An optimal stopping problem related to
cash-flows of investments under uncertainty
Boualem Djehiche
KTH, Stockholm
Joint work with S. Hamadene and M-A. Morlais
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Position of the problem
Let Y 1 and Y 2 denote the expected profit and cost yields
respectively. The constituants of the cash flows are:
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The profit yield per unit time dt is ψ 1 and the cost yield is ψ 2 ;
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When exiting/abandoning the project at time t, the incurred
cost is a(t) and the incurred profit is b(t) (usually a 6= b but
often non-negative).
Exit/abandonment strategy:
The decision to exit the project at time t, depends on whether
Yt1 ≤ Yt2 − a(t) or Yt2 ≥ Yt1 + b(t).
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
A Snell envelop formulation
If Ft denotes the history of the project up to time t, the expected
profit yield, at time t, is
Z τ
1
1
1
2
1
Yt = ess supτ ≥t E
ψ (s, Ys )ds + Yτ − a(τ ) 1[τ <T ] + ξ 1[τ =T ] |Ft
t
where, the sup is taken over all exit times τ from the project.
The optimal exit time related to the incurred cost Y 2 − a should be
τt∗ = inf{s ≥ t, Ys1 = Ys2 − a(s)} ∧ T .
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
The expected cost yield at time t, is
Z σ
2
2
1
2
2
Yt = ess infσ≥t E
ψ (s, Ys )ds + Yσ + b(σ) 1[σ<T ] + ξ 1[σ=T ] |Ft
t
where, the inf is taken over all exit times σ from the project.
The optimal exit time related to the incurred profit Y 1 + b should
be
σt∗ = inf{s ≥ t, Ys2 = Ys1 + b(s)} ∧ T .
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Problem formulation
Establish existence and uniqueness of (Y 1 , Y 2 ) which solves the
coupled system of Snell envelops
Yt1 = ess supτ ≥t E
Yt2 = ess infσ≥t E
R τ
t
R σ
t
ψ 1 (s, Ys1 )ds + Yτ2 − a(τ ) 1[τ <T ] + ξ 1 1[τ =T ] |Ft
ψ 2 (s, Ys2 )ds + Yσ1 + b(σ) 1[σ<T ] + ξ 2 1[σ=T ] |Ft
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Related problems
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One-sided obstacles: The switching problem;
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Fully two-sided obstacles: The switching games problem;
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The multiple-phases membrane problem.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Set up
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B := (Bt )0≤t≤T a Brownian motion on a probability space
(Ω, F, P).
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(Ft )0≤t≤T the completed natural filtration of B.
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X := (Xt )0≤t≤T a diffusion process which stands for factors
which determine the price of the underlying commodity we
wish to control such as e.g. the price of electricity in the
energy market.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
The Snell envelop versus reflected BSDEs
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S 2 denotes the set of all right-continuous with left limits
processes Y satisfying E supt∈[0,T ] |Yt2 | < ∞.
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Md,2 denotes the set of F-adapted and
d-dimensional
RT
2
processes Z such that E 0 |Zs | ds < ∞.
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A+ denotes the set of right-continuous with left limits and
increasing processes K .
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A+,2 the subset of A+ consisting of all the processes K
satisfying, in addition, E (KT2 ) < ∞.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Let ξ ∈ L2 (FT , P), f (t, ω, y , z) be uniformly Lipschitz in (y , z) and
is such that f (t.ω, 0, 0) ∈ M1,2 , and S := (St )t≤T an R-valued,
continuous and uniformly square integrable s.t. ST ≤ ξ. Assume
Ft -adaptation. Then
Theorem (El-Karoui et al., ’97) There exists a unique triple
(Yt , Zt , Kt )t≤T , valued in R 1+d+1 and Ft -adapted (K continuous
and increasing) such that
(
RT
RT
Yt = ξ + t f (s, Ys , Zs )ds + KT − Kt − t Zs dBs , t ≤ T ;
RT
Yt ≥ St and 0 (Yt − St )dKt = 0.
In addition, Y satisfies
Z τ
Yt = ess supτ ≥t E [
f (s, ω, Ys , Zs )ds + Sτ 1[τ <T ] + ξ1[τ =T ] |Ft ].
t
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
The Markovian framework: Connection with systems of
PDEs
Let (t, x) ∈ [0, T ] × R k and let (Xst,x )s≤T be the solution of the
following standard SDE.

Z s
Z s
 t,x
Xs = x +
b(u, Xut,x )du +
σ(u, Xut,x )dBu , s ∈ [t, T ]
t
t
 t,x
Xs = x, if s ≤ t.
Assume
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f (s, ω, y , z) = f (s, Xst,x (ω), y , z)
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ξ = g (XTt,x )
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Ss = h(s, Xst,x ).
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Then, again by a result in El-Karoui et al., (’97), there exists a
continuous deterministic function v (t, x) such that, for any
s ∈ [t, T ], Ys = v (s, Xst,x ). Moreover v is the unique viscosity
solution of
min{v − h, −Gv − f (t, x, v , σ(t, x)Dx v )} = 0;
v (T , x) = g (x),
where,
G = ∂t + L,
and L is the infinitesimal generator of X t,x .
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Cash-flows: A system of reflected BSDEs formulation
By El-Karoui et al. ’97, (Y 1 , Y 2 ) should solve the following system
of RBSDEs:

RT
RT

Yt1 = ξ 1 + t ψ 1 (s, Ys1 )ds + (KT1 − Kt1 ) − t Zs1 dBs ,






RT
RT


 Yt2 = ξ 2 + t ψ 2 (s, Ys2 )ds − (KT2 − Kt2 ) − t Zs2 dBs ,



Yt1 ≤ Yt2 − a(t), Yt2 ≥ Yt1 + b(t), 0 ≤ t ≤ T ,






RT
 RT
Yt1 − (Yt2 − a(t)) dKt1 = 0, 0 (Yt1 + b(t) − Yt2 )dKt2 = 0.
0
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Minimal and maximal solutions
We make the following assumptions:
(B1) For each i = 1, 2, the process ψ i depends explicitly on (t, Yti ).
Moreover, (t, y ) → ψ i (t, y )’s are Lipschitz continuous with
respect to y and satisfy,
Z T
i
2
|ψ (t, 0)| ds < ∞.
E
0
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
(B2) The obstacles a and b are continuous and in S 2 . Moreover,
they admit a semimartingale decomposition:
Z t
Z t
a(t) = a(0) +
Us1 ds +
Vs1 dBs ,
0
Z
0
t
b(t) = b(0) +
0
Us2 ds +
Z
t
Vs2 dBs ,
0
for some F-prog. meas. processes U 1 , V 1 , U 2 and V 2 .
(B3) ξ i ’s are in L2 (FT ) and satisfy
ξ 1 − ξ 2 ≥ max{−a(T ), −b(T )},
Boualem Djehiche KTH, Stockholm
P − a.s.
An optimal stopping problem related to cash-flows of investmen
The main result
Let the coefficients (ψ 1 , ψ 2 , a, b, ξ 1 , ξ 2 ) satisfy Assumptions
(B1)-(B3). Then the system of RBSDEs admits a minimal and a
maximal F-prog. meas. solutions (Y 1 , Y 2 , Z 1 , Z 2 , K 1 , K 2 ) and
(Ȳ 1 , Ȳ 2 , Z̄ 1 , Z̄ 2 , K̄ 1 , K̄ 2 ), respectively, which are in
(S 2 )2 × (Md,2 )2 × (A+,2 )2 .
Moreover,
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the processes Y i and Ȳ i , i = 1, 2 are P−a.s. continuous and
admit the above Snell representations.
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the random times τ ∗ and σ ∗ defined above and associated
with either Y i or Ȳ i , are optimal stopping times.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
A minimal solution through the increasing sequences
scheme
Start with the pair (Y 1,0 , Z 1,0 ) that solves uniquely the BSDE
Z T
Z T
1,0
1
1
1,0
Yt = ξ +
ψ (s, Ys )ds −
Zs1,0 dBs .
t
t
and introduce the following system of RBSDEs

dYs2,n+1 = ψ 2 (s, Ys2,n+1 )ds − dKs2,n+1 − Zs2,n+1 dBs ,







1,n+1


= ψ 1 (s, Ys1,n+1 )ds + dKs1,n+1 − Zs1,n+1 dBs ,
 dYs




Ys2,n+1 ≥ Ys1,n + b(s), Ys1,n+1 ≤ Ys2,n+1 − a(s), 0 ≤ s ≤ T ,




R T 1,n+1



− (Yt2,n+1 − a(t))dKt1,n+1 = 0, Yt1,n+1 = ξ 1 ;

0 (Yt





 R T 1,n
+ b(t) − Yt2,n+1 )dKt2,n+1 = 0, Yt2,n+1 = ξ 2 .
0 (Yt
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
This sequence of solutions satisfies the following properties:
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For any n ≥ 0, both (Y 1,n , Z 1,n , K 1,n ) and
(Y 2,n+1 , Z 2,n+1 , K 2,n+1 ) exist and are in S 2 × Md,2 × A+,2 .
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The two sequences (Y 1,n )n≥0 and (Y 2,n )n≥1 are increasing in
n, meaning that for all n ≥ 0,
Yt1,n ≤ Yt1,n+1 and Yt2,n+1 ≤ Yt2,n+2 P-a.s. and for all t.
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the limit process (Y 1 , Y 2 ) of (Yt1,n , Yt2,n ) is continuous, a
minimal solution of our system of RBSDEs and admits a Snell
envelop representation.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
A maximal solution through the decreasing sequences
scheme
Start with the pair (Ȳ 2,0 , Z̄ 2,0 ) that solves the standard BSDE
Z T
Z T
2,0
2
2,0
2
ψ (s, Ȳs )ds −
Ȳt = ξ +
Z̄s2,0 dBs ,
t
t
and introduce the following system of RBSDEs

d Ȳs1,n+1 = ψ 1 (s, Ȳs1,n+1 )ds + d K̄s1,n+1 − Z̄s1,n+1 dBs ,







2,n+1


= ψ 2 (s, Ȳs2,n+1 )ds − d K̄s2,n+1 − Z̄s2,n+1 dBs ,
 d Ȳt




Ȳs1,n+1 ≤ Ȳs2,n − a(s), Ȳs2,n+1 ≥ Ȳs1,n+1 + b(s), 0 ≤ s ≤ T ,




R T 1,n+1



− (Ȳt2,n − a(t))d K̄t1,n+1 = 0, ȲT1,n+1 = ξ 1 ,

0 (Ȳt





 R T 1,n+1
+ b(t) − Ȳt2,n+1 )d K̄t2,n+1 = 0, ȲT2,n+1 = ξ 2 .
0 (Ȳt
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
This sequence of solutions satisfies the following properties.
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For any n ≥ 0, both (Ȳ 2,n , Z̄ 2,n , K̄ 2,n ) and
(Ȳ 1,n+1 , Z̄ 1,n+1 , K̄ 1,n+1 ) exist and are in S 2 × Md,2 × A+,2 .
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The two sequences (Ȳ 1,n )n≥1 and (Y 2,n )n≥0 are decreasing in
n, meaning that for all n ≥ 0,
Ȳt1,n ≥ Ȳt1,n+1 and Ȳt2,n+1 ≥ Ȳt2,n+2 P-a.s. and for all t.
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the limit process (Ȳ 1 , Ȳ 2 ) of (Ȳt1,n , Ȳt2,n ) is continuous, a
maximal solution of our system of RBSDEs and admits a Snell
envelop representation.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Non-uniqueness: A counter example
Assume
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ψ 1 (t, ω, y ) = y and ψ 2 (t, ω, y ) = 2y ,
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a = b = 0 and ξ 1 = ξ 2 = 1.
The corresponding system of BSDEs is

RT
RT

Yt1 = 1 + t Ys1 ds − t Zs1 dBs + KT1 − Kt1 ,


RT
RT

Yt2 = 1 + 2 t Ys2 ds − t Zs2 dBs − KT2 − Kt2 ,

Y 1 ≥ Yt2 , t ≤ T ,

 R tT

Ys1 − Ys2 d(Ks1 + Ks2 ) = 0.
0
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
It can be ckecked that
e T −t , e T −t , 0, 0, 0, e T (1 − e −t )
and
e
2(T −t)
,e
2(T −t)
1 2T
−2t
, 0, 0, e (1 − e ), 0)
2
are solutions of the system of BSDEs.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
A uniqueness result
Theorem. Assume that
(i) the mappings ψ 1 and ψ 2 do not depend on y , i.e.,
ψi := (ψi (t, ω)), i = 1, 2,
(ii) the barriers a and b satisfy
Z
T
P − a.s.
1[a(s)=b(s)] ds = 0.
0
Then, the solution of the system of BSDE’s is unique.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
The Markovian framework. A PDE formulation
When the dependence of (Y 1 , Y 2 ) on the sources of uncertainty
(the diffusion process X t,x ) is explicit, we can show that there
exists two deterministic functions u 1 and u 2 such that
Ys1 = u 1 (s, Xst,x ),
Ys2 = u 2 (s, Xst,x ),
and are viscosity solutions of the following system of variational
inequalities:

 min{u 1 (t, x) − u 2 (t, x) + a(t), −Gu 1 (t, x) − ψ 1 (t, x, u 1 (t, x))} = 0,
max{u 1 (t, x) + b(t) − u 2 (t, x), Gu 2 (t, x) + ψ 2 (t, x, u 2 (t, x))} = 0,
 1
u (T , x) = g 1 (x), u 2 (T , x) = g 2 (x).
Through a counter-example, we can show that the system may
have infinitely many solutions.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen
Some references
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BD, S. Hamadène and M-E. Morlais (2009): Optimal
stopping of expected profit and cost yields in an investment
under uncertainty (Preprint).
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BD, S. Hamadène, A. Popier (2009): A Finite Horizon
Optimal Multiple Switching Problem (SIAM JCO).
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T. Arnarsson, BD, M. Poghosyan, H. Shahgholian (2009): A
PDE approach to regularity of solutions to finite horizon
optimal switching problems. Nonlinear Analysis: Theory,
Methods & Applications.
Boualem Djehiche KTH, Stockholm
An optimal stopping problem related to cash-flows of investmen