Erick Trevino-Aguilar

21 Pages·2016·0.27 MB·English

Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Semimartingale properties of the lower Snell
envelope in optimal stopping under model
uncertainty
Erick Trevino-Aguilar1
1UniversidaddeGuanajuato,M´exico.
Second Actuarial Science and Quantitative Finance. Cartagena
Colombia, Junio 2016
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Programa
1 Motivation
2 A few known results
3 Semimartingale properties
4 A (counter)-example
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Arbitrage prices in incomplete markets
Let X be a price process of a financial market and let P be its
class of martingale measures.
Theorem
Let H be the payoff process of an American option. Then, the set
of arbitrage free prices is an interval with boundaries
π (H) := inf supE [H ] and π (H) := sup supE [H ].
inf P τ sup P τ
P∈Pτ∈T P∈Pτ∈T
See El Karoui and Quenez [2], Karatzas and Kou [6], Kramkov
[7]...
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Optimal exercise under model uncertainty
1 In decision theory, Ellsberg’s paradox [3], highlights how the
ambiguity about the distribution are crucial in understanding
human decisions under risk and uncertainty.
The solution to the paradox is given by the so-called maxmin
preferences axiomatized by Gilboa and Schmeidler [5].
2 The axiomatic framework of Gilboa and Schmeidler [5] yields
for each preference a family of probability measures under
which utilities are quantified and the worst possible outcome
is the utility assigned and under which decisions are taken.
3 In the setting of [5], time consistency in an intertemporal
framework is axiomatized by Epstein and Schneider [4].
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Assumptions
Definition
Let τ ∈ T be a stopping time and Q and Q be probability
1 2
measures equivalent to P. The probability measure defined through
Q3(A) := EQ1[Q2[A | Fτ]],A ∈ FT
is called the pasting of Q and Q in τ.
1 2
Assumption
The family Q of equivalent probability measures is stable under
pasting.
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Assumptions
Assumption
The process H is a c`adl`ag positive F-adapted process which is of
class(D) with respect to each Q ∈ Q, i.e.,
lim supE [H ;H ≥ x] = 0.
Q τ τ
x→∞τ∈T
The stochastic process H is upper semicontinuous in expectation
from the left with respect to each probability measure Q ∈ Q.
That is, for any stopping time θ of the filtration F and an
increasing sequence of stopping times {θ } converging to θ, we
i i∈N
have
limsupE [H ] ≤ E [H ]. (1)
Q θi Q θ
i→∞
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Optimal stopping times
Theorem (Trevino [8])
Define
τQ := inf{s ≥ ρ | H ≥ UQ}. (2)
ρ s s
Then, the random time
(cid:110) (cid:111)
τ↓ := ess inf τQ | Q ∈ Q , (3)
ρ ρ
is a stopping time and it is optimal:
(cid:104) (cid:105)
ess sup ess inf E [H | F ] = ess inf E H | F .
τ∈T[ρ,T] Q∈Q Q τ ρ Q∈Q Q τρ↓ ρ
(4)
The lower Snell envelope is a Q-submartingale in stochastic
intervals of the form [ρ,τ↓].
ρ
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
The lower Snell envelope
Theorem (Trevino [9])
Under mild conditions, there exists an optional right-continuous
stochastic process U↓ := {Ut↓}0≤t≤T such that for any stopping
time τ ∈ T
U↓ = ess inf ess sup E [H | F ], P−a.s.
τ Q∈Q ρ∈T[τ,T] Q ρ τ
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
In a recent paper, Cheng and Riedel [1] investigate the robust
stopping problem
U↓ = inf supE [H ],
τ Q ρ
Q∈Qρ∈T
under g-expectations with backward differential stochastic
equations techniques.
Their solution consists in stopping as soon as the underlying
process touches its lower Snell envelope. Moreover, they
obtain a structural result which describes the lower Snell
envelope as the sum of a process of bounded variation and a
stochastic integral with respect to Brownian motion.
SemimartingalepropertiesofthelowerSnellenvelope [email protected]
Motivation
Afewknownresults
Semimartingaleproperties
A(counter)-example
References
Assumption
There exists a probability measure Q ∈ Q such that H is of the
form
H = H +SQ +LQ −NQ,
t 0 t t t
for SQ a Q-submartingale and LQ,NQ c`adl`ag non decreasing
processes with SQ = LQ = NQ = 0, and E [NQ] < ∞.
0 0 0 Q T
SemimartingalepropertiesofthelowerSnellenvelope [email protected]