DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES

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1 Communications on Stochastic Analysis Vol. 9, No. 4 (215) Serials Publications DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES HUI-HSIUNG KUO AND KIMIAKI SAITÔ* Abstract. We study the discrete parameter case of near-martingales, nearsubmartingales, and near-supermartingales. In particular, we prove Doob s decomposition theorem for near-submartingales. This generalizes the classical case for submartingales. 1. Motivation From Non-adapted Stochastic Integral Let B(t), t, be a Brownian motion starting at and {F t the filtration given by B(t), namely, F t = σ{b(s); s t, t. The Itô integral b f(t) db(t) a (see, e.g., the book [8]) is defined for {F t -adapted stochastic processes f(t) with almost all sample paths being in L 2 [a, b]. Several extensions of the Itô theory of stochastic integration to cover non-adapted integrands have been introduced and extensively studied by, just to mention a few names, Buckdahn [3], Dorogovtsev [4], Hitsuda [5], Itô [6], Kuo Potthoff [1], León Protter [12], Nualart Pardoux [13], Pardoux Protter [14], Russo Vallois [15], and Skorokhod [16]. In particular, in his lecture for the 1976 Kyoto Symposium, Itô [6] gave rather elegant ideas to define the following non-adaptive stochastic integral (I) B(1) db(s), t 1, (1.1) namely, enlarging the σ-field F t to G t = σ{b(1), B(s); s t, t 1, so that the integrand B(1) is adaptive and B(t) is a quasimartingale with respect to the filtration {G t. Then the stochastic integral in equation (1.1) is defined as a stochastic integral with respect to a quasimartingale and has the value (I) B(1) db(s) = B(1)B(t), t 1. (1.2) On the other hand, the Hitsuda Skorokhod integral (see [5] [16]) can be expressed in terms of a white noise integral (see the book [7]) and has the value (HS) B(1) db(s) = s B(1) ds = B(1)B(t) t, t 1. (1.3) Received ; Communicated by the editors. 21 Mathematics Subject Classification. Primary 6G42, 6G48; Secondary 6G5, 6H5. Key words and phrases. Brownian motion, stochastic integral, Hitsuda Skorokhod integral, conditional expectation, martingale, near-martingale, near-submartingale, near-supermartingale, Doob s decomposition theorem, instantly independent sequence. *This work was supported by JSPS Grant-in-Aid Scientific Research 15K

2 468 HUI-HSIUNG KUO AND KIMIAKI SAITÔ Being motivated by Itô s ideas and observing the different values of equations (1.2) and (1.3), we have defined in [1] [2] the stochastic integral B(1) db(s) in the following way. Decompose the integrand B(1) as B(1) = B(t) + ( B(1) B(t) ), where the first term B(t) is the Itô part of B(1) and the second term B(1) B(t) is the counterpart of B(1). For the Itô part, the evaluation points are the left endpoints of subintervals, while the evaluation points for the counterpart are the right endpoints of subintervals. Thus for t 1, we have [ ( )] B(1) db(s) = B(s) + B(1) B(s) db(s) = lim n i=1 = lim n i=1 = lim n n [ B(si 1 ) + ( B(1) B(s i ) )]( B(s i ) B(s i 1 ) ) n [ ( B(1) B(si ) B(s i 1 ) )]( B(s i ) B(s i 1 ) ) ( B(1) n ( B(si ) B(s i 1 ) ) i=1 n ( B(si ) B(s i 1 ) ) ) 2 = B(1)B(t) t, (1.4) where the limit is convergence in probability. Note that this value is the same as the Hitsuda Skorokhod integral in equation (1.3). There is an intrinsic difference between the stochastic processes X t = B(1)B(t) t, Y t = B(1)B(t), t 1, (1.5) given by equations (1.4) and (1.2), respectively. For any s t, we see that In particular, put t = s to get It follows from equations (1.6) and (1.7) that i=1 E[X t F s ] = B(s) 2 s. (1.6) E[X s F s ] = B(s) 2 s. (1.7) E[X t F s ] = E[X s F s ], s t. (1.8) On the other hand, it is easy to check that the stochastic process Y t = B(1)B(t) in equation (1.5) does not satisfy equation (1.8). This leads to the following concept introduced in [11]. Definition 1.1. A stochastic process X t with E X t < for a t b is called a near-martingale with respect to a filtration {F t if it satisfies the condition in equation (1.8). We can define near-submartingale and near-supermartingale with respect to a filtration {F t by the following respective conditions: and E[X t F s ] E[X s F s ], s t, (1.9) E[X t F s ] E[X s F s ], s t.

3 DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES 469 Observe that if a stochastic process X t is adapted to a filtration {F t, then nearmartingale, near-submartingale, and near-supermartingale reduce to martingale, submartingale, and supermartingale, respectively. In this paper we will study the discrete parameter case of near-martingales and near-submartingales. In particular, we will prove Doob s decomposition theorem for near-submartingales. 2. Near-martingales and Near-submartingales Let {F n ; 1 n N be a fixed filtration, i.e., an increasing sequence of σ-fields. Definition 2.1. A sequence X n, 1 n N, of integrable random variables is called a near-martingale with respect to {F n ; 1 n N if E[X n+1 F n ] = E[X n F n ], 1 n N 1. (2.1) Remark 2.2. It is easy to see that the equality in equation (2.1) is equivalent to the equality: E[X m F n ] = E[X n F n ], 1 n m N. (2.2) Similarly, we can define near-submartingale and near-supermartingale just by replacing the equality sign in equation (2.1) with and, respectively. They also have the corresponding equivalent conditions as in equation (2.2). Obviously, if a sequence X n, 1 n N, is adapted to {F n ; 1 n N, then near-martingale, near-submartingale, and near-supermartingale are martingale, submartingale, and supermartingale, respectively. Example 2.3. Take a sequence ξ 1, ξ 2,..., ξ N of independent random variables with mean. Let {F n be the filtration given by F n = σ{ξ k ; 1 k n. Put S n = ξ ξ n, X n = S N S n, 1 n N. (2.3) The sequence S n, 1 n N, is a martingale. On the other hand, E[X n+1 F n ] = E[ξ n ξ N F n ] = E(ξ n ξ N ) =. Similarly, we have E[X n F n ] =. Thus E[X n+1 F n ] = E[X n F n ], which shows that X n, 1 n N, is a near-martingale. Furthermore, suppose ξ n, n 1, is a sequence of independent random variables with mean. For fixed N, X n = S N S n, 1 n N, is a near-martingale as shown above. However, X n = S N S n, n N, is a martingale. Example 2.4. Let ξ 1, ξ 2,..., ξ N be a sequence of independent random variables with mean and var(ξ n ) = σ 2 n. Let F n = σ{ξ k ; 1 k n. Put S n = ξ ξ n, X n = S n S N n σk, 2 1 n N. (2.4)

4 47 HUI-HSIUNG KUO AND KIMIAKI SAITÔ It is easy to check that [ n+1 E[X n+1 F n ] = E S n+1 S N σ 2 k ] F n = E[(S n + ξ n+1 )(S n + ξ n+1 + ξ N ) n+1 F n ] n+1 = Sn 2 + σn+1 2 = S 2 n Similarly, we can easily derive σ 2 k n σk. 2 (2.5) E[X n F n ] = S 2 n σ 2 k n σk. 2 (2.6) It follows from equations (2.5) and (2.6) that E[X n+1 F n ] = E[X n F n ]. Hence the sequence X n = S n S N n σ2 k, 1 n N, is a near-martingale. Moreover, let ξ n, n 1, be a sequence of independent random variables with mean and var(ξ n ) = σn. 2 Take F n = σ{ξ k ; 1 k n. Define S n and X n as in equation (2.4). For fixed N, the sequence X n, 1 n N, is a near-martingale as shown above. On the other hand, the sequence X n, n N, is a martingale. Theorem 2.5. Let S n, 1 n N, be a square integrable martingale with respect to a filtration {F n ; 1 n N. Then is a near-martingale. Proof. Note that Hence we have V n = S n (S N S n ), 1 n N, V n+1 V n = (S n+1 S n )S N S 2 n+1 + S 2 n. (2.7) E[V n+1 V n F n ] = E[(S n+1 S n )S N F n ] E[S 2 n+1 F n ] + E[S 2 n F n ] = E { E[(S n+1 S n )S N F n+1 ] F n E[S 2 n+1 F n ] + S 2 n = E { (S n+1 S n )E[S N F n+1 ] F n E[S 2 n+1 F n ] + S 2 n = E { (S n+1 S n )S n+1 F n E[S 2 n+1 F n ] + S 2 n = S n E[S n+1 F n ] + S 2 n = S 2 n + S 2 n =. Hence E[V n+1 F n ] = E[V n F n ] and so V n, 1 n N, is a near-martingale.

5 DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES 471 Theorem 2.6. Suppose S n, n = 1, 2,..., is a square integrable martingale with respect to a filtration {F n ; 1 n N. For a fixed natural number N, let V n = S n (S N S n ), n = 1, 2,.... Then (1) V n, 1 n N, is a near-martingale, (2) V n, n N, is a supermartingale. Proof. The first assertion follows from Theorem 2.5. To prove the second assertion, we use equation (2.7) to show that for n N, E[V n+1 V n F n ] = S N E[(S n+1 S n ) F n ] E[S 2 n+1 F n ] + S 2 n = E[S 2 n+1 F n ] + S 2 n, since S 2 n is a submartingale. Thus E[V n+1 F n ] E[V n F n ] for n N. But the sequence V n, n N, is adapted to the filtration {F n. Therefore, we have E[V n+1 F n ] V n, n N. This shows that V n, n N, is a supermartingale. 3. Doob s Decomposition Theorem In this section we prove Doob s decomposition theorem for near-submartingales. Theorem 3.1. Let X n, n 1, be a near-submartingale with respect to a filtration {F n. Then there exists a unique decomposition with M n and A n satisfying the following conditions: (1) M n, n 1, is a near-martingale. (2) A 1 =. (3) A n is F n 1 -measurable for n 2. (4) A n is inceasing almost surely. X n = M n + A n, n 1, (3.1) Proof. Existence of a decomposition Define A 1 = and M 1 = X 1. Then we have equation (3.1) for n = 1. To find A 2 and M 2 such that X 2 = M 2 + A 2 with desired properties, we take conditional expectation with respect to F 1 : Therefore, we define E[X 2 F 1 ] = E[M 2 F 1 ] + E[A 2 F 1 ] = E[M 1 F 1 ] + A 2 = E[X 1 F 1 ] + A 2. A 2 = E[X 2 F 1 ] E[X 1 F 1 ], M 2 = X 2 A 2. Then we have equation (3.1) for n = 2. Observe that A 2 is F 1 -measurable and A 1 A 2 almost surely since {X n is a near-submartingale.

6 472 HUI-HSIUNG KUO AND KIMIAKI SAITÔ Inductively, we repeat the above arguments to define A n and M n for n 3 by n ( ) A n = E[X k F k 1 ] E[X k 1 F k 1 ], k=2 M n = X n A n. Then we have equation (3.1) for n 3. Notice that A n is F n 1 -measurable and A n 1 A n almost surely since {X n is a near-submartingale. Now, we need to show that M n, n 1, is a near-martingale with respect to {F n. Note that for n 2, we have M n = X n which yields the equality n k=2 ( ) E[X k F k 1 ] E[X k 1 F k 1 ], M n M n 1 = X n X n 1 E[X n F n 1 ] + E[X n 1 F n 1 ]. Then we take conditional expectation with respect to F n 1 to show that E[M n M n 1 F n 1 ] =, namely, E[M n F n 1 ] = E[M n 1 F n 1 ]. Hence M n, n 1, is a near-martingale with respect to {F n. Uniqueness of a decomposition Suppose we have two such decompositions Then we have X n = M n + A n = N n + B n, n 1. (3.2) M n N n = B n A n, n 1. (3.3) For n = 1, we have B 1 = A 1 =. Hence M 1 = N 1. For n 2, take the conditional expectation of equation (3.3) with respect to F n 1 to get E[M n N n F n 1 ] = E[B n A n F n 1 ] = B n A n, (3.4) where in the last equality we have used the fact that A n and B n are F n 1 - measurable. On the other hand, use equation (3.3) for n 1 and the fact that M n and N n are near-martingales to get E[M n N n F n 1 ] = E[M n 1 N n 1 F n 1 ] = E[B n 1 A n 1 F n 1 ] = B n 1 A n 1, (3.5) where the last equality holds since B n 1 and A n 1 are F n 2 -measurable and so are F n 1 -measurable. Thus by equations (3.4) and (3.5), B n A n = B n 1 A n 1, n 2. This equation together with A 1 = B 1 implies that A n = B n almost surely for all n 1. Then by equation (3.2) we have M n = N n almost surely for all n 1. Hence the decomposition is unique.

7 DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES 473 Example 3.2. Let ξ n, n 1, be a sequence of independent random variables with mean and var(ξ n ) = σ 2 n. Take F n = σ{ξ k ; 1 k n. Define S n = ξ ξ n. For fixed N, consider the sequence X n = S n S N, 1 n N. (3.6) First we show that the sequence X n, 1 n N, is a near-submartingale. It is easy to see that On the other hand, we have E[X n+1 F n ] = E[S n+1 S N F n ] = E[(S n + ξ n+1 )(ξ ξ N ) F n ] = E[(S n + ξ n+1 ) 2 F n ] = E[S 2 n + 2S n ξ n+1 + ξ 2 n+1 F n ] = S 2 n + σ 2 n+1. (3.7) E[X n F n ] = E[S n S N F n ] = S n E[S N F n ] = S 2 n. (3.8) By equations (3.7) and (3.8), we have E[X n+1 F n ] E[X n F n ] almost surely. Hence X n, 1 n N, is a near-submartingale. To find the Doob decomposition of X n, 1 n N, recall from Example 2.4 that the sequence n Z n S n S N σk, 2 1 n N, is a near-martingale. This motivates us to define M n and A n by and { S1 S N, if n = 1, M n = S n S N n k=2 σ2 k, if n 2. A n = {, if n = 1, n k=2 σ2 k, if n 2. Note that M n = Z n + σ 2 1. Hence M n is a near-martingale. Then we can easily see that the Doob decomposition of S n S N is given by S n S N = M n + A n, 1 n N. We need to point out a difference between martingale case and near-martingale case. Suppose X n is a square integrable martingale. It is well known that Xn 2 is a submartingale. However, for a square integrable near-martingale X n, it is not true in general that Xn 2 is a near-submartingale. For instance, the sequence X n = S N S n, 1 n N, in Example 2.3 is a near-martingale. However, it is easy to check that Xn, 2 1 n N, is not a near-submartingale. In fact, it is a near-supermartingale.

8 474 HUI-HSIUNG KUO AND KIMIAKI SAITÔ 4. Instantly Independent Sequences Note that martingales must be adapted with respect to an associated filtration. In [11], we introduced the concept of instantly independent stochastic processes, which play the counterpart role of adapted stochastic processes. Thus for the discrete case, we have instantly independent sequences of random variables. Definition 4.1. A sequence {Φ n of random variables is said to be instantly independent with respect to a filtration {F n if Φ n and F n are independent for each n. We have the following two basic properties of instantly independent sequences of random variables. Theorem 4.2. If X n is a near-martingale, then EX n is a constant (independent of n). Conversely, if EX n is a constant and X n is instantly independent, then X n is a near-martingale. Proof. Suppose X n is a near-martingale. Then we have E[X n+1 F n ] = E[X n F n ], n 1. Upon taking expectation, we immediately get EX n+1 = EX n for all n 1. Hence EX n is a constant. Conversely, suppose EX n is a constant and X n is instantly independent with respect to a filtration {F n. Then E[X n+1 F n ] = E { E[X n+1 F n+1 ] F n = E { EX n+1 F n = EX n+1 = c, where c is a constant. On the other hand, since X n and F n are independent, we have E[X n F n ] = EX n = c. Hence E[X n+1 F n ] = E[X n F n ] and so X n, n 1, is a near-martingale. Theorem 4.3. Suppose X n is a square integrable martingale and Φ n is a square integrable sequence of instantly independent random variables with EΦ n being a constant. Then the product X n Φ n is a near-martingale. Proof. Using the assumptions we can easily derive E[X n+1 Φ n+1 F n ] = E { E[X n+1 Φ n+1 F n+1 ] F n = E { X n+1 E[Φ n+1 F n+1 ] Fn = E { X n+1 EΦ n+1 F n = EΦ n+1 E[X n+1 F n ] = cx n, (4.1) where c = EΦ n is a constant. On the other hand, we have E[X n Φ n F n ] = X n E[Φ n F n ] = X n EΦ n = cx n. (4.2)

9 DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES 475 It follows from equations (4.1) and (4.2) that E[X n+1 Φ n+1 F n ] = E[X n Φ n F n ] almost surely. Hence X n is a near-martingale. Example 4.4. Let ξ 1, ξ 2,..., ξ N be a sequence of independent random variables with mean and finite variances. Let F n = σ{ξ k ; 1 k n. Put S n = ξ 1 + ξ ξ n. Then S n is a martingale with respect to the filtration {F n. Let θ be a real-vlaued function on R. For fixed N, assume that the random variables θ(s N S n ), 1 n N, are square integrable. Then the following sequence Φ n = θ(s N S n ) Eθ(S N S n ), 1 n N, is instantly independent with respect to the filtration {F n with mean. Hence by Theorem 4.3 the sequence ) Y n = S n (θ(s N S n ) Eθ(S N S n ), 1 n N, is a near-martingale. Acknowledgment. The mathematical concepts and the results in this paper were obtained in many discussions with K. Saitô during Kuo s visits to Meijo University since 211. Kuo would like to give his deepest appreciation to Professor Saitô for the invitations and for the warm hospitality. References 1. Ayed, W. and Kuo, H.-H.: An extension of the Itô integral, Communications on Stochastic Analysis 2, no. 3 (28) Ayed, W. and Kuo, H.-H.: An extension of the Itô integral: toward a general theory of stochastic integration, Theory of Stochastic Processes 16(32), no. 1 (21) Buckdahn, R.: Anticipative Girsanov transformations, Probab. Th. Rel. Fields 89 (1991) Dorogovtsev, A. A.: Itô Volterra equations with an anticipating right-hand side in the absence of moments, Infinite-dimensional Stochastic Analysis (Russian) 41 5, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, Hitsuda, M.: Formula for Brownian partial derivatives, Second Japan-USSR Symp. Probab. Th.2 (1972) Itô, K.: Extension of stochastic integrals, Proc. Intern. Symp. Stochastic on Differential Equations, K. Itô (ed.) (1978) 95 19, Kinokuniya. 7. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Kuo, H.-H.: Introduction to Stochastic Integration. Universitext (UTX), Springer, Kuo, H.-H.: The Itô calculus and white noise theory: A brief survey toward general stochastic integration, Communications on Stochastic Analysis 8, no. 1 (214) Kuo, H.-H. and Potthoff, J.: Anticipating stochastic integrals and stochastic differential equations; in: White Noise Analysis: Math. and Appl., T. Hida et al. (eds.), World Scientific (199) Kuo, H.-H., Sae-Tang, A., and Szozda, B.: A stochastic integral for adapted and instantly independent stochastic processes, in Advances in Statistics, Probability and Actuarial Science Vol. I, Stochastic Processes, Finance and Control: A Festschrift in Honour of Robert J. Elliott (eds.: Cohen, S., Madan, D., Siu, T. and Yang, H.), World Scientific, 212,

10 476 HUI-HSIUNG KUO AND KIMIAKI SAITÔ 12. León J. A. and Protter, P.: Some formulas for anticipative Girsanov transformations, in: Chaos Expansions, Multiple Wiener-Itô integrals and Their Applications, C. Houdré and V. Pérez-Abreu (eds.), CRC Press, Nualart, D. and Pardoux, E.: Stochastic calculus with anticipating integrands, Probab. Th. Rel. Fields 78 (1988) Pardoux, E. and Protter, P.: A two-sided stochastic integral and its calculus, Probab. Th. Rel. Fields 76 (1987) Russo, F. and Vallois, P.: Anticipative Stratonovich equation via Zvonkin method, Stochastic Processes and Related Topics (Siegmundsberg, 1994), , Stochastics Monogr., 1, Gordon and Breach, Yverdon, 1996, 16. Skorokhod, A. V.: On a generalization of a stochastic integral, Theory Probab. Appl. 2 (1975) Hui-Hsiung Kuo: Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 783, USA. address: kuo@@math.lsu.edu Kimiaki Saitô: Department of Mathematics, Meijo University, Tenpaku, Nagoya , Japan address: ksaito@meijo-u.ac.jp