Let T n = {t ∈ T : t ≥ n}, γ n (u) = ess sup t∈Tn E(X t (u) | F n ) for n ≥ 1, T N n = {t ∈ T n : t ≤ N}, γ N n (u) = ess sup t∈T N n E(X t (u) | F n ) for n = 1, . . . , N, γ ′ n (u) = lim N →∞ γ N n (u) for n ≥ 1. From (8), we have γn (u) ≥ X n (u) and γ∞ (u) = 0, because lim n→∞ Ṽ (u + τ n ) = 0 and X ∞ (u) = 0. Define W (u) = ∞ n=1 X n (u). Then from Remark 2.1 we have E(W (u)) < ∞. Since γn (u) ≤ X n (u) + E( ∞ k=n+1 X k (u) | F n ) for n ≥ 1, we have γn (u) ≤ E(W (u)|F n ).
Therefore, the assumptions of Lemma 4.9 from Chow et al. (1971, p. 79)) are satisfied and γn (u) ≤ γ n (u). Moreover, from Theorem 4.6 of Chow et al. (1971, p. 76)

(u) = γ n (u) for n ≥ 1. Hence, γn (u) = γ n (u) and Ṽ (u + τ n ) = V (u + τ n ). Moreover, V (u) = max{X 0 (u), E(γ 1 (u) | F 0 )} = E(γ 1 (u)) = E(γ 1 (u)) = Ṽ (u) for u ≥ 0.

Solution of the optimal stopping time problem with random starting time

Note that if I(M ≤ τ n ) = 1, then I(M ≤ τ n+1 ) = 1. Moreover, if I(M ≤ τ n ) = 0, then I(M ≤ τ k ) = 0 for k = n -1, . . . , 0. Hence, {Z * n , F * n } ∞ n=0 , where Z * n = (Y n , τ n , I(M ≤ τ n ))
, is a homogeneous Markov chain and X * n is a function of Z * n only. Therefore, using the fact that the Y i are i.i.d. random variables we deduce that E(ess ) and as a consequence

sup t * ∈T * n+1 E(X * t * | F * n+1 ) | F * n ) is a function of τ n and I(M ≤ τ n ) only, say V * (τ n , I(M ≤ τ n )). Moreover, since X * 0 = 0, the Markov time σ * = inf{n ≥ 1 : X * n ≥ V * (τ n , I(M ≤ τ n ))} is optimal in T * , where inf ∅ = +∞. Note that V * (τ i , I(M ≤ τ i )) > 0 for τ i < U. (9) Hence, {I(M > v) = 1} ∩ {X i ≥ V * (τ i , I(M ≤ τ i ))} ∩ {τ i ≤ v} = ∅ for v ∈ [0, U). Therefore, P (M > v, τ σ * ≤ v) = 0 for v ∈ [0, U). Hence, P (M > v) = P (M > v, τ σ * > v) for v ∈ [0, U
P (τ σ * > v) = P (M > v) + P (τ σ * > v, M ≤ v).
Moreover, note that

P (τ σ * > v) is continuous for v ≥ 0. Define f * (v) = P (τ σ * > v, M ≤ v) for 0 ≤ v < U.

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Note that f * (v) is continuous in each of the intervals (s i , s i+1 ), i = 0, . . . , k. Let v ∈ (s i , s i+1 ) and h > 0 be such that (vh, v + h) ⊆ (s i , s i+1 ). Note that

f * (v + h) = P (τ σ * > v + h, M ≤ v) + P (τ σ * > v + h, M ∈ (v, v + h]).
It is easy to see that

exp(-h)(F M (v + h) -F M (v)) ≤ P (τ σ * > v + h, M ∈ (v, v + h]) ≤ F M (v + h) -F M (v).
To compute

P (τ σ * > v + h, M ≤ v), note that I(M ≤ τ k )V * (τ k , I(M ≤ τ k )) = I(M ≤ τ k )V (τ k ), k = 1, 2, . . . . ( 10
)
To prove Equation ( 10), we first use the approximation by finite horizon problem (finite number of events in stream (τ n )) and Theorem 3.2 of Chow et al. (1971, pp. 50 and 68) and the properties of I(M ≤ τ k ). Next, we use Theorem 4.3 of Chow et al. (1971, pp. 50 and 68) (for more details see Appendix B). Now, note that from (9) for τ n < U we have

X * n ≥ V * (τ n , I(M ≤ τ n )) iff I(M ≤ τ n ) = 1 and g(Y n r(τ n )) ≥ V * (τ n , I(M ≤ τ n )). Moreover, X * n < V * (τ n , I(M ≤ τ n )) iff I(M ≤ τ n ) = 0 or I(M ≤ τ n ) = 1 and g(Y n r(τ n )) < V * (τ n , I(M ≤ τ n )). Note that if N(v + h) -N(v) = 1, then τ N (v)+1 ∈ (v, v + h].
Hence, using ( 10) and ( 2) we get

exp(-h)f * (v) + h exp(-h)F ( inf x∈(v,v+h] y(x))f * (v) + o(h) ≤ P (τ σ * > v + h, M ≤ v) ≤ exp(-h)f * (v) + h exp(-h)F ( sup x∈(v,v+h] y(x))f * (v) + o(h).

Using the fact that exp(

-h) = 1 -h + o(h) and f * (v) ∈ [0, 1] we obtain lim h→0 + f * (v + h) -f * (v) h = lim h→0 + F M (v + h) -F M (v) h -f * (v) F (y(v)). Moreover, taking f * (v) = f * (v -h + h) we obtain lim h→0 + f * (v-h)-f * (v) -h = lim h→0 + F M (v-h)-F M (v) -h -f * (v) F (y(v)). Since F M (v) is differentiable in each of the intervals (s i , s i+1 ), i = 0, . . . , k, we have lim h→0 F M (v+h)-F M (v) h = F ′ M (v). Therefore, df * (v) dv = F ′ M (v) -f * (v) F (y(v)). (11)

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Moreover, for each interval (s i , s i+1 ), i = 0, . . . , k, there exists exactly one solution of (11) with boundary condition f * (s i ) = b i , i = 0, ..., k, respectively. Hence, writing

E v t = exp(-v t F (y(x))dx) we have f * (v) = v s i F ′ M (t)E v t dt + b i E v s i . Note that b 0 = f * (m) = P (M ≤ m).
The remaining conditions f * (s i ) = b i , i = 1, . . . , k, are obtained recursively from the continuity of the function

P (τ σ * > v). Let FM (t) = 1 -F M (t). Define w = max{n ≥ 0 : s n ≤ v}. Hence, for v ∈ (m, U) P (τ σ * > v) = FM (v) + v m F ′ M (t)E v t dt + F M (m)E v m + w j=1 P (M = s j )E v s j .
Note that the above function is differentiable in each of the intervals (s i , s i+1 ), i = 0, . . . , k. Since P (M ≤ τ σ * ) = 1, we have sup

t * ∈T * E(X * t * ) = E(X * σ * ) = E(E(g(Y σ * r(τ σ * )) | τ σ * , σ * )).
Moreover, from ( 9) and ( 10), we find that for

τ σ * ∈ [m, U) the conditional distribution of g(Y σ * r(τ σ * )) given τ σ * , σ * is the same as the conditional distri- bution of g(Y r(τ σ * )) given {Y ≥ y(τ σ * )}
, where Y, τ σ * are independent, Y has the distribution function F and y(u) is defined by (2). Hence sup

t * ∈T * E(X * t * ) = U m ∞ y(v) g(xr(v))dF (x) • v m F ′ M (t)E v t dt + F M (m)E v m + w j=1 P (M = s j )E v s j dv. (12)
Remark 2.3 In case r(m) = 0, the function y(s) is defined by (2) on (m, U) instead of [m, U), but this does not affect the solutions of the problems under consideration. Moreover, we can consider the case m > 0 and U infinite or the problems in which the set S ∩ (m, U) is countable and does not have accumulation points in (m, U) but in this case an effective solution also does not exist.

Proposition 2.1 Suppose that the assumptions of Section 2 are satisfied.

Then

sup t∈T E(X t (0)F M (τ t )) ≤ sup t * ∈T * E(X * t * ) ≤ sup t∈T E(X t (0)).
Proof. Since T ⊂ T * and X * n ≤ X n (0), it follows that sup t∈T E(X * t ) ≤ sup t * ∈T * E(X * t * ) ≤ sup t * ∈T * E(X t * (0)). Moreover, X * n is not measurable with respect to F n , hence for t ∈ T we have E(X * t ) = E(X t (0)F M (τ t )). Since X n (0) is a function of Z n,0 only, sup t * ∈T * E(X t * (0)) = sup t∈T E(X t (0)).

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Let us note that if P (M ≤ m) = 1, then in Proposition 2.1 we have equalities instead of inequalities. In Example 1, we will show that in general the inequalities may be strict.

Stopping time problem with random starting time and random horizon

Suppose that a random variable T , which can be interpreted as a random horizon, has known distribution function F T and is independent of M, Y 's and τ 's. Denote FT (s) = 1 -F T (s). Let T be the set of Markov times adapted to { Fn } ∞ n=0 , where Fn = σ(F * n , I(T > τ 0 ), . . . ,

I(T > τ n )), Tn = { t ∈ T : t ≥ n}, n = 0, 1, . . . . Note that E(sup X * n I(T > τ n )) < ∞.
The theorem below is a generalization of the theorems presented by [START_REF] Samuel-Cahn | Optimal stopping with random horizon with application to the full-information best-choice problem with random freeze[END_REF] and Ferenstein and Krasnosielska (2009b). The idea of the transformation of an optimal stopping time problem with random horizon to an optimal stopping time problem with a modified structure of rewards is based on the paper of [START_REF] Samuel-Cahn | Optimal stopping with random horizon with application to the full-information best-choice problem with random freeze[END_REF].

Theorem 2.2 Let Y 1 , Y 2 , ... be i.i.d. nonnegative random variables independent of the jump times τ 1 , τ 2 , ... of a Poisson process, Y 0 = 0, τ 0 = 0. Let M and T be independent positive random variables independent of the sequences of Y 's and τ 's, with distribution functions F M and F T , respectively. Let g : R + 0 → R + 0 and r : R + 0 → R + 0 be Borel functions,

X * n = g(Y n r(τ n ))I(M ≤ τ n ), E(sup n X * n ) < ∞ and X * ∞ = lim sup n→∞ X * n . Then, for any k ∈ N ∪ {0}, sup t∈ Tk E(X * t I(T > τ t)) = sup t * ∈T * k E(X * t * FT (τ t * )). Proof. Set ξ n = I(M ≤ τ n ), ζ n = I(T > τ n ), Ẑn = (Y n , τ n , ξ n , ζ n ), n = 0, 1, . . . . Note that if ζ n = 1, then ζ k = 1 for k = n -1, . . . , 0. Moreover, if ζ n = 0, then ζ n+1 = 0. Hence, we see that { Ẑn , Fn } ∞ n=0 is a homogeneous Markov chain with the state space E = supp Y 1 × R + 0 × {0, 1} × {0, 1}. Therefore, E(ess sup t∈ Tn+1 E(X * t ζ t | Fn+1 ) | Fn ) is a function of τ n , ξ n and ζ n , say V * (τ n , ξ n , ζ n ). Hence, the Markov time σ * k = inf{n ≥ k : X * n ζ n ≥ V * (τ n , ξ n , ζ n )} is optimal in Tk . Define ϕ(y, s, w, v) = g(yr(s))wv and B = {(y, s, w, v) ∈ E : ϕ(y, s, w, v) ≥ V * (s, w, v)}. Hence σ * k = inf{n ≥ k : (Y n , τ n , ξ n , ζ n ) ∈ B}. Define: B = E \ B, C = {(y, s, w) : (y, s, w, 1) ∈ B}, C = {(y, s, w) : (y, s, w, 1) ∈ B} and R = {(y, s, w, 1) ∈ E}. Note that B ∩ R = C × {1} and B ∩ R = C × {1}. Using the fact that I(ζ n = 1) = I(ζ n = 1) • . . . • I(ζ 0 = 1), we obtain X * σ * k = X * σ * k , where σ * k = inf{n ≥ k : (Y n , τ n , ξ n ) ∈ C} and σ * k ∈ T * k . Since I(T > τ n ) is not measurable with respect to F * n , we have E(X * t * I(T > τ t * )) = E(X * t * FT (τ t * )) for t * ∈ T * .
From the above theorem, we see that to find sup t∈ T E(X * t I(T > τ t)) it is enough to find sup t * ∈T * E(X * t * FT (τ t * )). Without loss of generality we can assume that inf{s : F T (s) = 1} ≥ U and all points of discontinuity of F T (•)

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belong to S. Moreover, we assume that there exist functions ĝ1 and ĝ2 such that |g(yr(x 1 )) FT (x 1 )-g(yr(

x 2 )) FT (x 2 )| ≤ |ĝ 1 (y)ĝ 2 (x 1 , x 2 )| for x 1 , x 2 ≥ 0 and y ∈ supp Y 1 , E|ĝ 1 (Y 1 )| < ∞, ĝ1 (0) = 0 and ∞ 0 |ĝ 2 (x 1 + t, x 2 + t)|dt < ĉ|x 1 -x 2 |
for some constant ĉ ≥ 0. Note that the above conditions are satisfied if E(Y 1 ) < ∞ and g(x) = x and r(x) is non-increasing and continuous for x ≥ 0. Denote Êv t = exp(-v t F (ŷ(x))dx), where ŷ(s) = g -1 ( V (s)/ FT (s))/r(s) for s ∈ (m, U) and ŷ(s) = 0 elsewhere, and V (u) is continuous on [0, ∞) and for u ∈ (s jm, s j+1m), j = 0, ..., k, satisfies P (M = s j ) Êv s j dv.

Game with known starting time and known horizon

In this section we generalize the problem considered in Subsection 2.1 to a multi-person game considered in Ferenstein and Krasnosielska (2009a). In this case m can be interpreted as the time up to which the players have to wait for a permit to run their businesses. In this section we need additionally the following assumptions U < ∞, r(x + y) ≤ r(x)r(y) for x, y ≥ m, g(xy) ≤ g(x)g(y). These assumptions are needed to show that relevant stopping times are finite (see [START_REF] Stadje | An optimal k-stopping problem for the Poisson process[END_REF]). Note that these assumptions are satisfied if g(s) = s p , p > 0, and r(s) = exp(-s). Moreover, assume that there are j > 1 ordered players who sequentially observe offers Y n at times τ n , n = 1, 2, . . . . Players' indices correspond to their ordering, called priorities, so that 1 refers to the player with the highest priority and j to the player with the lowest one.

Each player who has just decided to make a selection at τ n gets the reward X n (0) if and only if he has not obtained any reward before and there is no player with higher priority who has also decided to take the current reward. As soon as the player gets the reward, he quits the game. The remaining players select rewards in the same manner, their priorities remain as previously.

Let V j,l (u) be the optimal mean reward for the game with reward sequence {g(Y n r(u + τ n ))}, u ≥ 0, for player l in the j-person game, where l is the

  (m + u))dF (x) with lim u→U V (u) = 0. Note that V (u) = sup t∈T E(X t (u) FT (u)). Moreover, sup t * ∈T * E(X * t * FT (τ t * )) =

  where I(A) is the indicator function of the event A. Let T and T * be the sets of Markov times adapted to {F n } ∞ n=0 and {F * n } ∞ n=0 , respectively. Write R + 0 = R + ∪ {0}. Our aim is to find sup t * ∈T * EX * t

* , where

  , we conclude that {γ ′ n (u), F n } ∞ n=1 is the minimal supermartingale dominating {X n (u), F n } ∞n=1 and from Theorem 4.4 ofChow et al. (1971, p. 69)), we have γ ′ n

Acknowledgment

The author would like to express her special thanks to Professor Elżbieta Ferenstein for her insightful comments on this article, and the anonymous referee for his careful reading and valuable suggestions.

(Anna Krasnosielska1 This work has been supported by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

player's priority, l = 1, . . . , j, j = 1, 2, . . . . Note that V 1,1 (u) = V (u) for u ≥ 0. As in Ferenstein and Krasnosielska (2009a) we find V j,l (u). Denote y l (s) = g -1 (V l,l (s))/r(s) for s ∈ [m, U) and 0 elsewhere, l = 1, . . . , j. Theorem 3.1 (i) V j,l (•) is continuous on [0, ∞) for l = 1, . . . , j.

(ii) V j,l (u) = V l,l (u), for l = 1, . . . , j -1, u ≥ 0. (iii) For u ∈ (s im, s i+1m), i = 0, . . . , k, j > 1, we have d du V j,j (u) = F (y j (u + m))V j,j (u) -V j-1,j-1 (u + m) F (y j-1 (u + m))

-

The above differential equation can be solved numerically with the condition Vj,j (u) → 0 as u → U -m in accordance with the rule presented in Remark 2.2.

Taking u = 0 we obtain the optimal mean rewards for all players in the game with rewards of the form X n (0).

Examples

Let g(s) = s for s ≥ 0 and {Y i } be a sequence of i.i.d. random variables with exponential distribution with mean 1. Then F (y) = exp(-y).

Example 4.1 Let r(u) = 1 for u ∈ [0, 1/2), r(u) = 2/3 for u ∈ [1/2, 2) and r(u) = 0 elsewhere, P (M = 0) = 1/2, P (M = 1) = 1/2. Then m = 0 and U = 2 and lim u→2 y(u) = lim u→2 (V (u)/r(u)) = 0 and Equation ( 12) is equivalent to dy(u)/du =exp(-y(u)) in each of the intervals (0, 1/2), (1/2, 2). Hence, for u ∈ (1/2, 2) and u ∈ (0, 1/2) we obtain y(u) = ln(1+c i -u), c i ∈ R, i = 1, 2, respectively. Using condition y(2) = 0 we get c 1 = 2 and y(u

and using (4), we find that sup t∈T E(X t (0)F M (τ t )) ≈ 0.5856. Hence, in Proposition 2.1 the inequalities cannot be replaced by equalities.

Example 4.2 Let P (M = 0) = 1, P (T = 3) = 1 and r(s) = 1 for s ∈ [1, 3) and r(s) = 0 elsewhere. Hence, m = 1, U = 3 and (4) becomes

for u ∈ (0, 1). Solving this equation numerically with the boundary condition

Note that |V (u)| < ∞ for all u ≥ 0. Now, we will show that for all u 1 ≥ 0 and u 2 ≥ 0 we have:

Note that σ(u 1 ) and σ(u 2 ) are the optimal stopping times for sequences X n (u 1 ) and X n (u 2 ), respectively, that is sup t∈T E(X t (u i )) = E(X σ(u i ) (u i )) for i ∈ {1, 2}. Moreover, without loss of generality, we can assume that

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where the last inequality we obtain from Equation (A2) and definition of supremum. Hence, we conclude that Equation (A1) is satisfied for all nonnegative u 1 and u 2 . Now, we will prove that V (u) is continuous for u ≥ 0. Let u 1 ≥ 0 and u 2 ≥ 0, then

Hence, V (•) satisfies the Lipschitz condition, and consequently it is continuous.

Appendix B. Derivation of Equation ( 10)

Using the property of I(M ≤ τ k ) and Theorem 3.2 from Chow, Robbins and Siegmund, 1971, p. 50, we have And hence, using the fact that γ n is F n -measurable and independent of M, we have for n ≥ k

To complete the proof it is enough to take n = k.


A S Alexandrov

Angle-resolved photoemission spectroscopy of band tails and anomalous kinetics of underdoped cuprates

Keywords: numbers: 71.38.-k, 74.40.+k, 72.15.Jf, 74.72.-h, 74.25.Fy

The electron-phonon interaction in cuprates with c-axis polarised optical phonons, which is roughly one order of magnitude stronger than superexchange, bounds holes into mobile bipolarons. Bipolarons pin the chemical potential within the charge-transfer gap of doped Mott insulators, accounting for unusual kinetics and thermodynamics of doped cuprates such as the Nernst and giant proximity effects, pseudo-gaps, and normal-state diamagnetism. We propose that "quasi-particle" peaks, "Fermi-arcs", and high-energy "waterfalls" in the photoemission spectra of cuprates originate from the photo-ionization matrix elements of disorder-localised band-tails in the charge-transfer gap.

Bipolaron thermodynamics and kinetics of cuprates

A growing number of observations point to the possibility that high-T c cuprate superconductors [1] may not be conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, but rather derive from the Bose-Einstein condensation (BEC) of real-space pairs, which are mobile bipolarons [2][START_REF] Alexandrov | Theory of Superconductivity: From Weak to Strong Coupling[END_REF][START_REF] Edwards | [END_REF]. A possible fundamental origin of such strong departure of the cuprates from conventional BCS behaviour is the unscreened Fröhlich electron-phonon interaction (EPI) with the polaron level shift of the order of 1 eV [5], routinely neglected in the Hubbard U and t -J models. EPI with c-axis polarized optical phonons is virtually unscreened since the upper limit for the outof-plane plasmon frequency ( 200 cm -1 [6]) is well below the characteristic frequency of optical phonons, ω 0 ≈ 400 -1000 cm -1 . As a result of poor screening the magnetic exchange interaction remains small compared with the Fröhlich EPI at any doping of cuprates. Consequently, in order to generate an adequate theory of hightemperature superconductivity, the finite-range Coulomb repulsion and unscreened EPI should be treated on an equal footing. When both interactions are strong compared with the kinetic energy of carriers, this Coulomb-Fröhlich model predicts the ground state in the form of mobile inter-site bipolarons [5,7].

A parameter-free estimate of the Fermi energy, F , using the magnetic-field penetration depth found a very low value, F 100 meV [8]. The low value of the carrier kinetic energy clearly supports a real-space (i.e individual) pairing in cuprate superconductors. Indeed, there is strong experimental evidence for a gap, Δ, in the normalstate single-particle density of states (DOS) of cuprates [2], which is known as the pseudogap. Experimentally measured pseudogaps are roughly 30 meV or larger [9]. If one accepts [10] that the pseudogap is about half of the pair binding energy, 2Δ, then the condition for real-space pairing, F 2πΔ, is readily satisfied in many cuprates [8].

Also magnetotransport and thermal magnetotransport data strongly support preformed bosons in cuprates. In particular, many high magnetic-field studies revealed a non-BCS upward curvature of the upper critical field H c2 (T ) (see [11] for a review of experimental data), in accordance with the theoretical prediction for the Bose-Einstein condensation of charged bosons in the magnetic field [12]. The Lorenz number, L = e 2 κ e /T σ differs significantly from the Sommerfeld value L e = π 2 /3 of the standard Fermi-liquid theory, if carriers are doublecharged bosons [13], as observed in the direct measurements of the Lorenz number using the thermal Hall effect [14,15]. The large Nernst effect [16] and diamagnetism, uncovered by torque magnetometery [17], as well as the giant proximity effect [18] have been readily explained as the normal state signatures of charged preformed bosons, Refs. [19][20][21] respectively. Most compelling evidence for (bi)polaronic carries in novel superconductors is the discovery of a substantial isotope effect on the carrier mass [22] predicted for the (bi)polaronic conductors in Ref. [23]. High resolution angle-resolved photoemission spectroscopy (ARPES) [24] provides another piece of evidence for the strong EPI in cuprates apparently with c-axis-polarised optical phonons, as suggested by us [5].

ARPES of impurity-band tails

Even though ARPES is continually strengthening our insights into the band structure and correlations in cuprates, it has also revealed many poorly understood phenomena, such as the incoherent "background", the sharp "quasi-particle" peaks near some points of the Brillouin zone, which form "arcs" of "Fermi surface" ( [25] and references therein), widely studied low-energy diphump and kink features (for review see [24]) and recently discovered steep downturn of the dispersion toward higher energies (the so-called "waterfall") [26,27].

ARPES of undoped and lightly doped cuprates [24,25,28] proved to be particularly critical in the assess- scribed by the Fermi-Dirac golden rule as

I im (k, E) = 2πe 2 m 2 e n(E) i | ψ f |A 0 • ∇| ψ i | 2 δ(E+Δ-E i ).
(1) We define all energies relative to the chemical potential, μ, which is situated at energy Δ above the top of the valence band within the impurity band as shown in Fig. 2. Only the impurity states with the binding energy E i below μ = 0 contribute at zero temperature. Here A 0 is the amplitude of X-ray vector potential, n(E) = 1/[exp(E/T )+1] is the Fermi-Dirac distribution function, and = k B = 1. While the LDA+GTB band dispersions fully accounts for the short-range Coulomb correlations in cuprates, the size of shallow impurity states is much larger than the lattice constant , so that their envelope function is roughly insensitive to the short-range correlations. Hence we can take the impurity wavefunction as, ψ i (r) = F i (r)ψ g (r), and the final state to be the normalized plane wave, ψ f (r) = (N v) -1/2 exp(ik • r). Here ψ g (r) is the itinerant state at the top of the valence band, renormalised by correlations, and F i (r) is a slowly varying envelope function (N is the number of unit cells of volume v in the crystal).

In the framework of GTB [32] one can expand ψ g (r) using the Wannier orbitals, ψ g (r) = N dipole matrix element in Eq.( 1) as

I im (k, E) = In(E) i |f i (k -g)| 2 δ(E + Δ -E i ), (2)
where I = 2π(ed/m e ) 2 (A 0 • k) 2 /v is proportional to the valence band matrix element squared, which is roughly a constant in a wide range of k near g, d = drw(r) exp(ig • r), and f i (q) = (N v) -1 dr exp(iq • r)F i (r) is the Fourier transform of the impurity envelope function. Since the size of the envelope is large compared with the lattice constant, its Fourier transform strongly depends on q, which explains the experimental EDC and MDC, Fig. 1.

We choose the impurity state to be hydrogen-like, F i (r) = (N v/πξ 3 i ) 1/2 exp(-r/ξ i ) as the hydrogen model accurately predicts many properties of shallow levels in heavily doped semiconductors, so that f i (q) = 8π(ξ 3 i /πN v) 1/2 (1 + q 2 ξ 2 i ) -2 for 3D impurity states, and f i (q) ∝ (1 + q 2 ξ 2 i ) -3/2 for 2D states like localised surface states. It is important to recognisee here that ξ i is related to the impurity binding energy as ξ -2 i = mE i , where m is roughly the hole effective mass. As a result we get

I im (k, E) = xIn(E)M (k -g, E) with M (k -g, E) = 64π vm 3/2 (E + Δ) 5/2 [E + Δ + (k -g) 2 /m] 4 ρ im (E+Δ). (3) Here ρ im (E) = N -1 i i δ(E -E i )
is the band-tail density of states (DOS) normalized to unity, and x = N i /N is the impurity concentration per cell proportional to doping. In the 2D case the result is similar,

M 2D (k, E) ∝ (E + Δ) 2 [E + Δ + (k -g) 2 /m] -3 ρ im (E + Δ).
We notice that due to a very sharp dependence on q of the matrix element in Eq.( 2) any uncertainty of k z does not smear out the strong dependence of I im (k, E) on the in-plane momentum component, k . Since the chemical potential shifts towards the band edge with doping, Δ in Eq.(3) becomes smaller. Hence, the band-tail peak, I im (k, E), which is proportional to x, not only increases but also becomes sharper with doping as observed [25]. To provide more insight into the shape and momentum dependence of experimental EDC we approximate the band-tail DOS by the simple form, ρ im (E) = [n/Γ(p/n + 1/n)](E/γ) p exp(-E n /γ n ), where Γ(x) is the gamma-function with exponents n, p depending on the dimensionality and the correlation length of the disorder potential. We can separate impurity and band contributions by subtracting normalized ARPES intensity of the parent cuprate from the intensity of the doped one. Then, the band-tail ARPES, Eq.( 2), averaged over k z as described in Ref. [31], fits reasonably well the experimental relative intensities at all momenta around g with m = m e , n = 2, and p = 7/2, Fig. 1. It describes the substantial loss of intensity with changing the momentum by only a few percent relative to g, as well as the shape of the relative EDC.

The band-tailing also contributes to the waterfall effect [31]. There are impurity tails near local maxima of the LDA+GTB valence band at Γ point (0, 0) and at g 1 = (π/a, π/a), as shown in Fig. 2. Adding all three tail contributions,

I im (k, E) ∝ n(E)[M (k , E+E 2 )+M (k - g, E) + M (k -g 1 , E + E 2 )]
, where E 2 is roughly the valence band-width, describes qualitatively steep downturn of the dispersion toward higher energies and reappearence of the impurity peaks near (0, 0) and g 1 at high binding energies, as observed in a number of doped cuprates [26,27].

Our theory proposes that the ARPES intensity near (π/2a, π/2a) is proportional to the square of the Fourier component, f i (q), of the impurity wave-function envelope, Eq.( 2). Therefore, we can find the real-space image of the function, F i (r), by taking the Fourier transform of the square root of the experimental intensities, which reveals some band-mass anisotropy, and the size of the localised state of about 20 lattice constants justifying the "envelope" approximation used for the impurity wavefunction [31].

In summary, when holes are bound into bipolarons, the chemical potential remains within the in-gap bandtails at the bipolaron mobility edge in doped chargetransfer Mott insulators. Placing the chemical potential within the impurity band explains many unusual thermodynamic and kinetic properties of underdoped cuprates [START_REF] Alexandrov | Theory of Superconductivity: From Weak to Strong Coupling[END_REF]. In the framework of our bipolaronic scenario we propose an explanation for ARPES "quasi-particle" peaks, "Fermi-arcs", and high-energy "waterfalls" as a consequence of photo-ionization matrix-elements of disorderlocalised band-tails in the charge-transfer gap of doped cuprates [31].

We are grateful to ZX Shen and Teppei Yoshida for providing us with their raw ARPES data [25] and enlightening comments. This work was supported by EP-

  FIG. 1: Band-tail EDC (solid lines) with pseudogap Δ = 300 meV and band-tail width γ = 300 meV compared with the experimental relative EDC (symbols) near (π/2a, π/2a). Relative intensities are obtained by subtracting ARPES intensities of the parent compound, La2Cu04 (a), shifted by δμ, from EDC of slightly doped La1.97Sr0.03CuO4 (b) as measured by Yoshida et al. [25]. Both intensities have been normalized by their values at E = -800 meV and the chemical potential shift between two samples has been taken as δμ = 70 meV.

A c c e p t e d m a n u s c r i p t FIG. 2 :

 t2 FIG.2: LDA+GTB valence band dispersion[32] amended with band tails (ladder lines) near Γ, (π/2, π/2) and (π, π) maxima (here k is measured in 1/a)

A c c e p t e d m a n u s c r i p t

 t SRC (UK) (grant number EP/C518365/1).


J P Hague

Kinks and d-waves from phonons: The intermediate coupling story

Keywords: Extended Eliashberg Theory, Superconductivity, Spectroscopy, Unconventional Pairing

I present results from an approach that extends the Eliashberg theory by systematic expansion in the vertex function; an essential extension at large phonon frequencies, even for weak coupling. In order to deal with computationally expensive double sums over momenta, a dynamical cluster approximation (DCA) approach is used to incorporate momentum dependence into the Eliashberg equations. First, I consider the effects of introducing partial momentum dependence on the standard Eliashberg theory using a quasi-local approximation; which I use to demonstrate that it is essential to include corrections beyond the standard theory when investigating d-wave states. Using the extended theory with vertex corrections, I compute electron and phonon spectral functions. A kink in the electronic dispersion is found in the normal state along the major symmetry directions, similar to that found in photo-emission from cuprates. The phonon spectral function shows that for weak coupling W λ < ω0, the dispersion for phonons has weak momentum dependence, with consequences for the theory of optical phonon mediated d-wave superconductivity, which is shown to be 2nd order in λ. In particular, examination of the order parameter vs. filling shows that vertex corrections lead to d-wave superconductivity mediated via simple optical phonons. I map out the order parameters in detail, showing that there is significant induced anisotropy in the superconducting pairing in quasi-2D systems.

Introduction

Angle-resolved photo-emission spectroscopy (ARPES) directly probes the dispersion of electrons; identifying a kink associated with the active optical phonon in the cuprates [1]. Neutron scattering has shown an anomalous change in the phonon spectrum at the superconducting transition, indicating an interesting role for phonons in the cuprates [2]. Estimates of the magnitude of the electron-phonon coupling, and the energy of the phonon mode put the problem outside the limited region of applicability for BCS theory, so schemes to cope with a wider range of parameters need to be developed. Moreover, since there a node in the superconduct- * Tel: +44-1509-228208. Fax: +44-1509-223986

Email address: J.P.Hague@lboro.ac.uk (J.P.Hague). 1 I acknowledge support by EPSRC grant EP/C518365/1 ing gap consistent with d-wave symmetry [3], any theory implicating phonons as the mechanism must be able to explain the unconventional order.

Electron-phonon (e-ph) interactions can be described by the following generic model,

H = kσ ǫ k c † k,σ c k,σ + kq g q √ ω q c † k-q,σ c k,σ (b † q + b q ) + q ω q (b † q b q + 1/2) (1)
Here, = 1 and ǫ k = -2t(cos(ak x ) + cos(ak y )) -2t ⊥ cos(ck z ) with t = 0.25 and t ⊥ = 0.01 (representing a quasi-2D system and taming the van-Hove singularities). The Holstein model is a special case where g q = g 0 and ω q = ω 0 , for which λ = g 2 0 /(ω 2 0 W ) is the dimensionless e-ph coupling (W is the half-band width). The momentumindependent phonon dispersion approximates an A partial density of states (DOS) is associated with each sub-zone. DCA enables computations in the thermodynamic limit, via an equivalent cluster impurity problem. Some treatments of the ARPES kink used a standard Eliashberg theory with anisotropic coupling and an assumed d-wave order parameter for fitting purposes [4,5]. The extended Eliashberg approach that I have developed differs significantly from previous work in the sense that it is fully self-consistent, with no prior assumptions about the form of the order parameter (i.e. any d-wave order arises directly from the self-consistency of the theory) [6,7].

Eliashberg theory and momentum: Failure for optical-phonon mediated d-wave states

In this section, I discuss how quasi-local Eliashberg equations can be constructed, and demonstrate how the lowest-order Fock diagram does not contribute to the gap equations. The calculation is intended to be demonstrative, and shows that standard Eliashberg equations fail for an optical phonon mediated d-wave state.

The Eliashberg equations are computed from the Fock contribution to the self-energy:

Σ(k, iω n ) = -T ω n ′ dq (2π) 3 g 2 q τ3 Ĝ(k -q, ω n ′ ) ×τ 3 K(q, iω n -iω n ′ ) (2)
and equate Σ with the approximate form,

Σ ≈ (1 -Z k )iω n τ0 + ∆ k (iω n )τ 1 + χτ 3 (3)
Normally the Eliashberg equations are computed using a local (momentum-independent) approximation. However, to examine d-wave states, the local approximation is not sufficient. Substituting the Green function Ĝ(k,

iω n ) -1 = iω n τ0 -ζ k τ3 -Σ in eq. 2 (ζ k = ǫ k -µ,
∆ n,K = λT n ′ Q dζ K(n, n ′ )D K-Q (ζ)∆ n ′ ,K-Q Z 2 K-Q ω 2 n ′ + ζ 2 + ∆ 2 n ′ ,K-Q (4)
When gap functions have d-wave symmetry, ∆ π,0 = -∆ 0,π and ∆ 0,0 = ∆ π,π = 0, and the partial DOS D(0, π) = D(π, 0) has the symmetry of the lattice, it is immediately clear that the O(λ) contribution to the quasi-local Eliashberg equations vanishes, and 2nd order terms must be considered. When there is some weak momentum dependence in the e-ph coupling and phonon frequency, this cancellation will not be exact, but the contribution from 2ndorder terms can still be expected to be the largest in the perturbation series. Thus a fully extended set of Eliashberg equations including vertex corrections needs to be constructed. Currently, it is not possible to construct a similar approximation to eq. 4 for the vertex corrected theory. However a numerical approach using the full DCA equations is available to enlighten some aspects of the e-ph problem. The rest of this article concerns results from an extended Eliashberg theory with momentum dependence and vertex corrections determined numerically using DCA. No form for the order parameter is assumed in advance and full forms for the Green functions and self-energies are used. Full details of the translation of the diagrams in Fig. 1(a) may be found in ref. [6].

Normal state spectroscopy

Figure 2 shows image plots of the electron spectral functions [8]. Vertex corrections are most pronounced for ω 0 > W λ, as expected from analysis of the parameter space [9]. In particular, the vertex corrected theory shows a kink at the energy scale relating to the phonon frequency measured at the (π, 0) point (i.e. renormalised zone edge phonon frequency) in agreement with the ARPES measurements in Ref. [1]. The effects of vertex corrections are most pronounced for ω 0 > W λ, as expected from analysis of the parameter space [9]. Horizontal dashed lines indicate the renormalised phonon frequency at the (π, 0) zone-edge as computed within the self-consistent theory (i.e. the one which would be measured in a neutron experiment). In particular, only the vertex corrected theory shows a kink at the energy scale relating to the phonon frequency measured at the (π, 0) point [8] as reported in Ref. [1].

Phonon spectral functions are shown in Fig. 3. The phonon renormalisation is extremely small for ω 0 = 0.4W . In contrast, modes at the (π, π) point in the Brillouin zone are strongly softened for ω 0 = 0.2W when vertex corrections are not taken into account. The second order terms act against the softening and reinforce the theory. A small reduction in spectral weight is seen mid way along the Γ-X and Γ-W lines when ω 0 > W λ. This might be a precursor to the strong softening seen in Ref. [2] on passing through the transition temperature, but further analysis of the superconducting state is necessary to be sure.

Evolution of the order parameter

d-wave superconductivity is one of the remarkable results from the theory extended with vertex corrections. Figure 4 shows the anomalous density as the chemical potential is varied first upwards from half-filling and then back from large filling showing that two solutions are stable. A small µ c = 1.2 is applied. λ = 0.6 and ω 0 /W = 0.4. Starting with a half-filled solution and increasing, then decreasing the chemical potential, first d-wave and then (on decreasing from high electron density) s-wave solutions to the self-consistent equations are found (as detailed in the caption of Fig. 4). Highly anisotropic solutions form as the Fermi-surface approaches the van-Hove points. In the d-wave state, the first order terms are not the leading order of the perturbation theory in λ. It was argued in Ref. [7] that this was in part due to the nearly flat phonon spectrum, so that the phonon propagator D(k, ω) ≈ D(ω) and the 1st order contribution to the anomalous self-energy W λ q D(q)F (kq) is small (F is the anomalous Green function). The anomalous contribution to the pseudopotential term is φ C = W µ C k F (k). Since F is modulated in the d-wave state, then the contribution of the pseudopotential term is zero. In the s-wave state, F is not modulated, and the resulting finite contribution destroys the s-wave order for large enough µ C , thus selecting a dominant d-wave contribution. The phonon spectra in Fig. 3 demonstrate that so long as ω 0 > W λ, then this condition is met, and 2nd order terms are essential. 2nd order terms remain the leading order correction to the anomalous self-energy, even for the larger λ used here. It is clear that the 2nd order terms are similar for both repulsive and attractive models, and thus it is perhaps not surprising that both attractive and repulsive models show a d-wave state.

Summary and Outlook

I have described spectroscopic results and order parameters from an extended Eliashberg-style theory with vertex and momentum corrections. The extended theory is essential for the investigation of d-wave superconductivity mediated by optical phonons, since the vertex corrections are the lowest order terms in the perturbation expansion in electron-phonon coupling λ. Indeed, d-wave superconductivity is immediately predicted with such an extension.

There are two remaining areas which require further investigation. The first involves a better treatment of the Coulomb repulsion, which is currently only treated as a simple Hartree-Fock pseudopotential. In particular, the ability to treat large repulsion and antiferromagnetism is essential for high-T C materials. Treatment of momentum dependent e-ph coupling would also be of interest. However, the overriding challenge at this stage is to convert c,d). There is an indication of reduction of spectral weight half way between the high symmetry points when ω 0 > W λ, perhaps with similar origin to that in the neutron results [2] Variation of chemical potential , there is only one solution with d-wave order. This persists to n = 1.175 where the order parameter spontaneously changes to s-wave order (b,c). The reduction in d-wave magnitude is reminiscent of a quantum critical transition. When decreasing µ from dense filling, the order parameter is a conventional s-wave (d), which persists below n = 1.175, i.e. there are 2 solutions (e). Below n = 1.05 there is only one solution. Close to n = 1.05, the s-wave solution has an extended-s character, as can be seen in panel (f).

the numerical results into a coherent analytic theory with vertex corrections.

A c c e p t e d m a n u s c r i p tFig. 1 .

 1 Fig. 1. (a) Overview of the diagrammatic representations of the self-energy used in this work. Σ is the electron, and Π the phonon self energy. The row labeled ME contains the vertex neglected and VC the corrected diagrams. (b) Schematic of the DCA (quasi-local) formalism for cluster size N C = 4. Within the squares Σ(k) → Σ(K) where K are represented by solid circles. Equivalently, ∆(k) → ∆(K).A partial density of states (DOS) is associated with each sub-zone. DCA enables computations in the thermodynamic limit, via an equivalent cluster impurity problem.

  τ are Pauli matrices and µ is the chemical potential), and treating the gap function as constant in sub-zones of k-space in the manner of DCA, one obtains a quasi-local approximation for the Eliashberg equations,

Fig. 2 .

 2 Fig.2. (Colour online) Electron spectral functions comparing VC and ME approximations when cluster size N C = 64. The effects of vertex corrections are most pronounced for ω 0 > W λ, as expected from analysis of the parameter space[9]. Horizontal dashed lines indicate the renormalised phonon frequency at the (π, 0) zone-edge as computed within the self-consistent theory (i.e. the one which would be measured in a neutron experiment). In particular, only the vertex corrected theory shows a kink at the energy scale relating to the phonon frequency measured at the (π, 0) point[8] as reported in Ref.[1].

Fig. 3 .

 3 Fig. 3. (Colour online) Phonon spectral functions. The phonon renormalisation is moderate for ω 0 = 0.4W , N C = 64. White points show the values computed from ω 2 -ω 2 0 (1 -Re[Π(ω)]) = 0 for clarification. (a) Modes at the W point in the Brillouin zone are almost completely softened in the theory without vertex corrections (a split mode is indicated in panel (b)). Vertex corrections act against that softening and reinforce the theory(c,d). There is an indication of reduction of spectral weight half way between the high symmetry points when ω 0 > W λ, perhaps with similar origin to that in the neutron results[2]

Fig. 4 .

 4 Fig.4. Anomalous density as µ is varied first upwards from half-filling and then back from large filling showing two types of solutions to the self-consistent equations. µ C = 1.2, λ = 0.6, N C = 4 and ω 0 /W = 0.4. (a) Around half-filling (n = 1), there is only one solution with d-wave order. This persists to n = 1.175 where the order parameter spontaneously changes to s-wave order (b,c). The reduction in d-wave magnitude is reminiscent of a quantum critical transition. When decreasing µ from dense filling, the order parameter is a conventional s-wave (d), which persists below n = 1.175, i.e. there are 2 solutions (e). Below n = 1.05 there is only one solution. Close to n = 1.05, the s-wave solution has an extended-s character, as can be seen in panel (f).