1. LATEX TikZposter
Capacity Expansion with Possible Default
Hessah Al Motairi, Jos´e Pasos, Mihail Zervos
j.e.pasos@lse.ac.uk
Capacity Expansion with Possible Default
Hessah Al Motairi, Jos´e Pasos, Mihail Zervos
j.e.pasos@lse.ac.uk
1. Economic motivation
Given an uncertain demand for a product, when and how much to
increase production capacity?
• Reducing capacity is not allowed.
• We face the risk of demand vanishing (default).
time
demand
2. Problem formulation
Given the linear diffusion
dXt = b(Xt) dt + σ(Xt) dWt, X0 = x > 0,
maximise
Jx,y(ξ) = Ex
T0
0
e−Λt
[h(Xt, Yt) dt − K(Yt) ◦ dξt] + e−ΛT0g(YT0
)1{T0<∞} ,
where
Yt = y + ξt, and Λt =
t
0
r(Xt) dt,
over all increasing, c`agl`ad and adapted processes {ξt}t≥0 satisfying the admissibility condition
Ex
T0
0
e−Λt
K(Yt) ◦ dξt < ∞.
3. Intuitive argument
Y
X
capacity
demand
G
W
I
4. Outline of our method
i
Characterise “investment” I and “waiting” W regions in
terms of an HJB equation.
ii
Describe the “free-boundary” G as the solution of an in-
tegral equation.
iii Construct a smooth solution w to the HJB equation.
iv
Prove that w coincides with the value function of the op-
timisation problem.
v Show that the intuitive “wait-and-invest” strategy is opti-
mal.
5. HJB equations
max {Lw(x, y) + h(x, y), wy(x, y) − K(y)} = 0, (x, y) ∈ S,
w(0, y) = g(y), y ∈ [0, ¯y] ∩ R.
6. Free-boundary
For each y ∈ (0, y∞) there exists a unique G(y) > 0 such that
G(y)
0
Ψ(s) [hy(s, y) − r(s)K(y)] ds = K(y) − g (y),
and G : (0, y∞) → (0, ∞) is C1
and strictly increasing.
7. Solution to the HJB
w(x, y) =



B(y)ϕ(x) + Rh(·,y)
(x, y), if (x, y) ∈ W ∩ (R+ × [y∞, ¯y]) ,
A(y)ψ(x) + B(y)ϕ(x) + Rh(·,y)
(x, y), if (x, y) ∈ W ∩ (R+ × [y0, y∞[) ,
w x, G−1
(x) −
G−1
(x)
y K(z) dz, if (x, y) ∈ I,
8. Optimal strategy
ξt = G−1
(X∗
t ) − y 1{X∗
t ≥G(y),t>0}, Yt = y + ξt = y ∨ G−1
(X∗
t ).
9. Basic example
When {Xt}t≥0 is a Brownian motion absorbed at zero, and
h(x, y) = eαx
yβ
, K(y) = K > 0,
r(x) = r > 0, g(y) = g y,
the equation for G in the variable z = eG
is given by
0 =
β
λr + α
z2λr+α
−
rK
λr
y1−β
z2λr
+
2rK
λr
− (K − g) y1−β
−
2βλr
λ2
r − α2
zλr
+
β
λr − α
zα
−
rK
λr
y1−β
, where λr =
√
2r.
r = 2, α = 1.5, β = 0.25,
g = 8, K = 10, 20, 50.
y
5 100
x
27
54
Capacity is low for current
demand: invest.
Capacity is high for current
demand: wait.
Smooth boundary separating
the two regions.
W expands as the invest-
ing cost increases.