# local martingale

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{T}},\mathbb{P})$ be a filtered probability space, where the time index set   $\mathbb{T}\subseteq\mathbb{R}$ has minimal element $t_{0}$. The most common cases are discrete-time, with $\mathbb{T}=\mathbb{Z}_{+}$, and continuous  time where $\mathbb{T}=\mathbb{R}_{+}$, in which case $t_{0}=0$.

A process $X$ is said to be a local martingale  if it is locally (http://planetmath.org/LocalPropertiesOfProcesses) a right-continuous martingale  . That is, if there is a sequence of stopping times $\tau_{n}$ almost surely increasing to infinity  and such that the stopped processes $1_{\{\tau_{n}>t_{0}\}}X^{\tau_{n}}$ are martingales. Equivalently, $1_{\{\tau_{n}>t_{0}\}}X_{\tau_{n}\wedge t}$ is integrable and

 $1_{\{\tau_{n}>t_{0}\}}X_{\tau_{n}\wedge s}=\mathbb{E}[1_{\{\tau_{n}>t_{0}\}}X_% {\tau_{n}\wedge t}\mid\mathcal{F}_{s}]$

for all $s. In the discrete-time case where $\mathbb{T}=\mathbb{Z}_{+}$ then it can be shown that a local martingale $X$ is a martingale if and only if $\mathbb{E}[|X_{t}|]<\infty$ for every $t\in\mathbb{Z}_{+}$. More generally, in continuous-time where $\mathbb{T}$ is an interval of the real numbers, then the stronger property that

 $\left\{X_{\tau}:\tau\leq t\textrm{ is a stopping time}\right\}$

is uniformly integrable for every $t\in\mathbb{T}$ gives a necessary and sufficient condition for a local martingale to be a martingale.

Local martingales form a very important class of processes in the theory of stochastic calculus. This is because the local martingale property is preserved by the stochastic integral  , but the martingale property is not. Examples of local martingales which are not proper martingales are given by solutions to the stochastic differential equation

 $dX=X^{\alpha}\,dW$

where $X$ is a nonnegative process, $W$ is a Brownian motion  and $\alpha>1$ is a fixed real number.

An alternative definition of local martingales which is sometimes used requires $X^{\tau_{n}}$ to be a martingale for each $n$. This definition is slightly more restrictive, and is equivalent      to the definition given above together with the condition that $X_{t_{0}}$ must be integrable.

Title local martingale LocalMartingale 2013-03-22 15:12:43 2013-03-22 15:12:43 skubeedooo (5401) skubeedooo (5401) 8 skubeedooo (5401) Definition msc 60G07 msc 60G48 Martingale LocalPropertiesOfProcesses