Local martingale

In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; however, in general a local martingale is not a martingale, because its expectation is distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.

Definition

Let (Ω, "F", P) be a probability space; let "F"_∗ = { "F"_"t" | "t" ≥ 0 } be a filtration of "F"; let X : [0, +∞) × Ω → "S" be an "F"_∗-adapted stochastic process. Then "X" is called an "F"_∗-local martingale if there exists a sequence of "F"_∗-stopping times "τ"_"k" : Ω → [0, +∞) such that
* the "τ"_"k" are almost surely increasing: P ["τ"_"k" < "τ"_"k"+1] = 1;
* the "τ"_"k" diverge almost surely: P ["τ"_"k" → +∞ as "k" → +∞] = 1;
* the stopped process

::$1\_\{\{\; au\_k>0X\_t^\{\; au\_\{k\; :=\; 1\_\{\{\; au\_k>0X\_\{min\; \{\; t,\; au\_k$

: is an "F"_∗-martingale for every "k".

Examples

Example 1

Let $W\_t$ be the Wiener process and $T\; =\; min\; \{\; t\; :\; W\_t\; =\; -1\; \}$ the time of first hit of -1. The stopped process $W\_t^T\; =\; W\_\{min(t,T)\}$ is a martingale; its expectation is 0 at all times, nevertheless its limit (as $t\; o\; infty$) is equal to -1 almost sure. A time change leads to a process: The process $X\_t$ is continuous almost sure, nevertheless, its expectation is discontinuous,: This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as $au\_k\; =\; min\; \{\; t\; :\; X\_t\; =\; k\; \}.$

Example 2

Let $W\_t$ be the Wiener process and $f$ a measurable function such that Then the following process is a martingale:: here: $displaystyle\; f\_s(x)\; =\; mathbb\{E\}\; f(x+W\_s)\; =\; int\; f(x+y)\; frac1\{sqrt\{2pi\; s\; mathrm\{e\}^\{-y^2/(2s)\}\; .$The Dirac delta function $delta$ (strictly speaking, not a function), being used in place of $f,$ leads to a process defined informally as $Y\_t\; =\; mathbb\{E\}\; (\; delta(W\_1)\; |\; F\_t\; )$ and formally as: where: $displaystyle\; delta\_s(x)\; =\; frac1\{sqrt\{2pi\; s\; mathrm\{e\}^\{-x^2/(2s)\}\; .$The process $Y\_t$ is continuous almost sure (since $W\_1\; e\; 0$ almost sure), nevertheless, its expectation is discontinuous,: This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as $au\_k\; =\; min\; \{\; t\; :\; Y\_t\; =\; k\; \}.$

References

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