On Optimal Stopping Problems with Power Function of Lévy Processes

Transcription

1 On Optimal Stopping Problems with Power Function of Lévy Processes Budhi Arta Surya Department of Mathematics University of Utrecht 31 August 2006 This talk is based on the joint paper with A.E. Kyprianou: Kyprianou, A. E., and Surya, B. A. On the Novikov-Shiryaev optimal stopping problems in continuous time. Elect. Comm. in Probab., 10 (2005),

2 Outline of Presentation The points of the talk: Problem formulation Some related works General theory of optimal stopping Appell polynomials Main results and sketch of the proof Numerical examples FELab Seminar, The University of Twente, 31 August

3 Setting of the Problem Let X be a real valued Lévy process defined on (Ω, F, P) with a filtration F t = σ(x s, 0 s t), F 0 = {, Ω}, t R +. The law of X is described by the Lévy-Khintchine formula E ( e iθx t ) = e iθx P ( X t dx ) = e tψ(θ), where Ψ(θ) is the characteristic exponent of X defined by Ψ(θ) = µθ 1 2 σ2 θ 2 + ( e iθy 1 iθy1 { y <1} ) Π(dy), (1) FELab Seminar, The University of Twente, 31 August

4 and σ 0, µ R, and Π is a measure concentrated on R\{0} satisfying the integrability condition R (1 y2 )Π(dy) <. When X has paths of bounded variation 1 it will be more convenient to write (1) in the form Ψ(θ) = idθ + (1 e iθx )Π(dx) (2) where d R is known as the drift. General optimal stopping problem We consider the optimal stopping problems: find the value function V(x) = sup τ T [0, ] E x ( e qτ G(X τ )1 {τ< } ), x R, q 0, 1 R If the Lévy measure Π satisfies the integral test ( 1,1) x Π(dx) <. FELab Seminar, The University of Twente, 31 August

5 where G(x) is a measurable function, and T [0, ] is the class of Markov stopping times τ (w.r.t F t ) taking values in [0, ]. τ is said to be an optimal stopping time if V(x) = E x (e qτ G(X τ )1 {τ < } ) = V(x), x R, q 0. We discuss here the case where the payoff function is G(x) = ( max{x, 0} ) n for n = 1, 2,. FELab Seminar, The University of Twente, 31 August

6 Some Related Works Darling, Liggett, Taylor (1972) (G(x) = x +, q = 0; G(x) = e x+ 1; q > 0, t Z + ) Dubins and Teicher (1967) (G(x) = x +, q 0, t Z + ) Mordecki (2002) (G(x) = e x+ 1, q 0, t Z + ) Boyarchenko and Levendorskii (2002a, 2002b) (G(x) = 1 e x+, q > 0, t R +, t Z + ) Novikov and Shiryaev (2004, 2005) (G(x) = (x + ) ν, q = 0, ν 0, t Z + ), (G(x) = 1 e x+, q = 0, t Z + ) Kyprianou and Surya (2005) (G(x) = (x + ) n, q 0, n = 1, 2,, t R + ) Beibel (1998) (Concave G(x), q = 0, t R + ) FELab Seminar, The University of Twente, 31 August

7 General Theory of Optimal Stopping Chow et al. (1971), (1991): Great expectation... Shiryaev (1969), (1978): Optimal stopping rules... = Dictate that V (x) is the smallest function U = U(x) with the properties that ( U(x) G(x), U(x) E x e qt U(X t ) ), for all t 0 and x R and under very general assumptions of G that τ = inf{t 0 : V (X t ) = G(X t )}. FELab Seminar, The University of Twente, 31 August

8 For the special case q = 0, G(x) = (x + ), t Z +, (see Darling et al. (1972)) the optimal stopping time τ is given by τ h = inf{k 0 : X k h} with the optimal stopping level (threshold) h is given by h = E ( X ), X := sup{s k : k 0} and the value function V is given by V (x) = E ( X + x E(X ) ) + under the assumption that E(X 1 ) < x and E(X + 1 )2 <. FELab Seminar, The University of Twente, 31 August

9 Appell Polynomials Let Y be a r.v., and u E(e uy ) < be differentiable at u = 0. Appell Polynomials (Sheffer Polynomials), Q n (x; Y ), n = 1, 2,..., generated by Y is defined through the Esscher transform: n=0 Q n (x; Y ) ( u)n n! = e ux E(e uy ). For a fix n > 0, it can be shown that the Appell polynomial Q n (x; Y ) solves the recursive differential equation d dx Q m(x; Y ) =mq m 1 (x; Y ) E(Q m (Y + x; Y )) =x m. m n FELab Seminar, The University of Twente, 31 August

10 Let κ 1, κ 2,... be cumulants 2 of Y. Then, Q 0 (x; Y ) = 0, Q 1 (x; Y ) = x κ 1, Q 2 (x; Y ) = (x κ 1 ) 2 κ 2 Q 3 (x; Y ) = (x κ 1 ) 3 3κ 2 (x κ 1 ) κ 3 ) (x n Example 1. If Y N(µ, σ 2 ) then Q n (x; Y ) = σ n H n σ, where H n (x) is the Hermitte polynomials. Example 2. If Y exp(λ) then Q n (x; Y ) = x n nλ 1 y n 1. The polynomials Q k (x; Y ), k n are defined under E Y <. For other examples, see for instance Schoutens (2000). 2 Defined as κ1 = E(Y ), κ j = E(Y κ 1 ) j, j > 1. FELab Seminar, The University of Twente, 31 August

11 Appell Polynomials Q n (x) x Figure 1: Appell polynomial Q n (x), n = 4, generated by upward jumps (also known as spectrally positive) compound Poisson process (CPP) with drift d = FELab Seminar, The University of Twente, 31 August

12 The Main Results Let e q be an independent exponentially distributed r.v. with parameter q 0. For q = 0 it is understood that e q = with probability one. We assume throughout the remaining of this talk that either q > 0 or (q = 0 and lim sup t X t < ). (3) Before we state the results, let us remind ourself that: ( V n (x) = sup E x e qτ( X + ) ) n1{τ< } τ, x R, q 0, τ T [0, ] and X t = sup { X s : 0 s t }. FELab Seminar, The University of Twente, 31 August

13 Theorem 1. (The optimal solution of the problem) Fix n > 0. Suppose that the assumption (3) as well as (1, ) x n+1 Π(dx) <, q =0 and (1, ) x n Π(dx) <, q > 0. Then the Appell polynomial Q n (x) has finite coefficients and there exists h n [0, ) being the largest root of the equation Q n (x) = 0. Let us define a stopping time Then the value function is given by τ + h n = inf{t 0 : X t h n}. V n (x) = sup τ T [0, ] E x ( e qτ ( X + τ ) n1{τ< } ) = Ex ( Qn (X eq )1 {Xeq >h n } ). FELab Seminar, The University of Twente, 31 August

14 Theorem 2. (The smooth pasting principle) Fix each n = 1, 2,..., the solution to the optimal stopping problem in the previous theorem is continuous and has the property that d dx V n(x n) = d dx G(x n) P(X eq = 0) d dx Q n(x n). Hence there is smooth pasting at x n if and only if P(X eq = 0) = P(X eq = 0) = 0 if and only if one of the following three conditions are fulfilled. (i) R ( 1,1) x Π(dx) = (so that X has unbounded variation). (ii) R ( 1,1) x Π(dx) < (so that X has bounded variation) and in the representation (2) we have d > 0. (iii) R ( 1,1) x Π(dx) < (so that X has bounded variation) and in the representation (2) we have d = 0 and further Z x Π(dx) =. (0,1) R(0,x) Π(, y)dy The latter conclusions being collectively due to Rogozin (1968), Shtatland (1965) and Bertoin (1997). FELab Seminar, The University of Twente, 31 August

15 Sketch of the Proof of the Main Results Proof of Theorem 1. For each n > 0 we first solve the optimal stopping problem (( ) V n (x) = sup E x X + n1(τh τh < )). h>0 From the problem setting, it is clear to see that V n (x) V n (x). Following the properties of Appell polynomials, we obtain (( ) V n (x) = E x X + n1(τ ) ( ) τ + + = h h n < ) Ex Qn (X eq )1 {Xe q n >h n } Finally, we show that V n (x) V n (x) τ + h n time. is the optimal stopping FELab Seminar, The University of Twente, 31 August

16 Preliminary Lemmas We assume that lim sup t X t < for the case q = 0. Lemma 1. (Moments of supremum) If n > 0, then x n+1 Π(dx) <, q = 0 = E(X ) n < ; (1, ) (1, ) x n Π(dx) <, q > 0 = E(X eq ) n <. (4) Lemma 2. (Mean value property) Fix n {1, 2, }. Suppose that Y is a non-negative random variable satisfying E(Y n ) <. Then if Q n is the n-th Appell polynomial generated by Y then we have that E ( Q n (x + Y ) ) = x n for all x R. FELab Seminar, The University of Twente, 31 August

17 Lemma 3. (Fluctuation identity) Fix n {1, 2, }. Suppose that (3) holds and the integral tests in (4) apply. Define a stopping time τ + h = inf{t 0 : X t > h}. Then for all h > 0 and x R we have E x ( e qτ + h X n τ + h ) = Ex ( Qn (X eq )1 (Xeq >h) ). Lemma 4. (Largest positive root) Fix n {1, 2, }. Suppose that (3) holds and the integral tests in (4) apply. As before, suppose that Q n is generated by X eq. Then the equation Q n (x) = 0 has an unique root h n such that Q n (x) < 0 for x [0, h n] and positive an increasing on [h n, ]. FELab Seminar, The University of Twente, 31 August

18 Proof of Theorem 1 First let us define for a fixed n {1, 2, } a function V n (x) = E x ( Qn (X eq )1 (Xe q >h n ) ). First note from Lemma 3 that ( qτ V n (x) = E + x e h n(x τ + h ) n ) 1 (τh < ) n n Hence ( V n (x), τ h n ) is a candidate solution pair to the problem. Secondly we prove that V n (x + ) n for all x R. FELab Seminar, The University of Twente, 31 August

19 Note that this statement is obvious for x (, 0] [h n, ) just from the definition of V n. Otherwise when x (0, x n) we have, using Lemma 2 that V n (x) = E x ( Qn (X eq )1 (Xe q >x n ) ) = x n E x ( Qn (X eq )1 (Xe q x n ) ) (x + ) n where the final inequality follows from Lemma 4 and specifically the fact that Q n (x) 0 on (0, x n]. Note in particular, embedded in this argument is the statement that V n (x ) = (x + ) n at x = h n. Thirdly have P x almost surely that Q n (X eq )1 (Xe q >x n ) 0. Using the latter together with the fact that, on the event that {e q > t} we have X eq is equal in distribution to X t + I where I is independent of F t and FELab Seminar, The University of Twente, 31 August

20 equal in distribution to X eq, it follows that ( ) V n (x) E x 1(eq >t)q n (X eq )1 (Xe q >h n ( ) = E x 1(eq >t)e x (Q n (X t + X eq )1 (Xt +X e q >h ) F t) ) n ( = E x e qt Vn (X t ) ). From this inequality together with the Markov property, it is easily shown that {e qt Vn (X t ) : t 0} is a supermartingale. Finally putting these three facts together the proof of the Theorem 1 follows. We leave the proof of Theorem 2 as an exercise. FELab Seminar, The University of Twente, 31 August

21 The Optimal Value Function of the Problem (x + ) 4 V x4 * (x) x Figure 2: The value function V n (x) associated with the payoff function G(x) = (x + ) n, n = 4, driven by downward jumps CPP with drift d = FELab Seminar, The University of Twente, 31 August

22 The Candidate Solution of the Problem x Figure 3: The candidate solution c V n (x; h), n = 4, under downward jumps CPP with drift d = +0.05, corresponding to stopping at non-optimal stopping boundary h. FELab Seminar, The University of Twente, 31 August

23 The Optimal Value Function of the Problem (x + ) 4 V x4 * (x) x Figure 4: The value function V n (x) associated with the payoff function G(x) = (x + ) n, n = 4, driven by upward jumps CPP with drift d = FELab Seminar, The University of Twente, 31 August

24 The Candidate Solution of the Problem x Figure 5: The candidate solution c V n (x; h), n = 4, under upward jumps CPP with drift d = 0.05, corresponding to stopping at non-optimal stopping boundary h. FELab Seminar, The University of Twente, 31 August

25 Main References Bertoin, J. Lévy Processes, Cambridge University Press, Bertoin, J. Regularity of half-line for Lévy processes. Bull. Sci. Math., 121 (1997), Boyarchenko, S. I. and Levendorskii, S. Z. Non-Gaussian Merton- Black-Scholes theory, World Scientific Publishing Co., Inc., River Edge, NJ., Darling, D. A., Ligget, T., and Taylor, H. M. Optimal stopping for partial sums. Ann. Math. Stat., 43 (1972), Kyprianou, A. E. Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer, FELab Seminar, The University of Twente, 31 August

26 Kyprianou, A. E., and Surya, B. A. On the Novikov-Shiryaev optimal stopping problems in continuous time. Elect. Comm. in Probab., 10 (2005), Mordecki, E. Optimal stopping and perpetual options Lévy processes. Finance and Stoch., 6 (2002), Novikov, A. A., and Shiryaev, A. N. On an effective solution of the optimal stopping problem for random walks. To appear in Theo. Probab. and Appl., Novikov, A. A., and Shiryaev, A. N. Some optimal stopping problems for random walks and Appell functions, talk presented at the School Optimal Stopping with Applications The Manchester University, January, Rogozin, B. A. The local behavior of processes with independent increments. Theory Prob. Appl., 13 (1968), FELab Seminar, The University of Twente, 31 August

27 Sato, K. Lévy processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, UK, Schoutens, W. Stochastic processes and orthogonal polynomials. Lecture Notes in Mathematics, nr Springer, Shiryayev, A. N. Optimal Stopping Rules, Springer-Verlag, New York Inc, Shtatland, E. S. On local properties of processes with independent increments. Theory Probab. Appl., 10 (1965), FELab Seminar, The University of Twente, 31 August