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A Short Proof of the Doob–Meyer Theorem

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A Short Proof of the Doob–Meyer Theorem Available online at www.sciencedirect.com Stochastic Processes and their Applications 122 (2012) 1204–1209 www.elsevier.com/locate/spa A short proof of the Doob–Meyer theorem Mathias Beiglbock¨ ∗, Walter Schachermayer, Bezirgen Veliyev Faculty of Mathematics, Vienna University, Nordbergstr. 15, 1090 Wien, Austria Received 1 March 2011; received in revised form 21 October 2011; accepted 1 December 2011 Available online 9 December 2011 Abstract Every submartingale S of class D has a unique Doob–Meyer decomposition S D M C A, where M is a martingale and A is a predictable increasing process starting at 0. We provide a short proof of the Doob–Meyer decomposition theorem. Several previously known arguments are included to keep the paper self-contained. ⃝c 2011 Elsevier B.V. Open access under CC BY-NC-ND license. MSC: 60G05; 60G07 Keywords: Doob–Meyer decomposition; Komlos lemma 1. Introduction Throughout this article we fix a probability space .Ω; F; P/ and a right-continuous complete filtration .Ft /0≤t≤T . An adapted process .St /0≤t≤T is of class D if the family of random variables Sτ where τ ranges through all stopping times is uniformly integrable [9]. The purpose of this paper is to give a short proof of the following. Theorem 1.1 (Doob–Meyer). Let S D .St /0≤t≤T be a cadl` ag` submartingale of class D. Then, S can be written in a unique way in the form S D M C A (1) where M is a martingale and A is a predictable increasing process starting at 0. ∗ Corresponding author. Tel.: +43 1 4277 50724; fax: +43 1 4277 50727. E-mail address: [email protected] (M. Beiglbock).¨ 0304-4149 ⃝c 2011 Elsevier B.V.Open access under CC BY-NC-ND license. doi:10.1016/j.spa.2011.12.001 M. Beiglbock¨ et al. / Stochastic Processes and their Applications 122 (2012) 1204–1209 1205 D 1 Doob [4] noticed that in discrete time an integrable process S .Sn/nD1 can be uniquely represented as the sum of a martingale M and a predictable process A starting at 0; in addition, the process A is increasing iff S is a submartingale. The continuous time analogue, Theorem 1.1, goes back to Meyer [9,10], who introduced the class D and proved that every submartingale S D .St /0≤t≤T can be decomposed in the form (1), where M is a martingale and A is a natural process. The modern formulation is due to Doleans-Dade´ [2,3] who obtained that an increasing process is natural iff it is predictable. Further proofs of Theorem 1.1 were given by Rao [11], Bass [1] and Jakubowski [5]. Rao works with the σ .L1; L1/-topology and applies the Dunford–Pettis compactness criterion to obtain the continuous time decomposition as a weak-L1 limit from discrete approximations. To obtain that A is predictable one then invokes the theorem of Doleans-Dade.´ Bass [1] gives a more elementary proof based on the dichotomy between predictable and totally inaccessible stopping times. Jakubowski [5] proceeds as Rao, but notices that predictability of the process A can also be obtained through an application of Komlos’ lemma [8]. This is also our starting point. Indeed the desired decomposition can be obtained from a trivial L2-Komlos lemma, making the Dunford–Pettis criterion obsolete. 2. Proof of Theorem 1.1 The proof of uniqueness is standard and we have nothing to add here; see for instance [7, Lemma 25.11]. For the remainder of this article we work under the assumptions of Theorem 1.1 and fix T D 1 for simplicity.1 n Denote by Dn and D the set of n-th resp. all dyadic numbers j=2 in the interval T0; 1U. For n D each n, we consider the discrete time Doob decomposition of the sampled process S .St /t2Dn , n n n VD that is, we define A ; M by A0 0, n n At − At−1=2n VD ETSt − St−1=2n jFt−1=2n U and (2) n n Mt VD St − At (3) n n so that .Mt /t2Dn is a martingale and .At /t2Dn is predictable with respect to .Ft /t2Dn . The idea of the proof is, of course, to obtain the continuous time decomposition (1) as a limit, or rather, as an accumulation point of the processes Mn; An; n ≥ 1. Clearly, in infinite dimensional spaces a (bounded) sequence need not have a convergent subsequence. As a substitute for the Bolzano–Weierstrass Theorem we establish the Komlos- type Lemma 2.1 in Section 2.1. n In order to apply this auxiliary result, we require that the sequence .M1 /n≥1 is uniformly integrable. This follows from the class D assumption as shown by Rao [11]. To keep the paper self-contained, we provide a proof in Section 2.2. Finally, in Section 2.3, we obtain the desired decomposition by passing to a limit of the discrete time versions. As the Komlos-approach guarantees convergence in a strong sense, predictability of the process A follows rather directly from the predictability of the approximating processes. This idea is taken from [5]. 1 The extension to the infinite horizon case is straightforward, in this case it is appropriate to assumethat S is of class DL rather than class D. 1206 M. Beiglbock¨ et al. / Stochastic Processes and their Applications 122 (2012) 1204–1209 2.1. Komlos’ lemma Following Komlos [8],2 it is sometimes possible to obtain an accumulation point of a bounded sequence in an infinite dimensional space if appropriate convex combinations are taken into account. A particularly simple result of this kind holds true if . fn/n≥1 is a bounded sequence in a Hilbert space. In this case A D sup inffkgk2 V g 2 convf fn; fnC1;:::gg n≥1 is finite and for each n we may pick some gn 2 convf fn; fnC1;:::g such that kgnk2 ≤ A C 1=n. If n is sufficiently large with respect to " > 0, then k.gk C gm/=2k2 > A − " for all m; k ≥ n and hence 1 2 kg − g k2 D 2kg k2 C 2kg k2 − kg C g k2 ≤ 4 A C − 4.A − "/2: k m 2 k 2 m 2 k m 2 n By completeness, .gn/n≥1 converges in k:k2. By a straight forward truncation procedure this Hilbertian Komlos lemma yields an L1-version which we will need subsequently.3 Lemma 2.1. Let . fn/n≥1 be a uniformly integrable sequence of functions on a probability space .Ω; F; P/. Then there exist functions gn 2 conv. fn; fnC1; : : :/ such that .gn/n≥1 converges in k:kL1.Ω/. 2 .i/ VD .i/ 2 2 Proof. For i; n N set fn fn1fj fn|≤ig such that fn L .Ω/. We claim that there exist for every n convex weights λn; : : : ; λn such that the functions n Nn λn f .i/ C···C λn f .i/ converge in L2.Ω/ for every i 2 . n n Nn Nn N To see this, one first uses the Hilbertian lemma to find convex weights λn; : : : ; λn such that n Nn n .1/ n .1/ (λ f C· · ·Cλ f / ≥ converges. In the second step, one applies the lemma to the sequence n n Nn Nn n 1 n .2/ n .2/ (λ f C···C λ f / ≥ , to obtain convex weights which work for the first two sequences. n n Nn Nn n 1 Repeating this procedure inductively we obtain sequences of convex weights which work for the first m sequences. Then a standard diagonalization argument yields the claim. .i/ By uniform integrability, limi!1 k fn − fnk1 D 0, uniformly with respect to n. Hence, once again, uniformly with respect to n, k n .i/ C···C n .i/ − n C···C n k D lim (λn fn λN f / (λn fn λN fNn / 1 0: i!1 n Nn n n n 1 Thus (λ f C···C λ f / ≥ is a Cauchy sequence in L .Ω/. n n Nn Nn n 1 2.2. Uniform integrability of the discrete approximations n Lemma 2.2 ([11]). The sequence .M1 /n≥1 is uniformly integrable. 2 Indeed, [8] considers Cesaro sums along subsequences rather then arbitrary convex combinations. But for our purposes, the more modest conclusion of Lemma 2.1 is sufficient. 3 Lemma 2.1 is also a trivial consequence of Komlos’ original result [8] or other related results that have been established through the years. Cf. [6, Chapter 5.2] for an overview. M. Beiglbock¨ et al. / Stochastic Processes and their Applications 122 (2012) 1204–1209 1207 Proof. Subtracting ETS1jFt U from St we may assume that S1 D 0 and St ≤ 0 for all 0 ≤ t ≤ 1. n D − n Then M1 A1, and for every .Ft /t2Dn -stopping time τ n n n Sτ D −ETA1jFτ UC Aτ : (4) n 1 ≥ We claim that .A1/nD1 is uniformly integrable. For c > 0; n 1 define n n τn.c/ D inf . j − 1/=2 V A j=2n > c ^ 1: From An ≤ c and (4) we obtain S ≤ −ETAnjF UC c. τn.c/ τn.c/ 1 τn.c/ Thus, Z Z Z n D T nj U ≤ − A1 dP E A1 Fτn.c/ dP cP τn.c/ < 1 Sτn.c/ dP: f n g f g f g A1>c τn.c/<1 τn.c/<1 f g ⊆ f c g Note τn.c/ < 1 τn. 2 / < 1 , hence, by (4) Z Z n n − c D − Sτn dP A1 A c dP c .
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