Martingales and Stopping Times

\label{app:martingalesStoppingTimes} Let [math]\mathcal F_t[/math] denote a [math]\sigma[/math]-algebra. A process [math]X_t[/math] is an [math]\mathcal F_t[/math]-martingale if and only if

[[math]]\mathbb E_tX_T = X_t\qquad\forall t\leq T\ ,[[/math]]

where [math]\mathbb E_t=\mathbb E[~\cdot~|\mathcal F_t][/math]. A process [math]X_t[/math] is a submartingale if and only if

[[math]]\mathbb E_tX_T \geq X_t\qquad\forall t\leq T\ ,[[/math]]

and a supermartingale if and only if

[[math]]\mathbb E_tX_T \leq X_t\qquad\forall t\leq T\ .[[/math]]

Note that true martingale is both a sub and supermartingale.

Stopping Times

In probability theory, a stopping time is a a stochastic time that is non-anticipative of the underlying process. For instance, for a stock price [math]S_t[/math] a stopping time is the first time the price reaches a level [math]M[/math],

[[math]]\tau = \inf\{t \gt 0|S_t\geq M\}\ .[[/math]]

The non-anticipativeness of the stopping is important because there are some events that are seemingly similar but are not stopping times, for instance

[[math]]\nu=\sup\{t \gt 0|S_t \lt M\}[[/math]]

is not a stopping time. Stopping times are useful when discussing martingales. For example, so-called stopped-processes inherit the sub or supermartingale property. Namely, [math]X_{t\wedge\tau}[/math] is a sub or supermartingale of [math]X_t[/math] is a sub or supermartingale, respectively. There is also the optional stopping theorem:

Theorem (Optional Stopping Theorem)

Let [math]X_t[/math] be a submartingale and let [math]\tau[/math] be a stopping time. If [math]\tau \lt \infty [/math] a.s. and [math]X_{t\wedge\tau}[/math] uniformly integrable, then [math]\mathbb EX_0\leq \mathbb EX_\tau[/math] with equality if [math]X_t[/math] is a martingale.

An example application of the optional stopping theorem is Gambler's ruin: Let

[[math]]\tau=\inf\{t \gt 0|W_t\notin (a,b)\}\ ,[[/math]]

where [math]0 \lt b \lt \infty[/math] and [math]-\infty \lt a \lt 0[/math]. Then [math]\mathbb P(\tau \lt \infty)=1[/math] and by optional stopping,

[[math]] \begin{align*} 0&=W_0=\mathbb EW_\tau\\ &= a\mathbb P(W_\tau=a)+b(1-\mathbb P(W_\tau=a))\\ &=(a-b)\mathbb P(W_\tau=a)+b\ , \end{align*} [[/math]]

which can be simplified to get

[[math]]\mathbb P(W_\tau=a) = \frac{b}{b-a}\ .[[/math]]

Local Martingales

First define a local martingale:

Definition (Local Martingale)

A process [math]X_t[/math] is a local martingale if there exists a sequence of finite and increasing stopping times [math]\tau_n[/math] such that [math]\mathbb P(\tau_n\rightarrow\infty\hbox{ as }n\rightarrow\infty)=1[/math] and [math]X_{t\wedge\tau_n}[/math] is a true martingale for any [math]n[/math].

Some remarks are in order:

In discrete time there are no local martingales; a martingale is a martingale.

A true martingale [math]X_t[/math] is a local martingale, and any bounded local martingale is in fact a true martingale.

The It\^o stochastic integral is in general a local martingale, not necessarily a true martingale. That is,

[[math]]I_t = \int_0^t\sigma_udW_u\ ,[[/math]]

is only a local martingale, but there exists stopping times [math]\tau_n[/math] such that

[[math]]I_{t\wedge\tau_n} = \int_0^{t\wedge\tau_n}\sigma_udW_u\ ,[[/math]]

is a true martingale. For It\^o integrals there is the following theorem for a sufficient (but not necessary) condition for true martingales:

Theorem

The It\^o integral [math] \int_0^t\sigma_udW_u[/math] is a true martingale on [math][0,T][/math] if

[[math]]\mathbb E\int_0^T\sigma_s^2ds \lt \infty \ ,[[/math]]

i.e., the It\^o isometry is finite.

For the stochastic integral [math]I_t = \int_0^t\sigma_udW_u[/math] we can define

[[math]]\tau_n = \inf\left\{t \gt 0\Bigg|\int_0^t\sigma_u^2ds\geq n\right\}\wedge T\ ,[[/math]]

for which we have a bounded It\^o isometry, and hence Theorem applies to make [math]I_{t\wedge\tau_n}[/math] a martingale on [math][0,T][/math]. An example of a local martingale is the constant elasticity of volatility (CEV) model,

[[math]]dS_t = \sigma S_t^{\alpha}dW_t\ ,[[/math]]

with [math]0\leq\alpha\leq 2[/math]; [math]S_t[/math] is strictly a local martingale for [math]1 \lt \alpha\leq 2[/math]. For [math]\alpha=2[/math] one can check using PDEs that the transition density is

[[math]]p_t(z|s) = \frac{s}{z^3\sqrt{2\pi t\sigma^2}}\left(e^{-\frac{\left(\frac1z-\frac1s\right)^2}{2t\sigma^2}}-e^{-\frac{\left(\frac1z+\frac1s\right)^2}{2t\sigma^2}}\right)\ .[[/math]]

One can check that [math]\mathbb ES_t^4=\infty[/math] for all [math]t \gt 0[/math] so that Theorem does not apply, but to see that it is a strict local martingale one must also check that [math]\mathbb E_tS_T \lt S_t[/math] for all [math]t \lt T[/math].

General references

Papanicolaou, Andrew (2015). "Introduction to Stochastic Differential Equations (SDEs) for Finance". arXiv:1504.05309 [q-fin.MF].