Optimality

Semiparametric transformation models 167
The first pair of equations for Ψ0 and Ψ2 in part (i) follows by setting g1(s, t) =
1 = g3(s, t). With s fixed, the equations
�
¯h1(s, t)− ¯h1(s, u+)b(du)c1((u, t)) = ¯g1(s, t),
have solutions
�
¯h3(s, t)−
[s,t)
(s,t]
�
¯h1(s, t) = ¯g1(s, t) +
�
¯h3(s, t) = ¯g3(s, t) +
¯h3(s, u−)c(du)b3([u, t]) = ¯g3(s, t),
[s,t)
(s,t]
¯g1(s, u+)b(du)Ψ1(u, t−),
¯g3(s, u−)c(du)Ψ3(u, t+).
The second pair of equations for Ψ0 and Ψ2 in part (i) follows by setting ¯g1(s, t)≡
1 ≡ ¯g3(s, t). Next, the “odd” functions can be represented in terms of “even”
functions using Fubini.
9. Gronwall’s inequalities
Following Gill and Johansen [18], recall that if b is a cadlag function of bounded
variation,�b�v≤ r1 then the associated product integralP(s, t) =π(s,t] (1+b(du))
satisfies the bound|P(s, t)|≤π(s,t] (1 +�b�v(dw))≤exp�b�v(s, t] uniformly in
0 < s < t≤τ. Moreover, the functions s→P(s, t), s≤t≤τ and t→P(s, t), t∈
(s, τ] are of bounded variation with variation norm bounded by r1er1 .
The proofs use the following consequence of Gronwall’s inequalities in Beesack
[3] and Gill and Johansen [18]. If b is a nonnegative measure and y∈ D([0, τ]) is a
nonnegative function then for any x∈D([0, τ]) satisfying
�
0≤x(t)≤y(t) + x(u−)b(du), t∈[0, τ],
we have
�
0≤x(t)≤y(t) +
Pointwise in t,|x(t)| is bounded by
max{�y�∞,�y − �
�∞}[1 +
(0,t]
(0,t]
(0,t]
y(u−)b(du)P(u, t), t∈[0, τ].
b(du)P(u, t)]≤{�y�∞,�y − � t
�∞} exp[ b(du)].
0
We also have�e −b |x|�∞≤ max{�y�∞,�y − �∞}. Further, if 0�≡ y∈ D([0, τ]) and b
is a function of bounded variation then the solution to the linear Volterra equation
x(t) = y(t) +
is unique and given by
�
x(t) = y(t) +
(0,t]
� t
0
x(u−)b(du)
y(u−)b(du)P(u, t).