• Search in book...
• Toggle Font Controls
176 PROOFS AND DERIVATIONS
Recall that by deﬁnition, b
(ω,ρ)
(X
1
1j
, s) is a log-optimal portfolio with respect
to the probability measure P
(ω,ρ)
j,s
. Let b
ω,ρ
(s) denote a log-optimal portfolio with
respect to the limit distribution P
(ω,ρ)
s
. Then, using Lemma B.2, we infer from
Equation B.2 that, as j tends to inﬁnity, we have the almost sure convergence
lim
j→∞
b
(ω,ρ)
(X
1
1j
, s), x
0
=b
ω,ρ
(s), x
0
,
for P
(ω,ρ)
s
(almost all x
0
) and hence for P
X
0
(almost all x
0
). Since s was arbitrary,
we obtain
lim
j→∞
b
(ω,ρ)
(X
1
1j
, X
1
&
), x
0
=b
ω,ρ
(X
1
ω
), x
0
almost surely, (B.3)
Next, we apply Lemma B.1 for the function
f
i
(x
−∞
) = logh
(ω,ρ)
(x
1
1i
), x
0
=logb
(ω,ρ)
(x
1
1i
, x
1
&
), x
0
deﬁned on x
−∞
= (...,x
1
, x
0
, x
1
). Note that
|f
i
(X
−∞
)|=|logh
(ω,ρ)
(X
1
1i
), x
0
|
d
j=1
|log X
(j)
0
|,
which has ﬁnite expectation, and
f
i
(X
−∞
) →b
ω,ρ
(X
1
ω
), X
0
almost surely as i →∞
by Equation B.3. As n →∞, Lemma B.1 yields
W
n
(
(ω,ρ)
) =
1
n
n
i=1
logh
(ω,ρ)
(X
i1
1
), X
i
=
1
n
n
i=1
f
i
(T
i
X
−∞
)
E{log b
ω,ρ
(X
1
ω
), X
0
}
def
= θ
ω,ρ
almost surely.
Therefore, by Equation B.1, we have
lim inf
n→∞
W
n
(B) sup
ω,ρ
θ
ω,ρ
sup
ω
lim inf
ρ
θ
ω,ρ
almost surely,
and it sufﬁces to show that the right-hand side is at least W
.
T&F Cat #K23731 — K23731_A002 — page 176 — 9/28/2015 — 20:47
PROOF OF CORN 177
(iii) To this end, ﬁrst, deﬁne, for Borel sets A, B R
d
+
,
m
A
(z) = P{X
0
A|X
1
ω
= z}
and
μ
ω
(B) = P{X
1
ω
B}.
Then, for any s support(μ
ω
), and for all A,
P
(ω,ρ)
s
(A) = P{X
0
A|E{(X
1
ω
s)
2
}≤2Var(s)(1 ρ)}
=
P{X
0
A,E{(X
1
ω
s)
2
}≤2Var(s)(1ρ)}
P{E{(X
1
ω
s)
2
}≤2Var(s)(1ρ)}
=
1
μ
ω
(S
s,2Var(s)(1ρ)
)
S
s,2Var(s)(1ρ)
m
A
(z)μ
ω
(dz)
m
A
(s) = P{X
0
A|X
1
ω
= s}
as ρ 1 and for μ
ω
, almost all s by Lebesgue density theorem (see Lemma B.4), and
therefore
P
(ω,ρ)
X
1
ω
(A) P{X
0
A|X
1
ω
}
as ρ 1 for all A. Thus, using Lemma B.2 again, we have
lim inf
ρ
θ
ω,ρ
= lim
ρ
θ
ω,ρ
= lim
ρ
E
log b
ω,ρ
(X
1
ω
), X
0
= E{log
1
b
ω
(X
1
ω
), X
0
2
}
(where b
ω
(·) is the log-optimum portfolio with respect
to the conditional probability P{X
0
A|X
1
ω
})
= E
max
b(·)
E{log
1
b(X
1
ω
), X
0
2
|X
1
ω
}
= E
E{log
1
b
ω
(X
1
ω
), X
0
2
|X
1
ω
}
def
= θ
ω
.
Next, to ﬁnish the proof, we appeal to the submartingale convergence theorem.
First, note that the sequence
Y
ω
def
= E{logb
ω
(X
1
ω
), X
0
|X
1
ω
}=max
b(·)
E{logb(X
1
ω
), X
0
|X
1
ω
}
of random variables forms a submartingale, that is, E{Y
ω+1
|Y
1
ω
Y
ω
}. To see this,
note that
E{Y
ω+1
|X
1
ω
}=E{E{logb
ω+1
(X
1
ω1
), X
0
|X
1
ω1
}|X
1
ω
}
E{E{logb
ω
(X
1
ω
), X
0
|X
1
ω1
}|X
1
ω
}
= E{logb
ω
(X
1
ω
), X
0
|X
1
ω1
}
= Y
ω
.
T&F Cat #K23731 — K23731_A002 — page 177 — 9/28/2015 — 20:47
178 PROOFS AND DERIVATIONS
This sequence is bounded by
max
b(·)
E{logb(X
1
−∞
), X
0
|X
1
−∞
},
which has a ﬁnite expectation. The submartingale convergence theorem (see Stout
1974) implies that a submartingale is convergence almost surely, and sup
ω
θ
ω
is ﬁnite.
In particular, by the submartingale property, θ
ω
is a bounded increasing sequence,
so that
sup
ω
θ
ω
= lim
ω→∞
θ
ω
.
Applying Lemma B.3 with the σ-algebras
σ(X
1
ω
) σ(X
1
−∞
)
yields
sup
ω
θ
ω
= lim
ω→∞
E
"
max
b(·)
E{logb(X
1
ω
), X
0
|X
1
ω
}
#
= E
"
max
b(·)
E{logb(X
1
−∞
), X
0
|X
1
−∞
}
#
= W
.
Then
lim inf
n→∞
W
n
(B) sup
ω,ρ
θ
ω,ρ
sup
ω
lim inf
ρ
θ
ω,ρ
= sup
ω
θ
ω
= W
almost surely,
and from the above three parts of proof, we can get that
lim
n→∞
1
n
log S
n
(B) = W
almost surely
and the proof of Theorem B.1 is ﬁnished.
B.2 Derivations of PAMR
B.2.1 Proof of Proposition 9.1
Proof First, if
t
= 0, then b
t
satisﬁes the constraint and is clearly the optimal
solution.
To solve the problem in case of
t
= 0, we deﬁne the Lagrangian for the
optimization problem (9.2) as
L(b, τ, λ) =
1
2
b b
t
2
+τ(x
t
·b ) +λ(b·1 1), (B.4)
where τ 0 is a Lagrange multiplier related to the loss function, λ is a Lagrange
multiplier associated with the simplex constraint, and 1 denotes a column vector of
m 1s. Note that the nonnegativity of portfolio b is not considered, since introducing
T&F Cat #K23731 — K23731_A002 — page 178 — 9/28/2015 — 20:47
DERIVATIONS OF PAMR 179
this term causes too much complexity, and alternatively we project the ﬁnal portfolio
into a simplex to enforce the constraint.
Setting the partial derivatives of L with respect to b to zero gives
0 =
L
b
= (b b
t
) +τx
t
+λ1.
Multiplying both sides by 1
, we can get λ =−τ
x
t
·1
m
. Moreover, since ¯x
t
=
x
t
·1
m
,
where ¯x
t
is the mean of t-th price relatives, or the market return, we can rewrite λ as
λ =−τ ¯x
t
. (B.5)
And the solution for L becomes
b = b
t
τ(x
t
−¯x
t
1). (B.6)
Plugging Equation B.5 and Equation B.6 to Equation B.4, we get
L(τ) =
1
2
τ
2
x
t
−¯x
t
1
2
τ
2
x
t
·(x
t
−¯x
t
1) +τ(b
t
·x
t
)
=−
1
2
τ
2
x
t
−¯x
t
1
2
+τ(b
t
·x
t
).
Note that in the above formula, we used the following formula:
x
t
−¯x
t
1
2
= x
t
·x
t
2 ¯x
t
(x
t
·1) x
2
t
(1 ·1) = x
t
·x
t
−¯x
t
(x
t
·1) = x
t
·(x
t
−¯x
t
1).
Setting the derivative of L(τ) with respect to τ to 0,weget
0 =
L
τ
=−τx
t
−¯x
t
1
2
+b
t
·x
t
.
Then τ can be set as
τ =
b
t
·x
t
x
t
−¯x
t
1
2
.
Since τ 0, we project τ to [0, ); thus,
τ = max
"
0,
b
t
·x
t
x
t
−¯x
t
1
2
#
=
t
x
t
−¯x
t
1
2
.
Note that in case of zero market volatility, that is, x
t
−¯x
t
1
2
= 0, we just set τ = 0.
We can summarize the update scheme for the case of
t
= 0 and the case of
t
> 0 by
setting τ. Thus, we simplify the notation following Equation 9.1 and show the uniﬁed
update scheme.
T&F Cat #K23731 — K23731_A002 — page 179 — 9/28/2015 — 20:47
180 PROOFS AND DERIVATIONS
B.2.2 Proof of Proposition 9.2
Proof We derive the solution of PAMR-1 following the same procedure as the
derivation of PAMR. If the loss is nonzero, we get a Lagrangian
L(b, ξ, τ, μ, λ) =
1
2
b b
t
2
+τ(x
t
·b ) +ξ(C τ μ) +λ(1 ·b1).
Setting the partial derivatives of L with respect to b to zero gives
0 =
L
b
= (b b
t
) +τx
t
+λ1,
Multiplying both sides by 1
, we can get λ =−τ
x
t
·1
m
=−τ
¯
x
t
. And the solution is
b = b
t
τ(x
t
−¯x
t
1).
Next, note that the minimum of the term ξ(C τ μ) with respect to ξ is zero when-
ever C τ μ = 0.IfC τ μ = 0, then the minimum can be made to approach
−∞. Since we need to maximize the dual, we can rule out the latter case and pose
the following constraint on the dual variables, C τ μ = 0. The KKT conditions
conﬁne μ to be nonnegative, so we conclude that τ C. We can project τ to the
interval [0,C] and get
τ = max
"
0, min
"
C,
b
t
·x
t
x
t
−¯x
t
1
2
##
= min
"
C,
t
x
t
−¯x
t
1
2
#
.
Again, we simplify the notation according to Equation 9.1 and show a uniﬁed update
scheme.
B.2.3 Proof of Proposition 9.3
Proof We derive the solution similar to the derivations of PAMR and PAMR-1. In
case that the loss is not 0, we can get the Lagrangian,
L(b, ξ, τ, μ, λ) =
1
2
b b
t
2
+τ(b ·x
t
) +Cξ
2
τξ +λ(1 ·b1).
Setting the partial derivatives of L with respect to b to zero gives
0 =
L
b
= (b b
t
) +τx
t
+λ1,
Multiplying both sides by 1
, we can get λ =−τ
x
t
·1
m
=−τ ¯x. And the solution is
b = b
t
τ(x
t
−¯x1).
Setting the partial derivatives of L with respect to ξ to zero gives
0 =
L
ξ
= 2Cξ τ =⇒ ξ =
τ
2C
.
T&F Cat #K23731 — K23731_A002 — page 180 — 9/28/2015 — 20:47
• No Comment
..................Content has been hidden....................