Local martingale

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_{min{ t, T }} 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

${\displaystyle {\displaystyle X_t=\{\begin{array}{ll}{W}_{min\left(\frac{t}{1-t},T\right)}& \text{for }0\le t<1,\\ -1& \text{for }1\le t<\mathrm{\infty }.\end{array}}}$

The process ${\displaystyle X_t}$ is continuous almost surely; nevertheless, its expectation is discontinuous,

${\displaystyle {\displaystyle \mathrm{E}{X}_t=\{\begin{array}{ll}0& \text{for }0\le t<1,\\ -1& \text{for }1\le t<\mathrm{\infty }.\end{array}}}$

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

Example 2

Let W_t be the Wiener process and ƒ a measurable function such that ${\displaystyle \mathrm{E}|f(W_1)|<\mathrm{\infty }.}$ Then the following process is a martingale:

${\displaystyle X_t=\mathrm{E}(f(W_1)\mid F_t)=\{\begin{array}{ll}{f}_{1-t}(W_t)& \text{for }0\le t<1,\\ f(W_1)& \text{for }1\le t<\mathrm{\infty };\end{array}}$

where

${\displaystyle f_s(x)=\mathrm{E}f(x+W_s)=\int f(x+y)\frac{1}{\sqrt{2\pi s}}{\mathrm{e}}^{-y^2/(2s)}dy.}$

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

${\displaystyle Y_t=\{\begin{array}{ll}{\delta }_{1-t}(W_t)& \text{for }0\le t<1,\\ 0& \text{for }1\le t<\mathrm{\infty },\end{array}}$

where

${\displaystyle {\delta }_s(x)=\frac{1}{\sqrt{2\pi s}}{\mathrm{e}}^{-x^2/(2s)}.}$

The process ${\displaystyle Y_t}$ is continuous almost surely (since ${\displaystyle W_1\ne 0}$ almost surely), nevertheless, its expectation is discontinuous,

${\displaystyle \mathrm{E}{Y}_t=\{\begin{array}{ll}1/\sqrt{2\pi }& \text{for }0\le t<1,\\ 0& \text{for }1\le t<\mathrm{\infty }.\end{array}}$

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as ${\displaystyle {\tau }_k=min\{t:Y_t=k\}.}$

Example 3

Let ${\displaystyle Z_t}$ be the complex-valued Wiener process, and

${\displaystyle X_t=\ln |Z_t-1|.}$

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

${\displaystyle \mathrm{E}{X}_t\to \mathrm{\infty }}$ as ${\displaystyle t\to \mathrm{\infty },}$

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

Martingales via local martingales

Let ${\displaystyle M_t}$ be a local martingale. In order to prove that it is a martingale it is sufficient to prove that ${\displaystyle M_t^{{\tau }_k}\to M_t}$ in L¹ (as ${\displaystyle k\to \mathrm{\infty }}$) for every t, that is, ${\displaystyle \mathrm{E}|M_t^{{\tau }_k}-M_t|\to 0;}$ here ${\displaystyle M_t^{{\tau }_k}=M_{t\wedge {\tau }_k}}$ is the stopped process. The given relation ${\displaystyle {\tau }_k\to \mathrm{\infty }}$ implies that ${\displaystyle M_t^{{\tau }_k}\to M_t}$ almost surely. The dominated convergence theorem ensures the convergence in L¹ provided that

${\displaystyle (\ast )\mathrm{E}\underset{k}{sup}|M_t^{{\tau }_k}|<\mathrm{\infty }}$ for every t.

Thus, Condition (*) is sufficient for a local martingale ${\displaystyle M_t}$ being a martingale. A stronger condition

${\displaystyle (\ast \ast )\mathrm{E}\underset{s\in [0,t]}{sup}|M_s|<\mathrm{\infty }}$ for every t

is also sufficient.

Caution. The weaker condition

${\displaystyle \underset{s\in [0,t]}{sup}\mathrm{E}|M_s|<\mathrm{\infty }}$ for every t

is not sufficient. Moreover, the condition

${\displaystyle \underset{t\in [0,\mathrm{\infty })}{sup}\mathrm{E}{\mathrm{e}}^{|M_t|}<\mathrm{\infty }}$

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

A special case:

${\displaystyle M_t=f(t,W_t),}$

where ${\displaystyle W_t}$ is the Wiener process, and ${\displaystyle f:[0,\mathrm{\infty })\times \mathbb{R}\to \mathbb{R}}$ is twice continuously differentiable. The process ${\displaystyle M_t}$ is a local martingale if and only if f satisfies the PDE

${\displaystyle (\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\frac{1}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2})f(t,x)=0.}$

However, this PDE itself does not ensure that ${\displaystyle M_t}$ is a martingale. In order to apply (**) the following condition on f is sufficient: for every ${\displaystyle \epsilon >0}$ and t there exists ${\displaystyle C=C(\epsilon ,t)}$ such that

${\displaystyle |f(s,x)|\le C{\mathrm{e}}^{\epsilon x^2}}$

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

Technical details