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CHAPTER 1 Martingale Theory We review basic facts from martingale theory. We start with discrete- time parameter martingales and proceed to explain what modifications are needed in order to extend the results from discrete-time to continuous-time. The Doob-Meyer decomposition theorem for continuous semimartingales is stated but the proof is omitted. At the end of the chapter we discuss the quadratic variation process of a local martingale, a key concept in martin- gale theory based stochastic analysis. 1. Conditional expectation and conditional probability In this section, we review basic properties of conditional expectation. Let (W, F , P) be a probability space and G a s-algebra of measurable events contained in F . Suppose that X 2 L1(W, F , P), an integrable ran- dom variable. There exists a unique random variable Y which have the following two properties: (1) Y 2 L1(W, G , P), i.e., Y is measurable with respect to the s-algebra G and is integrable; (2) for any C 2 G , we have E fX; Cg = E fY; Cg . This random variable Y is called the conditional expectation of X with re- spect to G and is denoted by E fXjG g. The existence and uniqueness of conditional expectation is an easy con- sequence of the Radon-Nikodym theorem in real analysis. Define two mea- sures on (W, G ) by m fCg = E fX; Cg , n fCg = P fCg , C 2 G . It is clear that m is absolutely continuous with respect to n. The conditional expectation E fXjG g is precisely the Radon-Nikodym derivative dm/dn. If Y is another random variable, then we denote E fXjs(Y)g simply by E fXjYg. Here s(Y) is the s-algebra generated by Y. The conditional probability of an event A is defined by P fAjG g = E fIAjG g . The following two examples are helpful. EXAMPLE 1.1. Suppose that the s-algebra G is generated by a partition: ¥ W = [i=1 Ai, Ai \ Aj = Æ if i 6= j, 1 2 1. MARTINGALE THEORY and P fAig > 0. Then E fXjG g is constant on each Ai and is equal to the average of X on Ai, i.e., 1 Z E fXjG g (w) = X dP, w 2 Ai. P fAig Ai EXAMPLE 1.2. The conditional expectation E fXjYg is measurable with respect to s(Y). By a well known result, it must be a Borel function f (Y) of Y. We usually write symbolically f (y) = E fXjY = yg . Suppose that (X, Y) has a joint density function p(x, y) on R2. Then we can take R R xp(x, y) dy f (y) = R . R p(x, y) dx The following three properties of conditional expectation are often used. (1) If X 2 G , then E fXjG g = X; more generally, if X 2 G then E fXYjG g = XE fYjG g . (2) If G1 ⊆ G2, then E fXjG1jG2g = E fXjG2jG1g = E fXjG1g . (3) If X is independent of G , then E fXjG g = E fXg . The monotone convergence theorem, the dominated convergence theo- rem, and Fatou’s lemma are three basic convergence theorems in Lebesgue integration theory. They still hold when the usual expectation is replaced by the conditional expectation with respect to an aribtary s-algebra. 2. Martingales A sequence F∗ = fFn, n 2 Z+g of increasing s-algebras on W is called a filtration (of s-algebras). The quadruple (W, F∗, F , P) with Fn ⊆ F is a filtered probability space. We usually assume that ¥ de f _ F = F¥ = Fn, n=0 the smallest s-algebra containing all Fn. Intuitively Fn represents the in- formation of an evolving random system under consideration accumulated up to time n. 2. MARTINGALES 3 A sequence of random variables X = fXng is said to be adapted to the filtration F∗ if Xn is measurable with respect to Fn for all n. The filtration X F∗ generated by the sequence X is X Fn = s fXi, i ≤ ng . It is the smallest filtration to which the sequence X is adapted. DEFINITION 2.1. A sequence of integrable random variables X = fXn, n 2 Z+g on a filtered probability space (W, F∗, P) is called a martingale with respect to F∗ if X is adapted to F∗ and E fXnjFn−1g = Xn for all n. It is called a submartingale or supermartingale if in the above relation = is replaced by ≥ or ≤, respectively. If the reference filtration is not explicitly mentioned, it is usually under- stood what it should be from the context. In many situations, we simply take the reference filtration is the one generated by the sequence itself. REMARK 2.2. If X is a submartingale with respect to some filtration F∗, X then it is also a martingale with respect to its own filtration F∗ . The definition of a supermartingale is rather unfortunate, for EXn ≤ EXn−1, that is, the sequence of expected values is decreasing. Intuitively a martingale represents a fair game. The defining property of a martingale can be written as E fXn − Xn−1jFn−1g = 0. We can regard the difference Xn − Xn−1 as a gambler’s gain at the nth play of a game. The above equation says that even after applying all the infor- mation and knowledge he has accumulated up to time n − 1, his expected gain is still zero. EXAMPLE 2.3. If fXng is a sequence of independent and integrable ran- dom variables with mean zero, then the partial sum Sn = X1 + X2 + ··· + Xn is a martingale. EXAMPLE 2.4. (1) If fXng is a martingale and f : R 7! R a convex func- tion such that each f (Xn) is integrable, then f f (Xn)g is a submartingale. (2) fXng is a submartingale and f : R 7! R a convex and increasing function such that each f (Xn) is integrable, then f f (Xn)g is a submartingale. EXAMPLE 2.5. (martingale transform) Let M be a martingale and Z = fZng adapted. Suppose further that each Zn is uniformly bounded. Define n Nn = ∑ Zi−1(Mi − Mi−1). i=1 4 1. MARTINGALE THEORY Then N is a martingale. Note that in the general summand, the multi- plicative factor Zi−1 is measurable with respect to the left time point of the martingale difference Mi − Mi−1. EXAMPLE 2.6. (Reverse martingale) Suppose that fXng is a sequence of i.i.d. integrable random variables and X + X + ··· X Z = 1 2 n . n n Then fZng is a reverse martingale, which means that E fZnjGn+1g = Zn+1, where Gn = s fSi, i ≥ ng . If we write Wn = Z−n and Hn = G−n for n = −1, −2, . .. Then we can write E fWnjHn−1g = Wn−1. 3. Basic properties Suppose that X = fXng is a submartingale. We have EXn ≤ EXn+1. Thus on average a submartingale is increasing. On the other hand, for a martingale we have EXn = EXn+1, which shows that it is purely noise. The Doob decomposition theorem claims that a submartingale can be decom- posed uniquely into the sum of a martingale and an increasing sequence. The following example shows that the uniqueness question for the decom- position is not an entirely trivial matter. EXAMPLE 3.1. Consider Sn, the sum of a sequence of independent and 2 square integrable random variables with mean zero. Then Sn is a sub- martingale. We have obviously 2 2 2 2 Sn = Sn − ESn + ESn. 2 n 2 2 2 From ESn = ∑i=1 EXi it is easy to verify that Mn = Sn − ESn is a martin- gale. Therefore the above identity is a decomposition of the submartingale 2 Sn into the sum of a martingale and an increasing process. On the other hand, n n 2 2 Sn = 2 ∑ Si−1Xi + ∑ Xi . i=1 i=1 The first sum on the right side is a martingale (in the form of a martingale transform). Thus the above gives another such decomposition. In general these two decompositions are different. The above example shows that in order to have a unique decomposition we need further restrictions. DEFINITION 3.2. A sequence Z = fZng is said to be predictable with respect to a filtration F∗ if Fn if Zn 2 Fn−1 for all n ≥ 0. 4. OPTIONAL SAMPLING THEOREM 5 THEOREM 3.3. (Doob decomposition) Let X be a submartingale. Then there is a unique increasing predictable process Z with Z0 = 0 and a martingale M such that Xn = Mn + Zn. PROOF. Suppose that we have such a decomposition. Conditioning on Fn−1 in Zn − Zn−1 = Xn − Xn−1 − (Mn − Mn−1), we have Zn − Zn−1 = E fXn − Xn−1jFn−1g . The right side is nonnegative if X is a submartingale. This shows that a Doob decomposition, if exists, must be unique. It is now clear how to pro- ceed to show the existence. We define n Zn = ∑ E fXi − Xi−1jFi−1g . i=1 It is clear that X is increasing, predictable, and Z0 = 0. Define Mn = Xn − Zn. We have Mn − Mn−1 = Xn − Xn−1 − E fXn − Xn−1jFn−1g , from which it is easy to see that E fMn − Mn−1jFn−1g = 0. This shows that M is a martingale. 4. Optional sampling theorem The concept of a martingale derives much of its power from the option- al sampling theorem we will discuss in this section. Most interesting events concerning a random sequence occurs not at a fixed constant time, but at a random time.
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