E. RANDOM WALKS AND THE WIENER-HOPF FACTORIZATION
E.
193
Random Walks and the Wiener-Hopf Factorization
Let {Xi } be a sequence of iid. random variables. Let U (x) be the distribution
function of Xi . Define
n
X
Sn =
Xi
i=1
and S0 = 0.
Lemma E.1.
i) If IIE[X1 ] < 0 then Sn converges to −∞ a.s..
ii) If IIE[X1 ] = 0 then a.s.
lim Sn = − lim Sn = ∞ .
n→∞
n→∞
iii) If IIE[X1 ] > 0 then Sn converges to ∞ a.s. and there is a strictly positive probability that Sn ≥ 0 for all n ∈ IIN.
Proof.
See for instance [40, p.396] or [66, p.233].
For the rest of this section we assume −∞ < IIE[Xi ] < 0. Let τ+ = inf{n >
0 : Sn > 0} and τ− = inf{n > 0 : Sn ≤ 0}, H(x) = IIP[τ+ < ∞, Sτ+ ≤ x] and
ρ(x) = IIP[Sτ− ≤ x]. Note that τ− is defined a.s. because Sn → −∞ as n → ∞, see
Lemma E.1. Define ψ0 (x) = 1I{x≥0} and
ψn (x) = IIP[S1 > 0, S2 > 0, . . . , Sn > 0, Sn ≤ x]
for n ≥ 1. Let
ψ(x) =
∞
X
ψn (x) .
n=0
Lemma E.2. We have for n ≥ 1
ψn (x) = IIP[Sn > Sj , (0 ≤ j ≤ n − 1), Sn ≤ x]
and therefore
ψ(x) =
∞
X
H ∗n (x) .
n=0
−1
Moreover, ψ(∞) = (1 − H(∞)) , IIE[τ− ] = ψ(∞) and IIE[Sτ− ] = IIE[τ− ]IIE[Xi ].
194
E. RANDOM WALKS AND THE WIENER-HOPF FACTORIZATION
Proof.
Let for n fixed
Sk∗
n
X
= Sn − Sn−k =
Xi .
i=n−k+1
Then {Sk∗ : k ≤ n} follows the same law as {Sk : k ≤ n}. Thus
IIP[Sn > Sj , (0 ≤ j ≤ n − 1), Sn ≤ x] = IIP[Sn∗ > Sj∗ , (0 ≤ j ≤ n − 1), Sn∗ ≤ x]
= IIP[Sn > Sn − Sn−j , (0 ≤ j ≤ n − 1), Sn ≤ x]
= IIP[Sj > 0, (1 ≤ j ≤ n), Sn ≤ x] = ψn (x) .
Denote by τn+1 := inf{k > 0 : Sk > Sτn } the n + 1-st ascending ladder time. Note
that Sτn has distribution H ∗n (x). We found that ψn (x) is the probability that Sn
is a maximum of the random walk and lies in the interval (0, x]. Thus ψn (x) is the
probability that there is a ladder height at n and Sn ≤ x. We obtain
∞
X
ψn (x) =
n=1
=
∞
X
n=1
∞
X
IIP[∃k : τk = n, 0 < Sn ≤ x] =
∞ X
∞
X
IIP[τk = n, 0 < Sτk ≤ x]
n=1 k=1
IIP[0 < Sτk ≤ x, τk < ∞] =
∞
X
H ∗k (x) .
k=1
k=1
Because IIE[Xi ] < 0 we have H(∞) < 1 (Lemma E.1) and H ∗n (∞) = (H(∞))n from
which ψ(∞) = (1 − H(∞))−1 follows.
For the expected value of τ−
IIE[τ− ] =
∞
X
IIP[τ− > n] =
n=0
∞
X
ψn (∞) = ψ(∞) .
n=0
Finally let τ− (n) denote the n-th descending ladder time and let Ln be then n-th
descending ladder height Sτ− (n−1) − Sτ− (n) . Then
−Sτ− (n) τ− (n)
L1 + · · · + Ln
=
n
τ− (n)
n
from which the last assertion follows by the strong law of large numbers.
Let now
ρn (x) = IIP[S1 > 0, S2 > 0, . . . , Sn−1 > 0, Sn ≤ x] .
Note that for x ≤ 0
ρ(x) =
∞
X
n=1
ρn (x) ,
E. RANDOM WALKS AND THE WIENER-HOPF FACTORIZATION
195
that ψ1 (x) = U (x) − U (0) if x > 0 and ρ1 (x) = U (x) if x ≤ 0. For n ≥ 1 one obtains
for x > 0
Z ∞
(U (x − y) − U (−y))ψn (dy)
ψn+1 (x) =
0
and for x ≤ 0
Z
∞
U (x − y)ψn (dy) .
ρn+1 (x) =
0
If we sum over all n we obtain for x > 0
Z ∞
ψ(x) − 1 =
(U (x − y) − U (−y))ψ(dy)
0−
and for x ≤ 0
∞
Z
U (x − y)ψ(dy) .
ρ(x) =
0−
Note that ρ(0) = ψ(0) = 1 and thus for x ≥ 0
Z ∞
U (x − y)ψ(dy) .
ψ(x) =
0−
The last two equations can be written as a single equation
ψ + ρ = ψ0 + ψ ∗ U .
Then
ψ − ψ0 + ρ ∗ H = ψ ∗ H + ρ ∗ H = H + ψ ∗ H ∗ U = H + ψ ∗ U − U
= H − U + ψ + ρ − ψ0
from which it follows that
U = H + ρ − H ∗ ρ.
The latter is called the Wiener-Hopf factorization.
E.1. Bibliographical Remarks
The results of this section are taken from [40].
(E.1)