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, and Girsanov theorem).

Definition

Let $(\Omega,F,P)$ be a probability space; let $F_*=\$ be a filtration of $F$; let X: adapted_stochastic_process_on_the_set_$S$._Then_$X$_is_called_an_$F_*$-local_martingale_if_there_exists_a_sequence_of_$F_*$-stopping_rule.html" ;"title="adapted process">adapted stochastic process on the set $S$. Then $X$ is called an $F_*$-local martingale if there exists a sequence of $F_*$-stopping rule">stopping times $\tau_k : \Omega \to [0,\infty)$ such that
* the $\tau_k$ are almost surely increasing: $P\left\=1$;
* the $\tau_k$ diverge almost surely: $P \left\=1$;
* the stopped process $X_t^ := X_$ is an $F_*$-martingale for every $k$.

Examples

Example 1

Let ''W''_''t'' be the Wiener process and ''T'' = min the time of first hit of −1. The stopped process ''W''_min 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

: $\displaystyle X_t = \begin W_ &\text 0 \le t < 1,\\ -1 &\text 1 \le t < \infty. \end$

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

: $\displaystyle \operatorname X_t = \begin 0 &\text 0 \le t < 1,\\ -1 &\text 1 \le t < \infty. \end$

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as $\tau_k = \min \$ 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 τ_''k'' is a martingale.
For the times before 1 it is a martingale since a stopped Brownian motion is. After the instant 1 it is constant. It remains to check it at the instant 1. By the bounded convergence theorem the expectation at 1 is the limit of the expectation at (''n''-1)/''n'' (as ''n'' tends to infinity), and the latter does not depend on ''n''. The same argument applies to the conditional expectation.

Example 2

Let ''W''_''t'' be the Wiener process and ''ƒ'' a measurable function such that $\operatorname , f(W_1), < \infty.$ Then the following process is a martingale:
: $X_t = \operatorname ( f(W_1) \mid F_t ) = \begin f_(W_t) &\text 0 \le t < 1,\\ f(W_1) &\text 1 \le t < \infty; \end$
here
: $f_s(x) = \operatorname f(x+W_s) = \int f(x+y) \frac1 \mathrm^ \, dy.$
The Dirac delta function $\delta$ (strictly speaking, not a function), being used in place of $f,$ leads to a process defined informally as $Y_t = \operatorname ( \delta(W_1) \mid F_t )$ and formally as
: $Y_t = \begin \delta_(W_t) &\text 0 \le t < 1,\\ 0 &\text 1 \le t < \infty, \end$
where
: $\delta_s(x) = \frac1 \mathrm^ .$
The process $Y_t$ is continuous almost surely (since $W_1 \ne 0$ almost surely), nevertheless, its expectation is discontinuous,
: $\operatorname Y_t = \begin 1/\sqrt &\text 0 \le t < 1,\\ 0 &\text 1 \le t < \infty. \end$
This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as $\tau_k = \min \.$

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 $\tau_k = \min \.$ Nevertheless, the expectation of this process is non-constant; moreover,
: $\operatorname X_t \to \infty$   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 $M_t^ \to M_t$ in ''L''¹ (as $k \to \infty$) for every ''t'', that is, $\operatorname , M_t^ - M_t , \to 0;$ here $M_t^ = M_$ is the stopped process. The given relation $\tau_k \to \infty$ implies that $M_t^ \to M_t$ almost surely. The dominated convergence theorem ensures the convergence in ''L''¹ provided that
: $\textstyle (*) \quad \operatorname \sup_k, M_t^ , < \infty$    for every ''t''.
Thus, Condition (*) is sufficient for a local martingale $M_t$ being a martingale. A stronger condition
: $\textstyle (**) \quad \operatorname \sup_ , M_s, < \infty$    for every ''t''
is also sufficient.

''Caution.'' The weaker condition
: $\textstyle \sup_ \operatorname , M_s, < \infty$    for every ''t''
is not sufficient. Moreover, the condition
: $\textstyle \sup_ \operatorname \mathrm^ < \infty$
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 : twice_continuously_differentiable._The_process_$_M_t_$_is_a_local_martingale_if_and_only_if_''f''_satisfies_the_ twice_continuously_differentiable._The_process_$_M_t_$_is_a_local_martingale_if_and_only_if_''f''_satisfies_the_Partial_differential_equation">PDE
:_$_\Big(_\frac_+_\frac12_\frac_\Big)_f(t,x)_=_0._$
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_.html" ;"title="Partial_differential_equation.html" ;"title="smooth function">twice continuously differentiable. The process $M_t$ is a local martingale if and only if ''f'' satisfies the Partial differential equation">PDE
: $\Big( \frac + \frac12 \frac \Big) f(t,x) = 0.$
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
: $\textstyle ">f(s,x), \le C \mathrm^$
for all $s \in [0,t$ and $x \in \mathbb.$

Technical details

References

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{{DEFAULTSORT:Local Martingale
Martingale theory