Optimal stopping, Appell polynomials and Wiener-Hopf factorization

Transcription

1 arxiv: v1 [math.pr] 19 Feb 2010 Optimal stopping, Appell polynomials and Wiener-Hopf factorization Paavo Salminen Åbo Aademi, Mathematical Department, Fänrisgatan 3 B, FIN Åbo, Finland, phsalmin@abo.fi Abstract In this paper we study the optimal stopping problem for Lévy processes studied by Noviov and Shiryayev in [10]. In particular, we are interested in finding the representing measure of the value function. It is seen that that this can be expressed in terms of the Appell polynomials. An important tool in our approach and computations is the Wiener-Hopf factorization. Keywords: optimal stopping problem, Lévy process, Wiener-Hopf factorization, Appell polynomial. AMS Classification: 60G40, 60J25, 60J30, 60J60, 60J75. 1

2 1 Introduction Let X {X t : t 0} denote a real-valued Lévy process and P x the probability measure associated with X when initiated from x. Furthermore, {F t } denotes the natural filtration generated by X and M the set of all stopping times τ with respect to {F t }. We are interested in the following optimal stopping problem: Find a function V and a stopping time τ such that Vx sup E x e rτ gx τ E x e rτ gxτ, 1.1 τ M where gx : x + n, n 1,2,..., and r 0. It is striing that the solution of this problem can be characterized fairly explicitly for a general Lévy process whose Lévy measure satisfies some integrability conditions. This solution is essentially due to Noviov and Shiryaev [10] who found it for random wals. Construction was lifted to the framewor of Lévy processes by Kyprianou and Surya [6] see also Kyprianou [5] Section 9. In Noviov and Shiryayev [9] the corresponding stopping problem with arbitrary power γ > 0, i.e., gx : x + γ is analyzed. For related problems for random wals and Lévy processes, see Darling et al. [4], and Mordeci [7]. The so called Appell polynomials in [9] a more general concept of Appell function is introduced play a central role in the development directed by Noviov and Shiryayev. In the solution, the function V is described as the expectation of a function of the maximum of the Lévy process up to an independent exponential time T. In the paper by Salminen and Mordeci [8] the appearance of a function of maximum is explained via the Wiener- Hopf-Rogozin factorization of Lévy processes and the Riesz decomposition and representation of excessive functions. The aim of this note is to study the Noviov-Shiryayev solution in light of the results in [8]. In particular, we focus on finding the representing measure of the excessive function V. In the spectrally positive case the representing measure has a clean expression in terms of the Appell polynomials of X T. In the general case we are able to find the Laplace transform of the representing measure but unable to find a "nice" inversion. If X is spectrally negative, this Laplace transform is expressed in terms of the Lévy-Khintchine exponent 2

3 of X. In the next section basic properties and examples of Appell polynomials are discussed. Therein is a short subsection on Lévy processes where we present notation, assumptions and features central and important for the application in optimal stopping. In Section 3 - to mae the paper more readable - the results from [10] and [8] are shortly recalled. After this we proceed with the main results of the paper concerning the representing measure of V. 2 Appell polynomials 2.1 Basic properties Let η be a random variable with some exponential moments, i.e., there exists u > 0 such that E e u η <. A family of polynomials {Q η ; 0,1,2,...}, the Appell polynomials associated with η, are defined via e ux Ee uη u! Qη x. 2.1 Putting here x η+z and taing expectations we obtain easily the so called mean value property of the Appell polynomials E Q η η +z z. 2.2 Writing and rearranging yields e ux Ee uη Q η m x i0 m u i i! xi u! Qη 0 m x Q η m Consequently, taing derivatives, it is seen that the Appell polynomials satisfy d dx Qη Q η 0 m x mq m 1x, η 2.4 x 1 for all x

4 The recursion in 2.4 if combined with the normalization cf. 2.2 EQ n η 0 n 1,2,... provides an alternative definition of the Appell polynomials. Recall that the cumulant function associated with η is defined as Ku : logee uη κ u!, 2.6 and the coefficients κ, 0,1,..., in the above McLaurin expansion are called the cumulants of η. We remar that but κ 1 Eη, κ 2 E η Eη 2, κ 3 E η Eη 3 κ 4 E η Eη 4. It is easily seen that the first Appell polynomials can be written as Q η 0 x 1, Qη 1 x x κ 1, Q η 2 x x κ 1 2 κ It is also possible to connect the cumulants and the origo moments via the Appell polynomials. To do this notice that where K u G u Gu, 2.8 Gu : Ee uη µ u! is the moment generating function of η and µ are the origo moments. Equation 2.8 taes the form u κ +1! u i µ i+1 i! i0 Q η u 0! leading to κ m+1 m m µ +1 Q η m

5 Notice also that if η 1 and η 2 are independent random variables with some exponential moments and if η : η 1 +η 2 then m m Q m η x+y Q η 1 xq η 2 m y This results after a straightforward computation from the identity e ux+y Ee uη eux Ee uη 1 e uy Ee uη 2. Remar 2.1. It is perhaps amusing to notice that 2.10 when combined with 2.2 leads to Newton s binomial formula. Indeed, Q η m η 1 +η 2 +x+y and taing expectations yield m x+y m m Q η 1 η 1 +xq η 2 m η 2 +y, m m x y m. Example 2.2. Let η be normally distributed with mean 0 and variance 1. Then e ux ux Ee uη exp 12 u2 ux u 2 /2 l! l! l0 1 l x l u 2l+!l!2 l Consequently, Q η n m0 l0 u m m! m! [m 2] x m! l0 5 [ m 2] l0 1 l x m 2l m 2l!l!2 l. 1 l x m 2l m 2l!l!2 l,

6 and, hence, The Appell polynomials associated with a N0, 1-distributed random variable are the Hermite polynomials He n, n 0,1,..., see Schoutens [11] p. 52, and Abramowitz and Stegun [1] p. 775, p Example 2.3. We calculate the Appell polynomials of an exponentially distributed random variable. To pave the way to the applications below, consider a standard Brownian motion B {B t : t 0} starting from 0 and its running maximum M t : sup s t B s. Let T be an exponentially with parameter r > 0, distributed random variable independent of B. Then M T is exponentially distributed with mean 1/ 2r. We find the Appell polynomials associated with M T. Since we have E e um T 2r, u < 2r. 2r u e ux 2r u e ux Ee um T 2r u! x 1+ Hence, the Appell polynomials are Q M T n x 1 u! u +1 x 2r! x x 1. 2r x n 2r x n 1, n 0,1,... Recall also that B T is Laplace-distributed with parameter 2r and, hence, it holds E e ub 2r T 2r u 2, u < 2r. Proceeding as above we find the Appell polynomial associated with B T : e ux 2r u2 Ee ub T 2r u! 6 e ux x 2 1 x 2. 2r

7 Let B be an independent copy of B, and introduce I t : inf s t B s. Since d M T I T it holds Q I T n x x+ n 2r x n 1, n 0,1,... It is well nown the Wiener-Hopf factorization that X T d M T +I T, and it is straightforward to chec cf m Q X T m m x+y Q M T xq I T m y. 2.2 Lévy processes Let X be a Lévy process as introduced above and T an exponential parameter r > 0 random variable independent of X. Our basic rather restrictive assumption on X is that X T has some exponential moments, i.e., for some λ > 0 Eexpλ X T < Define M T sup X t and I T inf X t t<t 0 t<t By the Wiener-Hopf-Rogozin factorization X T d M T +I T, 2.13 where I T is an independent copy of I T and d means that the variable on the left hand side is identical in law with the variable on the right hand side. From assumption 2.11 it follows using 2.13 that also M T and I T have some exponential moments. Moreover, under assumption 2.11, we have where ψγ aγ b2 γ 2 + EexpλX T R 1 r ψλ e γx 1 γx1 { x 1} Πdx

8 is the Lévy-Khinchine exponent of X, i.e., EexpλX t exptψλ. Notation Π stands for the Lévy measure, i.e., a non negative measure defined on R\{0} such that 1 x 2 Πdx < +. Clearly, it follows from 2.11 that for all a > 0 and γ > 0 a e γx Πdx+ but it is still possible to have or a 0 a e γx Πdx <, Π{0,a+Π{ a,0 xπdx+ 0 a x Πdx. In case X is spectrally negative, i.e., Π0,+ 0, the process moves continuously to the right or upwards and M T is exponentially distributed under P 0 with mean 1/Φr where Φr is the unique positive root of the equation ψλ r see Bertoin [2] p. 190 or Kyprianou [5] p From Example 2.3 it is seen that the Appell polynomials associated with M T are Q M T n x x n Φr x n 1, n 0,1, In case X is spectrally positive, i.e., Π,0 0, the process moves continuously to the left or downwards and I T is exponentially distributed under P 0 with mean 1/ˆΦr where ˆΦr is the unique positive root of the equationψ λ r. From Example 2.3 it is seen that the Appell polynomials associated with I T are Q I T n x x+ n ˆΦr 3 Optimal stopping problem 3.1 Review of three theorems x n 1, n 0,1, We recall now the solution of the optimal stopping problem 1.1 from [10] and [6]. We assume that r > 0, and recall that T is an exponentially with parameter r distributed random variable independent of X. 8

9 Theorem 3.1. Assume Then EM n T < and and 1,+ Vx : sup E x e rτ X τ + τ M x n Πdx <. n E0 Q M n M T +x1 {MT +x>x n} τ n inf{t 0 : X t > x n } is an optimal stopping time, where Q M is the Appell polynomial associated with M T and x n is its largest non-negative root. Remar 3.2. The proof of Theorem 3.1 uses the fact that Q M n x < 0 for all x 0,x n 3.1 see [10]. This property is based on the fluctuation identity E x e rh a XH n a 1 {Ha< } E0 Q M n M T +x1 {MT +x>a}, 3.2 where a > 0 and H a : inf{t : X t a}. Notice that 3.1 is not valid in general for Appell polynomials of a non-negative random variable. Indeed, from 2.7 we have, e.g., Q η 2 0 > 0 if 2Eη 2 > Eη 2. Example 3.3. Using the result in Theorem 3.1 it is easy to find the explicit solution of the problem for spectrally negative Lévy processes. Indeed, recall from previous section that M T is exponentially distributed with parameter Φr, the Appell polynomials associated with M T are given in 2.15, and x n : n/φr. With this data we obtain supe x e rτ X τ + τ n Ex Q M T n E x M T n n/φr e xφr y n Φr n/φr n n e n e xφr. Φr M T ; M T x n M n 1 T ; M T n Φr Φr y n 1 Φre Φry x dy y n Φr e yφr dy 9 n/φr ny n 1 e yφr dy

10 Therefore, the value function of the optimal stopping problem is given by x n, if x n/φr, Vx n n Φr e n e xφr, if x < n/φr. Clearly, the solution has the smooth-fit property. Next we recall the result from [8] which characterizes the value function via its representing measure. The theorem is proved in [8] for Hunt processes satisfying a set of assumptions. In particular, it is assumed that there exists a duality measure m such that the resolvent ernel has a regular density with respect to m for the assumptions and a discussion of their validity for Lévy processes, see [8]. This density is denoted by G r, and it holds P x X T dy rg r x,ymdy rg r 0,y xmdy Theorem 3.4. Consider a Lévy process {X t } satisfying the assumptions made in [8], a non-negative continuous reward function g, and a discount rate r > 0 such that E x supe rt gx t <. 3.3 t 0 Assume that there exists a Radon measure σ with support on the set [x, such that the function Vx : G r x,yσdy 3.4 satisfies the following conditions: a V is continuous, b Vx 0 when x. c Vx gx when x x, d Vx gx when x < x. [x, Let τ inf{t 0: X t x }

11 Thenτ is an optimal stopping time andv is the value function of the optimal stopping problem for {X t } with the reward function g, in other words, In case τ + Vx sup E x e rτ gx τ E x e rτ gx τ, x R. τ M e rτ gx τ : limsupe rt gx t. t Next result, also from [8], gives a more explicit form for value function V via the maximum variable M T. The result is derived by exploiting the Wiener-Hopf-Rogozin factorization. For simplicity, we assume that I T has a density. Theorem 3.5. Assume that the conditions of Theorem 3.4 hold. Then, there exists a function H: [x, R such that the value function V in 3.4 satisfies Vx E x HM T ; M T x, x x. Moreover, the function H has the explicit representation Hz r 1 z where f I is the density of the distribution of I T. 3.2 Representing measure x f I y zσdy, z x. 3.6 Let now V denote the value function for problem 1.1 as given in Theorem 3.1: Vx E 0 Q M n M T +x1 {MT +x>x n }. 3.7 This function has the properties a-c of Theorem 3.4; a and b follow from monotone convergence, c and d from the properties of the Appell polynomials or from the fact that V is indeed the value function of the problem. Therefore, it is natural to as whether it is possible to find a measure σ n, say, such that 3.6 holds, i.e., Q M n z r 1 z x n f I y zσ n dy, z x n. 3.8 From this equation it is possible to derive an expression for the Laplace transform of σ. 11

12 Proposition 3.6. There exists a measure σ n such that 3.8 holds. The Laplace transform σ n of this measure is given by where σ n γ : q n γ,x n : x n e γy σ n dy r q n γ,x n/e e γi T 3.9 γ! q nγ,x n QI 0, 3.10 x n n i0 with Q M n i x n 0, i 0,1,...,n. e γz Q M n z dz n! n i! e γx n QM γi+1 n i x n 3.11 Proof. Multiply equation 3.8 with e γz and integrate over x, to obtain to simply the notation we omit the subindex n z dze γz x e γz Q M zdz r 1 σdyf I y z x x r 1 σdy dz e γz f I y z x y r 1 σdy e γy dz e γz y f I y z x y r 1 σdy e γy dz e γu f I u. x In other words, σ is well defined by the assumption that I T has some exponential moments and satisfies σγ r qγ,x /E e γi T, which is 3.9. To obtain 3.10 use the definition of Appell polynomials see 2.1, and identity 3.11 is a straightforward integration using 2.4. Finally, notice that the positive zeros x n of QM n, respectively, satisfy x 1 x

13 This follows readily again from 2.4 and the fact that x is the unique positive zero of Q M. Hence, Q M x n 0 for 0,1,...,n. In case X is spectrally negative the value function can be determined explicitly as demonstrated in Example 3.3. However, the Laplace transform of the representing measure does not seem to have an explicit inversion. To discuss this case more in detail, recall that M T is exponentially distributed with parameter Φr. Hence, we can relax our assumptions on exponential moments. It holds for γ 0 E e γx r T r ψγ, and, by the Wiener-Hopf factorization, E e γi T E e γx T /E e γm T rφr γ Φrr ψγ Notice also that Q M x n n n /Φr. To conclude this discussion, we formulate the following Corollary 3.7. If {X t } is spectrally negative then σ n dx r q n γ,x n /E e γi T qn γ,x ψγ n Φrr Φr γ and q n γ,x n : x n n i0 e γz Q M n z dz n! e γx n n i! γ i+1 i Φr n i n Φr For spectrally positive Lévy processes we do not, in general, have any explicit nowledge of the Appell polynomials Q M and, hence, also the value function V remains hidden. However, in this case we may charaterize σ n nicely. This is possible since now I T is exponentially distributed and we see from 3.8, roughly speaing, that Q M is the Laplace transform of σ n. We obtain also a new expression for the value function. 13

14 Corollary 3.8. If X is spectrally positive then and, hence, σdx rq X n xdx, x x n, 3.14 Vx E Q X n X T +x; X T +x x n Proof. The Appell polynomials for I T are given explicitly in 2.16, and we have Q I T 0 0 1, Q I T 1 0 1/ˆΦr, and Q I T n 0 0 for n 2,3,... From 3.10 we obtain Consider γ qγ σγ r qγ+ r γ qγ ˆΦr γ e γz Q M n z dz x γ e γz x x d dx QM n x e γx dx x x nq M n 1 x e γx dx, d dx QM n x dx dz where we have changed the order of integration, used the fact x n is a zero of Q M n, and applied the differentiation formula 2.4. Consequently, we have done the inversion and it holds σz rq M n z+n r ˆΦr QM n 1z r Q M n zq I 0 0+nQ M n r n rq X n z, Q M n zqi 0 n 1zQ I 1 0 since Q I 0 0 for 2,3,... and, in the last step, formula 2.10 is applied. 14

15 We conclude by studying Brownian motion from the point of view of Corollary 3.8. Example 3.9. The resolvent ernel with respect to the Lebesgue measure is given by see, e.g., Borodin and Salminen [3] p. 120 G r x,y 1 2r e 2r x y, and, from Example 2.3, x n n/ 2r and Q X T n x x 2 nn 1 2r x n 2. It is straightforward to chec that V as given in Example 3.3 satisfies References Vx r x n G r x,yq X T n ydy. E x Q X T n X T ; X T x n. [1] M. Abramowitz and I. Stegun. Mathematical functions, 9th printing. Dover publications, Inc., New Yor, [2] J. Bertoin. Lévy Processes. Cambridge University Press, Cambridge, [3] A.N. Borodin and P. Salminen. Handboo of Brownian Motion Facts and Formulae, 2nd edition. Birhäuser, Basel, Boston, Berlin, [4] D.A. Darling, T. Liggett, and H.M. Taylor. Optimal stopping for partial sums. Ann. Math. Stat., 43: , [5] A. Kyprianou. Introductory Lectures on Fluctuations of Lévy processes with applications. Springer Verlag, Berlin, Heidelberg, [6] A. E. Kyprianou and B. A. Surya. On the Noviov-Shiryaev optimal stopping problems in continous time. Electronic Communications in Probability, 10: ,

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