Local martingale

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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 ( Ω , F , P ) {\displaystyle (\Omega ,F,P)} {\displaystyle (\Omega ,F,P)} be a probability space; let F ∗ = { F t ∣ t ≥ 0 } {\displaystyle F_{*}=\{F_{t}\mid t\geq 0\}} {\displaystyle F_{*}=\{F_{t}\mid t\geq 0\}} be a filtration of F {\displaystyle F} {\displaystyle F}; let X : [ 0 , ∞ ) × Ω → S {\displaystyle X\colon [0,\infty )\times \Omega \rightarrow S} {\displaystyle X\colon [0,\infty )\times \Omega \rightarrow S} be an F ∗ {\displaystyle F_{*}} {\displaystyle F_{*}}-adapted stochastic process on the set S {\displaystyle S} {\displaystyle S}. Then X {\displaystyle X} {\displaystyle X} is called an F ∗ {\displaystyle F_{*}} {\displaystyle F_{*}}-local martingale if there exists a sequence of F ∗ {\displaystyle F_{*}} {\displaystyle F_{*}}-stopping times τ k : Ω → [ 0 , ∞ ) {\displaystyle \tau _{k}\colon \Omega \to [0,\infty )} {\displaystyle \tau _{k}\colon \Omega \to [0,\infty )} such that

  • the τ k {\displaystyle \tau _{k}} {\displaystyle \tau _{k}} are almost surely increasing: P { τ k < τ k + 1 } = 1 {\displaystyle P\left\{\tau _{k}<\tau _{k+1}\right\}=1} {\displaystyle P\left\{\tau _{k}<\tau _{k+1}\right\}=1};
  • the τ k {\displaystyle \tau _{k}} {\displaystyle \tau _{k}} diverge almost surely: P { lim k → ∞ τ k = ∞ } = 1 {\displaystyle P\left\{\lim _{k\to \infty }\tau _{k}=\infty \right\}=1} {\displaystyle P\left\{\lim _{k\to \infty }\tau _{k}=\infty \right\}=1};
  • the stopped process X t τ k := X min { t , τ k } {\displaystyle X_{t}^{\tau _{k}}:=X_{\min\{t,\tau _{k}\}}} {\displaystyle X_{t}^{\tau _{k}}:=X_{\min\{t,\tau _{k}\}}} is an F ∗ {\displaystyle F_{*}} {\displaystyle F_{*}}-martingale for every k {\displaystyle k} {\displaystyle k}.

Examples

Example 1

Illustration for local martingale. Up Panel: Multiple simulated paths of the process X t {\displaystyle X_{t}} {\displaystyle X_{t}} which is stopped upon hitting − 1 {\displaystyle -1} {\displaystyle -1}. This shows gambler's ruin behavior, and is not a martingale. Down Panel: Paths of X t {\displaystyle X_{t}} {\displaystyle X_{t}} with an additional stopping criterion: the process is also stopped when it reaches a magnitude of k = 2.0 {\displaystyle k=2.0} {\displaystyle k=2.0}. This no longer suffers from gambler's ruin behavior, and is a martingale.

Let Wt be the Wiener process and T = min{ t : Wt = −1 } the time of first hit of −1. The stopped process Wmin{ 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

X t = { W min ( t 1 − t , T ) for  0 ≤ t < 1 , − 1 for  1 ≤ t < ∞ . {\displaystyle \displaystyle X_{t}={\begin{cases}W_{\min \left({\tfrac {t}{1-t}},T\right)}&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}} {\displaystyle \displaystyle X_{t}={\begin{cases}W_{\min \left({\tfrac {t}{1-t}},T\right)}&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}}

The process X t {\displaystyle X_{t}} {\displaystyle X_{t}} is continuous almost surely; nevertheless, its expectation is discontinuous,

E ⁡ X t = { 0 for  0 ≤ t < 1 , − 1 for  1 ≤ t < ∞ . {\displaystyle \displaystyle \operatorname {E} X_{t}={\begin{cases}0&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}} {\displaystyle \displaystyle \operatorname {E} X_{t}={\begin{cases}0&{\text{for }}0\leq t<1,\\-1&{\text{for }}1\leq t<\infty .\end{cases}}}

This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as τ k = min { t : X t = k } {\displaystyle \tau _{k}=\min\{t:X_{t}=k\}} {\displaystyle \tau _{k}=\min\{t:X_{t}=k\}} if there is such t, otherwise τ k = k {\displaystyle \tau _{k}=k} {\displaystyle \tau _{k}=k}. This sequence diverges almost surely, since τ k = k {\displaystyle \tau _{k}=k} {\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 τk is a martingale.[details 1]

Example 2

Let Wt be the Wiener process and ƒ a measurable function such that E ⁡ | f ( W 1 ) | < ∞ . {\displaystyle \operatorname {E} |f(W_{1})|<\infty .} {\displaystyle \operatorname {E} |f(W_{1})|<\infty .} Then the following process is a martingale:

X t = E ⁡ ( f ( W 1 ) ∣ F t ) = { f 1 − t ( W t ) for  0 ≤ t < 1 , f ( W 1 ) for  1 ≤ t < ∞ ; {\displaystyle X_{t}=\operatorname {E} (f(W_{1})\mid F_{t})={\begin{cases}f_{1-t}(W_{t})&{\text{for }}0\leq t<1,\\f(W_{1})&{\text{for }}1\leq t<\infty ;\end{cases}}} {\displaystyle X_{t}=\operatorname {E} (f(W_{1})\mid F_{t})={\begin{cases}f_{1-t}(W_{t})&{\text{for }}0\leq t<1,\\f(W_{1})&{\text{for }}1\leq t<\infty ;\end{cases}}}

where

f s ( x ) = E ⁡ f ( x + W s ) = ∫ f ( x + y ) 1 2 π s e − y 2 / ( 2 s ) d y . {\displaystyle f_{s}(x)=\operatorname {E} f(x+W_{s})=\int f(x+y){\frac {1}{\sqrt {2\pi s}}}\mathrm {e} ^{-y^{2}/(2s)}\,dy.} {\displaystyle f_{s}(x)=\operatorname {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 δ {\displaystyle \delta } {\displaystyle \delta } (strictly speaking, not a function), being used in place of f , {\displaystyle f,} {\displaystyle f,} leads to a process defined informally as Y t = E ⁡ ( δ ( W 1 ) ∣ F t ) {\displaystyle Y_{t}=\operatorname {E} (\delta (W_{1})\mid F_{t})} {\displaystyle Y_{t}=\operatorname {E} (\delta (W_{1})\mid F_{t})} and formally as

Y t = { δ 1 − t ( W t ) for  0 ≤ t < 1 , 0 for  1 ≤ t < ∞ , {\displaystyle Y_{t}={\begin{cases}\delta _{1-t}(W_{t})&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty ,\end{cases}}} {\displaystyle Y_{t}={\begin{cases}\delta _{1-t}(W_{t})&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty ,\end{cases}}}

where

δ s ( x ) = 1 2 π s e − x 2 / ( 2 s ) . {\displaystyle \delta _{s}(x)={\frac {1}{\sqrt {2\pi s}}}\mathrm {e} ^{-x^{2}/(2s)}.} {\displaystyle \delta _{s}(x)={\frac {1}{\sqrt {2\pi s}}}\mathrm {e} ^{-x^{2}/(2s)}.}

The process Y t {\displaystyle Y_{t}} {\displaystyle Y_{t}} is continuous almost surely (since W 1 ≠ 0 {\displaystyle W_{1}\neq 0} {\displaystyle W_{1}\neq 0} almost surely), nevertheless, its expectation is discontinuous,

E ⁡ Y t = { 1 / 2 π for  0 ≤ t < 1 , 0 for  1 ≤ t < ∞ . {\displaystyle \operatorname {E} Y_{t}={\begin{cases}1/{\sqrt {2\pi }}&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty .\end{cases}}} {\displaystyle \operatorname {E} Y_{t}={\begin{cases}1/{\sqrt {2\pi }}&{\text{for }}0\leq t<1,\\0&{\text{for }}1\leq t<\infty .\end{cases}}}

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

Example 3

Let Z t {\displaystyle Z_{t}} {\displaystyle Z_{t}} be the complex-valued Wiener process, and

X t = ln ⁡ | Z t − 1 | . {\displaystyle X_{t}=\ln |Z_{t}-1|\,.} {\displaystyle X_{t}=\ln |Z_{t}-1|\,.}

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

E ⁡ X t → ∞ {\displaystyle \operatorname {E} X_{t}\to \infty } {\displaystyle \operatorname {E} X_{t}\to \infty }   as t → ∞ , {\displaystyle t\to \infty ,} {\displaystyle t\to \infty ,}

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

Martingales via local martingales

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

( ∗ ) E ⁡ sup k | M t τ k | < ∞ {\displaystyle \textstyle (*)\quad \operatorname {E} \sup _{k}|M_{t}^{\tau _{k}}|<\infty } {\displaystyle \textstyle (*)\quad \operatorname {E} \sup _{k}|M_{t}^{\tau _{k}}|<\infty }    for every t.

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

( ∗ ∗ ) E ⁡ sup s ∈ [ 0 , t ] | M s | < ∞ {\displaystyle \textstyle (**)\quad \operatorname {E} \sup _{s\in [0,t]}|M_{s}|<\infty } {\displaystyle \textstyle (**)\quad \operatorname {E} \sup _{s\in [0,t]}|M_{s}|<\infty }    for every t

is also sufficient.

Caution. The weaker condition

sup s ∈ [ 0 , t ] E ⁡ | M s | < ∞ {\displaystyle \textstyle \sup _{s\in [0,t]}\operatorname {E} |M_{s}|<\infty } {\displaystyle \textstyle \sup _{s\in [0,t]}\operatorname {E} |M_{s}|<\infty }    for every t

is not sufficient. Moreover, the condition

sup t ∈ [ 0 , ∞ ) E ⁡ e | M t | < ∞ {\displaystyle \textstyle \sup _{t\in [0,\infty )}\operatorname {E} \mathrm {e} ^{|M_{t}|}<\infty } {\displaystyle \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:

M t = f ( t , W t ) , {\displaystyle \textstyle M_{t}=f(t,W_{t}),} {\displaystyle \textstyle M_{t}=f(t,W_{t}),}

where W t {\displaystyle W_{t}} {\displaystyle W_{t}} is the Wiener process, and f : [ 0 , ∞ ) × R → R {\displaystyle f:[0,\infty )\times \mathbb {R} \to \mathbb {R} } {\displaystyle f:[0,\infty )\times \mathbb {R} \to \mathbb {R} } is twice continuously differentiable. The process M t {\displaystyle M_{t}} {\displaystyle M_{t}} is a local martingale if and only if f satisfies the PDE

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

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

| f ( s , x ) | ≤ C e ε x 2 {\displaystyle \textstyle |f(s,x)|\leq C\mathrm {e} ^{\varepsilon x^{2}}} {\displaystyle \textstyle |f(s,x)|\leq C\mathrm {e} ^{\varepsilon x^{2}}}

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

Technical details

  1. 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.

References