Motivation A few known results Semimartingale properties A (counter)-example References Semimartingale properties of the lower Snell envelope in optimal stopping under model uncertainty Erick Trevino-Aguilar1 1Universidad de Guanajuato, M´exico. Second Actuarial Science and Quantitative Finance. Cartagena Colombia, Junio 2016 Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Programa 1 Motivation 2 A few known results 3 Semimartingale properties 4 A (counter)-example Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties 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 πinf (H) := inf sup EP [Hτ ] and πsup(H) := sup sup EP [Hτ ]: P2P τ2T P2P τ2T See El Karoui and Quenez [2], Karatzas and Kou [6], Kramkov [7]... Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties 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]. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Assumptions Definition Let τ 2 T be a stopping time and Q1 and Q2 be probability measures equivalent to P. The probability measure defined through Q3(A) := EQ1 [Q2[A j Fτ ]]; A 2 FT is called the pasting of Q1 and Q2 in τ. Assumption The family Q of equivalent probability measures is stable under pasting. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Assumptions Assumption The process H is a c`adl`agpositive F-adapted process which is of class(D) with respect to each Q 2 Q, i.e., lim sup EQ [Hτ ;Hτ ≥ x] = 0: x!1 τ2T The stochastic process H is upper semicontinuous in expectation from the left with respect to each probability measure Q 2 Q. That is, for any stopping time θ of the filtration F and an increasing sequence of stopping times fθi gi2N converging to θ, we have lim sup EQ [Hθi ] ≤ EQ [Hθ]: (1) i!1 Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Optimal stopping times Theorem (Trevino [8]) Define Q Q τρ := inffs ≥ ρ j Hs ≥ Us g: (2) Then, the random time # n Q o τρ := ess inf τρ j Q 2 Q ; (3) is a stopping time and it is optimal: h i ess sup ess infQ2QEQ [Hτ j Fρ] = ess infQ2QEQ H # j Fρ : τ2T [ρ,T] τρ (4) The lower Snell envelope is a Q-submartingale in stochastic # intervals of the form [ρ, τρ ]. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References The lower Snell envelope Theorem (Trevino [9]) Under mild conditions, there exists an optional right-continuous # # stochastic process U := fUt g0≤t≤T such that for any stopping time τ 2 T # Uτ = ess infQ2Qess supρ2T [τ;T]EQ [Hρ j Fτ ]; P − a:s: Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References In a recent paper, Cheng and Riedel [1] investigate the robust stopping problem # Uτ = inf sup EQ [Hρ]; Q2Q ρ2T 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. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Assumption There exists a probability measure Q 2 Q such that H is of the form Q Q Q Ht = H0 + St + Lt − Nt ; for SQ a Q-submartingale and LQ ; NQ c`adl`agnon decreasing processes with SQ = LQ = NQ = 0, and E [NQ ] < 1. 0 0 0 Q T Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References The lower Snell envelope for semimartingales Theorem (Trevino [10]) Suppose the Assumptions 2.2 and 3.1 holds true. Define Q # Q V := U + N . Let τ1; τ2 be two stopping times with 0 ≤ τ1 ≤ τ2 ≤ T. Then Q Q EQ [Vτ1 ] ≤ EQ [Vτ2 ]: Thus, V Q is a Q-submartingale. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References sketch of the proof For δ > 0 we set (1) θδ := (τ1 + δ) ^ τ2 (2) # θδ := τ (1) ^ τ2: θδ Now for i > 2 we define recursively (i) (i−1) θδ := (θδ + δ) ^ τ2; for i odd; (i+1) # θδ := τ (i) ^ τ2; for i + 1 even: θδ T For N > δ we have N 1 X Q Q X Q Q EQ V − V = EQ V − V : θ(i+1) θ(i) θ(i+1) θ(i) i=1 δ δ i=1 δ δ Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References sketch of the proof, cont. For i + 1 even we have Q Q EQ V (i+1) − V (i) ≥ 0; (5) θδ θδ due to the Q-submartingale property of Theorem 3. For i + 2 odd we show that V Q − V Q ≥ SQ − SQ + LQ − LQ : (i+2) (i+1) (τ # +δ)^τ τ # ^τ (τ # +δ)^τ τ # ^τ θδ θδ (i) 2 (i) 2 (i) 2 (i) 2 θ θ θ θ δ δ δ δ (6) Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Assumptions There is a \uniform-version" of Theorem 5 under the stronger condition of the next Assumption There exists a Q-submartingale fSt g0≤t≤T with S0 = 0 such that H is of the form Ht = H0 + St + Lt − Nt ; for L; N c`adl`agnon decreasing processes with L0 = N0 = 0, and EQ [NT] < 1; for each Q 2 Q. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Uniform decomposition of the lower Snell envelope Theorem Assume H satisfies the Assumptions 2.2 and 3.2. Then V := U# + N is a Q-submartingale. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Let B be a Brownian motion defined in the probability space (Ω; F; P) in this section we assume that F is the augmented filtration generated by B. Definition γ The class W consists of probability measures Q determined by density processes satisfying the Dooleans-Dade stochastic equation dZt = γt Zt dBt ; (7) with γ a progressively measurable process satisfying 1 γt ≤ 0 for t 2 [0; T], and h i 2 R T 2 E 0 γs ds < 1. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Theorem Take a non negative continuous deterministic function V defined in the interval [0; T]. Let H = B + V . Then, the lower Snell envelope of the process H is equal to H itself. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Theorem There exists no continuous uniform decomposition as the difference of a Q-submartingale and a non decreasing process for the Brownian motion B with respect to W. Semimartingale properties of the lower Snell envelope [email protected] - UG Motivation A few known results Semimartingale properties A (counter)-example References Theorem Let U be a non negative E#-supermartingale of class(D) with respect to each element Q of Q. Let Ξ be the class of finite partitions with points in D, a countable dense subset of [0; T]. Let DM(Ξ) be the class of random variables defined by n−1 X V Π = U − E#[U j F ] ; (8) T ti ti+1 ti i=0 for Π = f0 = t0 < t1 < : : : < tn = Tg a partition in Ξ. If the class DM(Ξ) is uniformly integrable with respect to a Q0 2 Q, then there exists a right-continuous non decreasing process V such that Z := U + V is a Q0-submartingale.