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; 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, a local martingale is not in general a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.

Local martingales are essential in stochastic analysis (see Itô calculus, semimartingale, and Girsanov theorem).

Definition
Let $$(\Omega,F,P)$$ be a probability space; let $$F_*=\{F_t\mid t\geq 0\}$$ be a filtration of $$F$$; let $$X\colon [0,\infty)\times \Omega \rightarrow S$$ be an $$F_*$$-adapted stochastic process on the set $$S$$. Then $$X$$ is called an $$F_*$$-local martingale if there exists a sequence of $$F_*$$-stopping times $$\tau_k \colon \Omega \to [0,\infty)$$ such that
 * the $$\tau_k$$ are almost surely increasing: $$P\left\{\tau_k < \tau_{k+1} \right\}=1$$;
 * the $$\tau_k$$ diverge almost surely: $$P \left\{\lim_{k\to\infty} \tau_k =\infty \right\}=1$$;
 * the stopped process $$ X_t^{\tau_k} := X_{\min \{ t, \tau_k \}}$$ is an $$F_*$$-martingale for every $$k$$.

Example 1
Let Wt be the Wiener process and T = min{ t : Wt = &minus;1 } the time of first hit of &minus;1. The stopped process Wmin{ t, T } is a martingale. Its expectation is 0 at all times; nevertheless, its limit (as t &rarr; &infin;) is equal to &minus;1 almost surely (a kind of gambler's ruin). A time change leads to a process


 * $$\displaystyle X_t = \begin{cases}

W_{\min\left(\tfrac{t}{1-t},T\right)} &\text{for } 0 \le t < 1,\\ -1 &\text{for } 1 \le t < \infty. \end{cases} $$

The process $$ X_t $$ is continuous almost surely; nevertheless, its expectation is discontinuous,


 * $$\displaystyle \operatorname{E} X_t = \begin{cases}

0 &\text{for } 0 \le t < 1,\\ -1 &\text{for } 1 \le t < \infty. \end{cases} $$

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as $$ \tau_k = \min \{ t : X_t = k \} $$ if there is such t, otherwise $$\tau_k = k$$. This sequence diverges almost surely, since $$\tau_k = k$$ for all k large enough (namely, for all k that exceed the maximal value of the process X). The process stopped at τk is a martingale.

Example 2
Let Wt be the Wiener process and &fnof; a measurable function such that $$ \operatorname{E} |f(W_1)| < \infty. $$ Then the following process is a martingale:
 * $$ X_t = \operatorname{E} ( f(W_1) \mid F_t ) = \begin{cases}

f_{1-t}(W_t) &\text{for } 0 \le t < 1,\\ f(W_1) &\text{for } 1 \le t < \infty; \end{cases} $$ where
 * $$ f_s(x) = \operatorname{E} f(x+W_s) = \int f(x+y) \frac1{\sqrt{2\pi s}} \mathrm{e}^{-y^2/(2s)} \, dy. $$

The Dirac delta function $$ \delta $$ (strictly speaking, not a function), being used in place of $$ f, $$ leads to a process defined informally as $$ Y_t = \operatorname{E} ( \delta(W_1) \mid F_t ) $$ and formally as
 * $$ Y_t = \begin{cases}

\delta_{1-t}(W_t) &\text{for } 0 \le t < 1,\\ 0 &\text{for } 1 \le t < \infty, \end{cases} $$ where
 * $$ \delta_s(x) = \frac1{\sqrt{2\pi s}} \mathrm{e}^{-x^2/(2s)} . $$

The process $$ Y_t $$ is continuous almost surely (since $$ W_1 \ne 0 $$ almost surely), nevertheless, its expectation is discontinuous,
 * $$ \operatorname{E} Y_t = \begin{cases}

1/\sqrt{2\pi} &\text{for } 0 \le t < 1,\\ 0 &\text{for } 1 \le t < \infty. \end{cases} $$ This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as $$ \tau_k = \min \{ t : Y_t = k \}. $$

Example 3
Let $$ Z_t $$ be the complex-valued Wiener process, and
 * $$ X_t = \ln | Z_t - 1 | \, . $$

The process $$ X_t $$ is continuous almost surely (since $$ Z_t $$ does not hit 1, almost surely), and is a local martingale, since the function $$ u \mapsto \ln|u-1| $$ is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as $$ \tau_k = \min \{ t : X_t = -k \}. $$ Nevertheless, the expectation of this process is non-constant; moreover,
 * $$ \operatorname{E} X_t \to \infty $$  as $$ t \to \infty, $$

which can be deduced from the fact that the mean value of $$ \ln|u-1| $$ over the circle $$ |u|=r $$ tends to infinity as $$ r \to \infty $$. (In fact, it is equal to $$ \ln r $$ for r ≥ 1 but to 0 for r ≤ 1).

Martingales via local martingales
Let $$ M_t $$ be a local martingale. In order to prove that it is a martingale it is sufficient to prove that $$ M_t^{\tau_k} \to M_t $$ in L1 (as $$ k \to \infty $$) for every t, that is, $$ \operatorname{E} | M_t^{\tau_k} - M_t | \to 0; $$ here $$ M_t^{\tau_k} = M_{t\wedge \tau_k} $$ is the stopped process. The given relation $$ \tau_k \to \infty $$ implies that $$ M_t^{\tau_k} \to M_t $$ almost surely. The dominated convergence theorem ensures the convergence in L1 provided that
 * $$\textstyle (*) \quad \operatorname{E} \sup_k| M_t^{\tau_k} | < \infty $$   for every t.

Thus, Condition (*) is sufficient for a local martingale $$ M_t $$ being a martingale. A stronger condition
 * $$\textstyle (**) \quad \operatorname{E} \sup_{s\in[0,t]} |M_s| < \infty $$   for every t

is also sufficient.

Caution. The weaker condition
 * $$\textstyle \sup_{s\in[0,t]} \operatorname{E} |M_s| < \infty $$   for every t

is not sufficient. Moreover, the condition
 * $$\textstyle \sup_{t\in[0,\infty)} \operatorname{E} \mathrm{e}^{|M_t|} < \infty $$

is still not sufficient; for a counterexample see Example 3 above.

A special case:
 * $$\textstyle M_t = f(t,W_t), $$

where $$ W_t $$ is the Wiener process, and $$ f : [0,\infty) \times \mathbb{R} \to \mathbb{R} $$ is twice continuously differentiable. The process $$ M_t $$ is a local martingale if and only if f satisfies the PDE
 * $$ \Big( \frac{\partial}{\partial t} + \frac12 \frac{\partial^2}{\partial x^2} \Big) f(t,x) = 0. $$

However, this PDE itself does not ensure that $$ M_t $$ is a martingale. In order to apply (**) the following condition on f is sufficient: for every $$ \varepsilon>0 $$ and t there exists $$ C = C(\varepsilon,t) $$ such that
 * $$\textstyle |f(s,x)| \le C \mathrm{e}^{\varepsilon x^2} $$

for all $$ s \in [0,t] $$ and $$ x \in \mathbb{R}. $$