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| In [[mathematics]], a '''local martingale''' is a type of [[stochastic process]], satisfying the [[Stopping time#Localization|localized]] version of the [[Martingale (probability theory)|martingale]] property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a [[Itō diffusion|driftless diffusion process]] is a local martingale, but not necessarily a martingale.
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| Local martingales are essential in [[stochastic analysis]], see [[Itō calculus#Local_martingales|Itō calculus]], [[semimartingale]], [[Girsanov theorem]].
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| ==Definition==
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| Let (Ω, ''F'', '''P''') be a [[probability space]]; let ''F''<sub>∗</sub> = { ''F''<sub>''t''</sub> | ''t'' ≥ 0 } be a [[filtration (abstract algebra)|filtration]] of ''F''; let X : [0, +∞) × Ω → ''S'' be an ''F''<sub>∗</sub>-[[adapted process|adapted stochastic process]] on set ''S''. Then ''X'' is called an ''F''<sub>∗</sub>'''-local martingale''' if there exists a sequence of ''F''<sub>∗</sub>-[[stopping rule|stopping times]] ''τ''<sub>''k''</sub> : Ω → [0, +∞) such that
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| * the ''τ''<sub>''k''</sub> are [[almost surely]] [[increasing]]: '''P'''[''τ''<sub>''k''</sub> < ''τ''<sub>''k''+1</sub>] = 1;
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| * the ''τ''<sub>''k''</sub> diverge almost surely: '''P'''[''τ''<sub>''k''</sub> → +∞ as ''k'' → +∞] = 1;
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| * the [[stopped process]]
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| ::<math>X_t^{\tau_{k}} := X_{\min \{ t, \tau_k \}}</math>
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| : is an ''F''<sub>∗</sub>-martingale for every ''k''.
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| ==Examples ==
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| ===Example 1===
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| Let ''W''<sub>''t''</sub> be the [[Wiener process]] and ''T'' = min{ ''t'' : ''W''<sub>''t''</sub> = −1 } the [[hitting time|time of first hit]] of −1. The [[stopped process]] ''W''<sub>min{ ''t'', ''T'' }</sub> is a martingale; its expectation is 0 at all times, nevertheless its limit (as ''t'' → ∞) is equal to −1 almost surely (a kind of [[gambler's ruin]]). A time change leads to a process
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| : <math>\displaystyle X_t = \begin{cases}
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| W_{\min(\frac{t}{1-t},T)} &\text{for } 0 \le t < 1,\\
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| -1 &\text{for } 1 \le t < \infty.
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| \end{cases} </math>
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| The process <math> X_t </math> is continuous almost surely; nevertheless, its expectation is discontinuous,
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| : <math>\displaystyle \mathbb{E} X_t = \begin{cases}
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| 0 &\text{for } 0 \le t < 1,\\
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| -1 &\text{for } 1 \le t < \infty.
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| \end{cases} </math>
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| This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as <math> \tau_k = \min \{ t : X_t = k \} </math> if there is such ''t'', otherwise τ<sub>''k''</sub> = ''k''. This sequence diverges almost surely, since τ<sub>''k''</sub> = ''k'' for all ''k'' large enough (namely, for all ''k'' that exceed the maximal value of the process ''X''). The process stopped at τ<sub>''k''</sub> is a martingale.<ref group="details">
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| For the times before 1 it is a martingale since a stopped Brownian motion is. After the instant 1 it is constant. It remains to check it at the instant 1. By the [[bounded convergence theorem]] the expectation at 1 is the limit of the expectation at (''n''-1)/''n'' (as ''n'' tends to infinity), and the latter does not depend on ''n''. The same argument applies to the conditional expectation.
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| </ref>
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| ===Example 2===
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| Let ''W''<sub>''t''</sub> be the [[Wiener process]] and ''ƒ'' a measurable function such that <math> \mathbb{E} |f(W_1)| < \infty. </math> Then the following process is a martingale:
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| : <math>\displaystyle X_t = \mathbb{E} ( f(W_1) | F_t ) = \begin{cases}
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| f_{1-t}(W_t) &\text{for } 0 \le t < 1,\\
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| f(W_1) &\text{for } 1 \le t < \infty;
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| \end{cases} </math>
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| here
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| : <math>\displaystyle f_s(x) = \mathbb{E} f(x+W_s) = \int f(x+y) \frac1{\sqrt{2\pi s}} \mathrm{e}^{-y^2/(2s)} . </math>
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| The [[Dirac delta function]] <math> \delta </math> (strictly speaking, not a function), being used in place of <math> f, </math> leads to a process defined informally as <math> Y_t = \mathbb{E} ( \delta(W_1) | F_t ) </math> and formally as
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| : <math>\displaystyle Y_t = \begin{cases}
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| \delta_{1-t}(W_t) &\text{for } 0 \le t < 1,\\
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| 0 &\text{for } 1 \le t < \infty,
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| \end{cases} </math>
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| where
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| : <math>\displaystyle \delta_s(x) = \frac1{\sqrt{2\pi s}} \mathrm{e}^{-x^2/(2s)} . </math>
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| The process <math> Y_t </math> is continuous almost surely (since <math> W_1 \ne 0 </math> almost surely), nevertheless, its expectation is discontinuous,
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| : <math>\displaystyle \mathbb{E} Y_t = \begin{cases}
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| 1/\sqrt{2\pi} &\text{for } 0 \le t < 1,\\
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| 0 &\text{for } 1 \le t < \infty.
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| \end{cases} </math>
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| This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as <math> \tau_k = \min \{ t : Y_t = k \}. </math>
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| ===Example 3===
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| Let <math> Z_t </math> be the [[Wiener process#Complex-valued Wiener process|complex-valued Wiener process]], and
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| : <math>\displaystyle X_t = \ln | Z_t - 1 | \, . </math>
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| The process <math> X_t </math> is continuous almost surely (since <math> Z_t </math> does not hit 1, almost surely), and is a local martingale, since the function <math> u \mapsto \ln|u-1| </math> is [[harmonic function|harmonic]] (on the complex plane without the point 1). A localizing sequence may be chosen as <math> \tau_k = \min \{ t : X_t = -k \}. </math> Nevertheless, the expectation of this process is non-constant; moreover,
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| : <math>\displaystyle \mathbb{E} X_t \to \infty </math> as <math> t \to \infty, </math>
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| which can be deduced from the fact that the mean value of <math> \ln|u-1| </math> over the circle <math> |u|=r </math> tends to infinity as <math> r \to \infty </math>. (In fact, it is equal to <math> \ln r </math> for ''r'' ≥ 1 but to 0 for ''r'' ≤ 1).
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| == Martingales via local martingales ==
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| Let <math> M_t </math> be a local martingale. In order to prove that it is a martingale it is sufficient to prove that <math> M_t^{\tau_k} \to M_t </math> [[convergence of random variables#Convergence in mean|in ''L''<sup>1</sup>]] (as <math> k \to \infty </math>) for every ''t'', that is, <math> \mathbb{E} | M_t^{\tau_k} - M_t | \to 0; </math> here <math> M_t^{\tau_k} = M_{t\wedge \tau_k} </math> is the stopped process. The given relation <math> \tau_k \to \infty </math> implies that <math> M_t^{\tau_k} \to M_t </math> almost surely. The [[dominated convergence theorem]] ensures the convergence in ''L''<sup>1</sup> provided that
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| : <math>\textstyle (*) \quad \mathbb{E} \sup_k| M_t^{\tau_k} | < \infty </math> for every ''t''.
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| Thus, Condition (*) is sufficient for a local martingale <math> M_t </math> being a martingale. A stronger condition
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| : <math>\textstyle (**) \quad \mathbb{E} \sup_{s\in[0,t]} |M_s| < \infty </math> for every ''t''
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| is also sufficient.
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| ''Caution.'' The weaker condition
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| : <math>\textstyle \sup_{s\in[0,t]} \mathbb{E} |M_s| < \infty </math> for every ''t''
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| is not sufficient. Moreover, the condition
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| : <math>\textstyle \sup_{t\in[0,\infty)} \mathbb{E} \mathrm{e}^{|M_t|} < \infty </math>
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| is still not sufficient; for a counterexample see [[Local martingale#Example 3|Example 3 above]].
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| A special case:
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| : <math>\textstyle M_t = f(t,W_t), </math>
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| where <math> W_t </math> is the [[Wiener process]], and <math> f : [0,\infty) \times \mathbb{R} \to \mathbb{R} </math> is [[smooth function|twice continuously differentiable]]. The process <math> M_t </math> is a local martingale if and only if ''f'' satisfies the [[Partial differential equation|PDE]]
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| : <math> \Big( \frac{\partial}{\partial t} + \frac12 \frac{\partial^2}{\partial x^2} \Big) f(t,x) = 0. </math>
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| However, this PDE itself does not ensure that <math> M_t </math> is a martingale. In order to apply (**) the following condition on ''f'' is sufficient: for every <math> \varepsilon>0 </math> and ''t'' there exists <math> C = C(\varepsilon,t) </math> such that
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| : <math>\textstyle |f(s,x)| \le C \mathrm{e}^{\varepsilon x^2} </math>
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| for all <math> s \in [0,t] </math> and <math> x \in \mathbb{R}. </math>
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| ==Technical details==
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| <references group="details" />
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| ==References==
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| * {{cite book | last=Øksendal | first=Bernt K. | authorlink=Bernt Øksendal | title=Stochastic Differential Equations: An Introduction with Applications | edition=Sixth edition | publisher=Springer | location=Berlin | year=2003 | isbn=3-540-04758-1}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Local Martingale}}
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| [[Category:Martingale theory]]
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| [[Category:Stochastic processes]]
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