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, in general a local martingale is not 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 Itō calculus, semimartingale, Girsanov theorem.

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 on set S. 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

$X_t^{\tau_{k}} := X_{\min \{ t, \tau_k \}}$

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_{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

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

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

$\displaystyle \mathbb{E} X_t = \begin{cases} 0 &\text{for } 0 \le t < 1,\\ -1 &\text{for } 1 \le t < \infty. \end{cases}$

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 τ_k = k. This sequence diverges almost surely, since τ_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 $\mathbb{E} |f(W_1)| < \infty.$ Then the following process is a martingale:

$\displaystyle X_t = \mathbb{E} ( f(W_1) | F_t ) = \begin{cases} f_{1-t}(W_t) &\text{for } 0 \le t < 1,\\ f(W_1) &\text{for } 1 \le t < \infty; \end{cases}$

here

$\displaystyle f_s(x) = \mathbb{E} f(x+W_s) = \int f(x+y) \frac1{\sqrt{2\pi s}} \mathrm{e}^{-y^2/(2s)} .$

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

$\displaystyle Y_t = \begin{cases} \delta_{1-t}(W_t) &\text{for } 0 \le t < 1,\\ 0 &\text{for } 1 \le t < \infty, \end{cases}$

where

$\displaystyle \delta_s(x) = \frac1{\sqrt{2\pi s}} \mathrm{e}^{-x^2/(2s)} .$

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

$\displaystyle \mathbb{E} Y_t = \begin{cases} 1/\sqrt{2\pi} &\text{for } 0 \le t < 1,\\ 0 &\text{for } 1 \le t < \infty. \end{cases}$

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

$\displaystyle X_t = \ln | Z_t - 1 | \, .$

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,

$\displaystyle \mathbb{E} X_t \to \infty$   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, $\mathbb{E} | M_t^{\tau_k} - M_t | \to 0;$ 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

$\textstyle (*) \quad \mathbb{E} \sup_k| M_t^{\tau_k} | < \infty$    for every t.

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

$\textstyle (**) \quad \mathbb{E} \sup_{s\in[0,t]} |M_s| < \infty$    for every t

is also sufficient.

Caution. The weaker condition

$\textstyle \sup_{s\in[0,t]} \mathbb{E} |M_s| < \infty$    for every t

is not sufficient. Moreover, the condition

$\textstyle \sup_{t\in[0,\infty)} \mathbb{E} \mathrm{e}^{|M_t|} < \infty$

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 $x \in \mathbb{R}.$