Martingale differences and Doob decomposition

A martingale difference sequence consists of integrable time-k measurable variables whose conditional means given time k minus one are zero. Summing these differences from an integrable initial value gives a martingale. Conversely, the increments of every martingale form such a sequence. The differences need not be independent or identically distributed; their conditional distributions may change with the past. Zero unconditional mean follows by taking expectations, but zero unconditional mean alone does not establish the stronger conditional property.

When differences are square integrable, distinct increments are orthogonal in L2. For an earlier increment and a later one, condition their product on the information immediately before the later increment. The earlier factor is measurable there and the later factor has zero conditional mean. Square integrability justifies the product's integrability by Cauchy–Schwarz. Hence the variance of a sum of martingale differences is the sum of their variances, even though independence may fail. Conditional variances can still be random and will later form the predictable bracket.

For an integrable adapted process X, define the predictable component A by summing the conditional means of its increments. At time n this sum is measurable for the information at n minus one, because each summand uses no later information. Set the martingale component to X minus A. Its next increment is precisely the conditionally centered difference, so it is a martingale. All terms are integrable at each finite time. No square-moment hypothesis is needed for this construction.

To see uniqueness, compare two decompositions with predictable components starting at zero. The difference of their martingale components is also predictable and begins at zero. Its time-k increment is known at time k minus one, while the martingale property says its conditional mean is zero. A known integrable variable equals its own conditional expectation, so every such increment is zero. This argument identifies exactly why predictability, rather than merely adaptedness, is needed for a canonical decomposition.

If X is a submartingale, each conditional drift increment is nonnegative, so A is increasing. If X is a supermartingale, the same convention makes A decreasing. For a general integrable adapted process, A may change direction, and describing it as increasing would be false. The decomposition is algebraically available in all three cases; the submartingale inequality is the extra condition that makes its drift monotone. This sign distinction is useful when deciding whether a proposed compensation has the correct orientation.

For a fair random walk, its square has one unit of expected increase per step, so subtracting time produces a martingale. A cubic function has a drift proportional to the present position, whose sign changes. Such examples show that a nonlinear function of a martingale need not be a martingale itself. Convex functions give submartingales when integrability permits conditional Jensen, while other functions can create drift without a fixed sign. Compensation must be calculated from the actual conditional increment rather than guessed from an ordinary derivative.