(TBD) Martingale

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Originated from gambling, a player will make, on average, no profit with only a limited budget. Paul Lévy proved that the famous martingale startegy is fool. In sprit of Levy’s work, a massive breakthrough was achieved by Joseph Doop. It is a fascinating framework that supports study of stochastic processes, and moreover, a modern probability theory.

I

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An adapted stochastic process $X$ on a filtered probability space $(\Omega, \mathcal{F}, F, P)$ is called a martingale if, for all $s, t \in T$ and $s \leq t$, it satisfies both (i) $X_{t} \in L^1$; (ii) $\operatorname{E}(X_{t} \vert \mathcal{F}_{s}) = X_{s}$ a.s.; The slower a filtration $t \to \mathcal{F}_t$ grows, the easier it is for $X$ to be a martingale. In continuous time, the continuity of the sample paths $t \to X_t$ and the filtration $t \to \mathcal{F}_t$ must be provided. //

The tower property and the stability of $\operatorname{E}$ yields $\operatorname{E}(X_{t} \,\vert\, \mathcal{F}_{s}) = \operatorname{E}[\operatorname{E}(X_{t} \,\vert\, \mathcal{F}_{t}) \,\vert\, \mathcal{F}_{s}] = \operatorname{E}[\operatorname{E}(X \,\vert\, \mathcal{F}_{s}) \,\vert\, \mathcal{F}_{t}]$. It implies the 2nd martingale property $\operatorname{E}(X_{t} \,\vert\, \mathcal{F}_{s}) = \operatorname{E}(\operatorname{E}(X_{t} \,\vert\, \mathcal{F}_{t-h}) \,\vert\, \mathcal{F}_s) = \operatorname{E}(X_{t-h} \,\vert\, \mathcal{F}_s) = \dots = \operatorname{E}(X_{s+h} \,\vert\, \mathcal{F}_s) = X_s$ with some $h \geq 0$. In particular, all martingale has a constant mean $\operatorname{E}X_{t} = \operatorname{E}X_{0}$ for all $t \in T$. By construction, a martingale may be seen as a probabilistic flat line which can be used to model a fair game. A partial sum process $S = (S_t)_{t \in T}$ consists of $S_{t} = \Sigma_{s=0}^{t}X_s$ with ind. $X_s$ whose $\operatorname{E}X_s = 0$ is a popular example of a discrete-time martingale (i.e. the Poisson process and the Wiener process/Brownian motion).

A few adjustments of definitions induce another martingales. We let the martingale differences (i.e increments) be $D_{0} = X_{0}$ and $D_{t} = X_{t} - X_{t-1}$ for all $t \in \mathbb{Z}^{+}$ (#1), so that $X_{t}$ is the sum of $D_{s}$ for all $s \leq t$. An adapted stochastic process $D = (D_{t})_{t \in T}$ is a martingale difference sequence if, for all $s, t \in T$ and $s \leq t$, it satisfies both (i) $D_{t} \in L^1$; (ii) $\operatorname{E}(D_{t} \,\vert\, \mathcal{F}_{s}) = 0$ a.s.; This is almost trivial that $\operatorname{E}D_{t} = 0$ and $\operatorname{E}(D_{t} \,\vert\, \mathcal{F}_{s}) = 0$ a.s. for all $t, s \in \mathbb{Z}^{\geq}$ with $s<t$. That is, a stochastic process $X$ with ind. increments and a constant mean can be refined as a martingale $D$ with a mean zero, where $X$ and $D$ are adpated to a common filtration $F$.

II

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If, for all $0 \leq s \leq t$, the identity is replaced by $\operatorname{E}(X_{t} \,\vert\, \mathcal{F}_{s}) \geq X_{s}$ a.s., then $X$ is called a sub-martingale. Also, if $\operatorname{E}(X_{t} \,\vert\, \mathcal{F}_s) \leq X_s$ a.s., then $X$ is called a super-martingale. (#3). For example, $X$ with ind. increments and a constant mean is a martingale, and is a sub-martingale (super-martingale) if its mean is increasing (decreasing). Note that a simple random walk with $p = 1/2$, in which a mean of each summand is given by $2p-1$, is a martingale (4). That is, a martingale corresponds to fair games, a sub-margingale corresponds to favorable games, and a super-margingale corresponds to unfavorable games.

These three unique types of stochastic process have the following properties. // The martingale properties are preserved under sums of the stochastic processes (thus the collection of martingales with respect to a fixed filtration $F$ forms a vector space) // The sub-martingale and super-martingale properties are preserved under multiplication by a positive constant. // Jensen’s inequality turns martingales into sub-martingales under appropriate conditions.

The Doob decomposition theorem: decomposes a basic stochastic process into a martingale and a predictable process (a definition of predictable process will be followed). // In discrete time, a process $X = (X_t)_{t \in T}$ being adapted to $F = (\mathcal{F}_t)_{t \in T}$ is predictable if $\sigma(X_t) \subset \mathcal{F}_{t-1}$ for all $t \in T$. // The proof came up in 1953 and showed both the existence and the uniqueness. // the Krickeberg decom. (p.490) // the Riesz decom. (p.491) // I.e. Gaussian stochastic processes.

III

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// stopping time theorem::: expectation of stopped martingale is not in general equal to that of a given martingale (i.e. the problem is that $\tau$ is too large), in fact, we have $\operatorname{E}\tau = \infty$ // The optional stopping theorem (or Doob’s optional sampling theorem) which applies to doubling strategies states that

// Doob’s (maximal) inequality::: generalised Kolmogorov’s inequality.

// Doob’s martingale convergence theorems::: If you’re more analysis oriented, martingales, provide (IMO) the most natural proof of the Radon-Nikodym theorem (ok, for separable sigma algebras). The proof is an application of the Martingale Convergence Theorem) // martingale clt

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(#1) I.e. A partial sum process associated with ind. sequences is a martingale. (4) Hence it is a sub-martingale (a super-martingale) for $p > 1/2$ ($p < 1/2$).

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