The name is absent

We are now equipped to show the differentiability of the value function with the help of the
ʌ

theorem by [1]. Define l(Xc) as the optimal strategy if the threshold is Xc. Now define

W(X)  =  k [^^(Xo) + B [X0 - X]] -X²

'--------------------------_v--------------------------'

î

Thus, all perturbations in X around X0 are immediately offset in the first period by an adjustment
in the loading equal to B [X₀ - X]. The advantage of W is that the payoff in future periods is
independent of X and only depends on X₀, as by definition c(X, l) = BX + l + b = BX +
l(X0) + B[Xo - X] + b = BXo + l(Xo) + b.

Note that W (X) is defined on a neighborhood around X0. Clearly, W (X) is concave, dif-
ferentiable at Xo, and W(X) ≤ V (X) as W(X) uses just one feasible strategy ^ll out of the set of
possible strategies whose maximum yields V (X). Furthermore, by construction W(X_o) = V (X_o).
Using the above result that V (X) is concave, as well as the theorem of [1], it follows that V (X) is
differentiable at X₀ as well and V'(X₀) = W'(X₀) = -Bk — 2X₀                           ∣

Corollary: The critical level Xc influences the value function only as an additive constant α
Given that V^'(X) = -Bk - 2X, we also know that the value function is given by V(X) =
α — BkX — X²                                                        I.

Proof of Proposition 3 The optimal combined loading is given by

k [1 ^{- b}∖ ^- r [^{1 - Ф} X~-—c) 1 ^- T~^φ (~—c) [B^kr ⁺ 2X^cr ^{+ r}2 ^{+ σ}22 ^{- σ}UιI
2 δ                          σv          2σv       σv                                ^{2      1}

Proof: Maximizing the right hand side of the Bellman equation by setting the derivative with
respect to cl equal to zero we get

( ——cλ f v(χc + u)fι(u)du + δ f c   f v'(_δ + v + u)fι(u)dug(v)dv

^σv    -∞                  -∞   -∞

c—^c---) V V(X_c + r + u)f2(u)du + δ [ V V'(c + r + v + u)f2(u)dug(v)dv

^{σ σ}v — ∞ — ∞                       X--C — -∞

+δ /       /   V'(c + v + u)fι(u)du g(v)dv + δ /      /   V'(c + r + v + u)f2(u)du g(v)dv

- -∞ - — ∞                            Jx c—c - -∞

29

<<   27   28   29   30   31   32   33   >>

More intriguing information