Partial product martingale and Kakutani's theorem without independence

Question

Let $S_0, \ldots, S_n$ be positive random variables with $S_0 = 1$ and of the form \begin{equation} \tag{1} S_n = S_0 \prod_{i=1}^{n} (1+X_i). \end{equation} The standard assumption (for example in the Kakutani's theorem) is that if the $(X_i)$ are independent and centered, then $(S_n)$ is a martingale. But this is just a sufficient condition. More precisely, we have the following necessary and sufficient condition: $(S_n)$ is a martingale if and only if $\prod_{i=1}^{n} (1+X_i)$ is integrable and \begin{equation} \tag{2} \mathbb{E}[X_n \, \vert \, \sigma(S_0, \ldots, S_{n-1}) ] = 0. \end{equation} My feeling is that there are examples where $(2)$ is satisfied and $(S_n)$ is a martingale, but the $X_n$ are not independent. Indeed, a constant conditional expectation does not imply that the random variable has to be independent of the $\sigma$-algebra. However, in this specific situation with positive random variables $S_n$ and the product form $(1)$, I fail to construct an example. Are there any good examples of dependent sequences $X_n$ having this property? I guess we need dependent but probably uncorrelated $X_n$ such that the $S_n$ are still positive and a martingale.

Answer 1

It seems that what is required of $X_n$ is that it is a martingale difference sequence, i.e. $X_n = Y_n- Y_{n-1}$, where $Y_n$ is a martingale. You can google that term (martingale difference) to find tons of examples. One of my personal favorites is generated by $Y_n = \sum_{j=1}^n A_j\epsilon_j$, where $A_j = f(\epsilon_{j-1},...,\epsilon_1)$, and the $\epsilon_i$ are iid mean zero random variables. $A_j$ can be thought of as a betting strategy having observed the past outcomes of a fair game. Another similar example is an ARCH/GARCH process.

In your product example, positivity is preserved so long as $X_i >-1$. For example, if $X_n = A_n \epsilon_n$, then so long as $-1 < A_n,\epsilon_n < 1$, $S_n$ is a positive martingale.

Answer 2

Given a (discrete time) filtration $(\mathcal F_n)$ on a probability space $(\Omega,\mathcal F,P)$ (with $\mathcal F_0=\{\emptyset,\Omega)$, say), let $\{Y_1, Y_2,\ldots\}$ be any adapted sequence of strictly positive integrable random variables. Define $$ Z_k:={Y_k\over E[Y_k\mid\mathcal F_{k-1}]},\qquad k\ge 1, $$ and $$ S_n:=\prod_{k=1}^n Z_k,\qquad n\ge 0, $$ $(S_0=1)$. Finally, assume $E[S_n]<\infty$ for each $n\ge 1$. (This is automatic when the $Y_k$ are independent and generate $(\mathcal F_n)$.) Then $(S_n)$ is a martingale.