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Optimal Stopping Time Problem for Random Walks with Polynomial Reward Functions Udc 519.21

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Optimal Stopping Time Problem for Random Walks with Polynomial Reward Functions Udc 519.21 Teor Imovr. ta Matem. Statist. Theor. Probability and Math. Statist. Vip. 86, 2012 No. 86, 2013, Pages 155–167 S 0094-9000(2013)00895-3 Article electronically published on August 20, 2013 OPTIMAL STOPPING TIME PROBLEM FOR RANDOM WALKS WITH POLYNOMIAL REWARD FUNCTIONS UDC 519.21 YU. S. MISHURA AND V. V. TOMASHYK Abstract. The optimal stopping time problem for random walks with a drift to the left and with a polynomial reward function is studied by using the Appel polynomials. An explicit form of optimal stopping times is obtained. 1. Introduction The stopping time problem for stochastic processes has several important applications for the modelling of the optimal behavior of brokers and dealers trading securities in financial markets. The classical approach to solving the optimal stopping time problem is based on the excessive functions needed to determine the so-called reference set that defines explicitly the optimal stopping time [1–3]. An entirely different approach to the solution of the optimal stopping time problem is used in this paper. Namely, our method is based on an application of the Appel polynomials (see [4, 5]). This paper is a continuation of studies initiated in [6] and is devoted to a generalization of some results obtained in [4]. Let ξ,ξ1,ξ2,... be a sequence of independent identically distributed random variables defined on a probability space (Ω, , P)andsuchthatE ξ<0. Consider a homogeneous Markov chain X =(X1,X2,X3,...) related to the sequence {ξi} as follows: k X0 = x ∈ R,Xk = x + Sk,S0 =0,Sk = ξi,k≥ 1. i=1 We denote by Px the probability distribution generated by the process X.Thus,Px, x ∈ R, together with X defines a Markov family with respect to the filtration (k)k≥0, where 0 = {∅, Ω} and k = σ{ξ1,...,ξk}, k ≥ 1. The optimal stopping problem is to determine a reward function V (x)= sup Ex g(Xτ )I{τ<∞},x∈ R, ∈ ¯ ∞ τ M0 {·} ¯ ∞ where g(x) is a given measurable function, I is an indicator function, and M0 is the class of all Markov times τ assuming values in [0, ∞]. The random variable ∗ τ =argmaxEx g(Xτ )I{τ<∞} ∈ ¯ ∞ τ M0 is called the optimal stopping time. 2010 Mathematics Subject Classification. Primary 60G40, 60G50. Key words and phrases. Appel polynomials, random walks, reward functions, stopping times. c 2013 American Mathematical Society 155 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 156 YU. S. MISHURA AND V. V. TOMASHYK The power reward function g(x)=(x+)k is considered in the paper [4]. In the current paper, we consider polynomial reward functions g(x) of the following form: N + k g(x)= Ck(x ) ,Ck ∈ R. k=1 The optimal stopping time problem for a random walk X is considered in the paper [6] for the case of polynomial reward functions of an arbitrary order and with nonnegative coefficients; an explicit expression for the optimal stopping time and that for the price function are also found in [6] for this case. The aim of this paper is to determine the optimal stopping time for a random walk X =(X1,X2,X3,...), where both a reward function and a price function are polynomial. Since the solution of the optimal stopping time problem is cumbersome for general coefficients of the polynomial reward and price functions, we state the result below for the general case and provide the detailed proof for a particular case where the order of the polynomial reward function does not exceed 3. 2. Auxiliary results and definitions We need several auxiliary results and definitions for the proof of the main result of this paper. Definition 1. Let η be a random variable such that E exp(λ|η|) < ∞ for some λ>0. The polynomials defined by ∞ exp(uy) uk = Q (y) E exp(uη) k! k k=0 are called the Appel (or Sheffer; see [5]) polynomials, Qk(y)=Qk(y, η), k =0, 1, 2,... The polynomials Qk(y) are expressed in terms of the cumulants χ1,χ2,... of the random variable η as follows: 2 Q0(y)=1,Q1(y)=y − χ1,Q2(y)=(y − χ1) − χ2, 3 Q3(y)=(y − χ1) − 3χ2(y − χ1) − χ3, − 2 3 − k where χ1 = μ1, χ2 = μ1 + μ2,andχ3 =2μ1 3μ1μ2 + μ3;hereμk = E η . Note that the polynomials Qk(y), k =1,...,n, are uniquely defined if one assumes that E |η|n < ∞.Moreover, d (1) Q (y)=kQ − (y),k≤ n, dy k k 1 in this case. This equality is sometimes used to define the Appel polynomials recursively [4]. Note that Qk(y) is a polynomial whose order equals k; see [5]. In particular, this meansthateverysetofn Appel polynomials Q1(y),Q2(y),...,Qn(y) is a system of linearly independent functions. Throughout in this paper we deal with Appel polynomials generated by the random variable M =supk≥0 Sk,thatis,Qk(y)=Qk(y, M), k =0, 1, 2,... The cases where the distribution of the random variable M can be written explicitly are considered in [7–10]. The following result is proved in [4]. Lemma 1 provides us with sufficient conditions for the Appel polynomials generated by the random variable M to be well defined. Lemma 1. a) Let E eλξ < 1 for some λ>0.Then E euM < ∞ for all u ≤ λ. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OPTIMAL STOPPING TIME PROBLEM 157 b) For an arbitrary p>0, E(ξ+)p+1 < ∞ =⇒ E M p < ∞. Remark 1. It is proved in [4] that M ≥ 0, P{M<∞} =1, P{M =0} > 0, and M law=(M + ξ)+. The proof of the following result is also given in [4]. Lemma 2. 1) Let E(ξ+)n+1 < ∞.Then n a) E Qn(M + x)=x ; b) if τa =inf{k ≥ 0: Xk ≥ a},then { ∞} n { ≥ } Ex I τa < Xτa = E I M + x a Qn(M + x) for all a ≥ 0. ≥ ∗ ≤ 2) The polynomial Qn(y), n 1, has a unique positive root an; moreover Qn(y) 0 for ≤ ∗ ≥ ∗ 0 y<an, and Qn(y) increases for y an. 3) Let f(x)=E I{M + x ≥ a∗}G(M + x) < ∞, where the function G(x) is such that G(y) ≥ G(x) ≥ G(a∗)=0for all y ≥ x ≥ a∗ ≥ 0.Thenf(x) ≥ E f(ξ + x) for all x. 4) Let f(x) and g(x) be two nonnegative functions such that (2) f(x) ≥ g(x) and f(x) ≥ E f(ξ + x) for all x.Then f(x) ≥ sup E I{τ<∞}g(Sτ + x) ∈ ¯ ∞ τ M0 for all x. ∗ It is proved in [4] that the roots an of the Appel polynomials Qn(y) are increasing, ∗ ∗ ∗ namely 0 <a1 <a2 <a3 <... The representation of the Appel polynomials in terms of the cumulants and property 2) of Lemma 2 imply that ≥ ≥ ≥ ≥ 2 3 − ≥ χ1 0,χ2 0,χ3 0,χ2 χ1,χ1 3χ2χ1 + χ3 0. Thecaseofχ1 = 0 is degenerate in the sense that the random walk moves to the left in this case and hence η = 0. This implies that the Appel polynomials are power functions in the case of χ1 = 0. Thus throughout below we assume that χ1 > 0. Definition 2. We say that P (y)isafunctionoftypeA(a) if there exists a number a>0 such that P (a)=0,P(y) ≤ 0in[0,a), (3) P (y) increases in [a, ∞). The following result contains a simple criteria for a function P2(y)=−C1Q1(y)+C2Q2(y),C1,C2 > 0, to be of type A(a). 2 Lemma 3. Let χ2 >χ1. The polynomial P2(y)=−C1Q1(y)+C2Q2(y),C1,C2 > 0, is a function of type A(a) if and only if the coefficients C1 and C2 are such that C1 χ2 ≤ − χ1. C2 χ1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 158 YU. S. MISHURA AND V. V. TOMASHYK In this case, the number a for which the polynomial P2(y) satisfies property (3) is given by 2 ∗ 1 C1 1 C1 a = a2 = + 2 + χ2 + χ1. 2 C2 4 C2 Proof. We represent the polynomial P2(y) in terms of the cumulants 2 P2(y)=C2(y − χ1) − C1(y − χ1) − C2χ2. Then the roots of the polynomial P (y)aregivenby 2 2 ∗ 1 C1 ± 1 C1 a1,2 = χ1 + 2 + χ2. 2 C2 4 C2 Since the maximal root of this polynomial is positive and P2(y)isincreasinginthe semiaxis on the right of the maximal root, the system of conditions (3) is equivalent to the inequality a∗ ≤ 0. We write the latter condition explicitely as 1 2 1 C1 − 1 C1 ≤ χ1 + 2 + χ2 0, 2 C2 4 C2 or, equivalently, − 2 ≥ (4) C2(χ2 χ1) C1χ1. The left hand side of inequality (4) is positive by the assumptions of the lemma. This implies the following restrictions on the coefficients C1 and C2: C1 χ2 ≤ − χ1. C2 χ1 Lemma 3 is proved. + k ∞ ∈{ } 2 Theorem 1. Let E ξ<0 and E(ξ ) < for k 1, 2 . Assume that χ2 >χ1 and that the coefficients C1 and C2 satisfy the assumptions of Lemma 3. Then the stopping ∗ { ≥ | ≥ ∗} time τ2 =inf k 0 Xk a2 is optimal for the random walk X =(X0,X1,X2,...) + + 2 with the reward function g(x)=−C1x + C2(x ) . The proof of this result is given in [6]. 3. Main result Consider the reward function + + 2 + n g(x)=C1x + C2(x ) + ...+ Cn(x ) ,Cn > 0.
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