Erick Trevino-Aguilar PDF
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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]
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