# Local martingale

A local martingale (random process whose expected future value equals its present value) is an adapted stochastic process \( (M_t)_{t \ge 0} \) that becomes a true martingale when stopped at a suitable sequence of stopping times increasing to infinity: there exist stopping times \( (T_n) \) with \( T_n \to \infty \) almost surely such that each stopped process \( (M_{t \wedge T_n})_{t \ge 0} \) is a martingale<sup>[1](https://fabricebaudoin.blog/2012/09/17/lecture-20-local-martingales-semimartingales-and-integrators/)</sup>. The sequence \( (T_n) \) is said to localize or reduce \( M \)<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>. Every martingale is a local martingale, since the fixed times \( T_n = n \) serve as a localizing sequence, but the converse is false; a local martingale that is not a martingale is called a strict local martingale<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup><sup> • </sup><sup>[3](https://www.sciencedirect.com/science/article/pii/S030441491400266X)</sup>.

| Key fact | Detail |
|---|---|
| Definition | An adapted process \( (M_t) \) for which some increasing sequence of stopping times \( T_n \to \infty \) a.s. makes each stopped process \( (M_{t \wedge T_n}) \) a martingale<sup>[1](https://fabricebaudoin.blog/2012/09/17/lecture-20-local-martingales-semimartingales-and-integrators/)</sup> |
| Relation to martingales | Every martingale is a local martingale; the converse is false, and strict local martingales do exist<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup> |
| Martingale criterion | A local martingale is a martingale if and only if it is of class (DL), meaning the family of values at bounded stopping times is uniformly integrable<sup>[4](https://almostsuremath.com/2009/12/24/local-martingales/)</sup> |
| Nonnegative case | A nonnegative local martingale is always a supermartingale, so \( E[M_t] \le E[M_0] \)<sup>[4](https://almostsuremath.com/2009/12/24/local-martingales/)</sup> |
| Canonical example | The inverse Bessel process, the reciprocal of the norm of a 3-dimensional Brownian motion, is a local martingale but not a martingale, and is bounded in \( L^2 \)<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup> |
| Main application | In the local martingale theory of bubbles, an asset price bubble is characterized by a price process that is a strict local martingale under an equivalent local martingale measure<sup>[5](https://onlinelibrary.wiley.com/doi/abs/10.1111/irfi.12404)</sup> |

## Definition and localization

The concept arises from extending the Itō integral beyond square-integrable integrands. For a process \( u \) that is square integrable only locally, one constructs stopping times \( (T_n) \) increasing almost surely to infinity such that \( E\left(\int_0^{T_n} u_s^2 ds\right) < \infty \); the integral \( \int_0^t u_s dB_s \) is then defined piecewise, and the resulting process is a local martingale<sup>[1](https://fabricebaudoin.blog/2012/09/17/lecture-20-local-martingales-semimartingales-and-integrators/)</sup>. Localization is a truncation by cutoff that preserves both the continuity of the process and the martingale structure through the Doob stopping theorem<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>.

For a continuous process issued from the origin, the standard localizing choice is \( T_n = \inf\{t \ge 0 : |M_t| \ge n\} \), which increases almost surely to infinity and makes each stopped process a bounded martingale<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>. The set of local martingales forms a real vector space: if \( M \) and \( M' \) are localized by \( (T_n) \) and \( (T'_n) \), then \( (T_n \wedge T'_n) \) localizes both, and \( M + M' \) stopped at these times is a square-integrable martingale<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>.

**What localization does not preserve.** For a local martingale, no integrability is guaranteed for \( M_t \) at a fixed deterministic time \( t \); \( M_t \) may fail to be in \( L^1 \). Moreover, for a stopping time \( T \), the stopped process \( (M_{t \wedge T}) \) is again a local martingale, but the Doob stopping theorem does not hold in general even when \( T \) is bounded<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>.

## When a local martingale is a martingale

Class (DL) is the exact criterion. A process is of class (DL) if the family \( \{X_T\} \), where \( T \) ranges over bounded stopping times, is uniformly integrable. A local martingale is a martingale if and only if it is of class (DL)<sup>[4](https://almostsuremath.com/2009/12/24/local-martingales/)</sup>. Equivalently, an adapted right-continuous integrable process \( X \) is a martingale exactly when \( E[X_T] = E[X_0] \) for every bounded stopping time \( T \)<sup>[6](https://www.normalesup.org/~thevenin/LectureNotes/Lecture_fin.pdf)</sup>.

Several sufficient conditions guarantee the martingale property. If a local martingale \( M \) satisfies \( |M_t| \le Z \) for all \( t \), where \( Z \in L^1 \), then \( M \) is a true martingale<sup>[6](https://www.normalesup.org/~thevenin/LectureNotes/Lecture_fin.pdf)</sup>. In particular, if a continuous local martingale is dominated by an integrable random variable in the sense that \( E\left(\sup_{t \ge 0} |M_t|\right) < \infty \), it is a uniformly integrable martingale<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>. A local martingale with independent increments is a martingale<sup>[7](https://www.sciencedirect.com/topics/mathematics/local-martingale)</sup>. In discrete time, a local martingale with \( E(|X_t|) < \infty \) for all \( t \) is automatically a true martingale<sup>[8](https://www.statslab.cam.ac.uk/~mrt31/AFM/notes.pdf)</sup>.

These conditions cannot be dropped entirely: there exist continuous local martingales that are bounded in \( L^2 \), and hence uniformly integrable, yet are not martingales<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>. What does survive without integrability is the supermartingale property: a nonnegative local martingale is always a supermartingale, so its expectation is nonincreasing in time<sup>[4](https://almostsuremath.com/2009/12/24/local-martingales/)</sup>.

## Examples of strict local martingales

**The inverse Bessel process.** Let \( (B_t) \) be a 3-dimensional [Brownian motion](https://www.edgepedia.org/brownian-motion) with \( B_0 = x \ne 0 \). The process \( (|B_t|) \) is a Bessel process, and its reciprocal \( (|B_t|^{-1}) \) is a local martingale, localized by the stopping times \( T_n = \inf\{t \ge 0 : |B_t| \le 1/(|x| + n)\} \), but it is not a martingale; it is bounded in \( L^2 \) and therefore uniformly integrable<sup>[2](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)</sup>. This process is the canonical example of a strict local martingale: the 1963 Class D counterexample of Johnson and Helms, known today as the inverse Bessel process<sup>[9](https://pi.math.cornell.edu/~soumik/strictlocmgle4.pdf)</sup>.

**A construction via h-transforms.** Delbaen and Schachermayer, while discussing arbitrage possibilities in Bessel processes, described an effective way to generate strict local martingales that are positive throughout. Changing the Wiener measure of a Brownian motion using, as Radon–Nikodým derivative, the martingale given by the Brownian motion absorbed upon hitting zero yields the law of a three-dimensional Bessel process; this change of measure is an example of Doob's h-transform. They showed that the reciprocal of every positive local martingale can be obtained by a similar procedure, transforming the law of a nonnegative martingale by the process itself<sup>[9](https://pi.math.cornell.edu/~soumik/strictlocmgle4.pdf)</sup>.

**A discrete-time construction.** A local martingale that is not a martingale can be built from the double-loss gambling strategy, the origin of the word martingale in 18th-century France. A gambler doubles his stake after each loss on a fair coin toss, so that he eventually wins one dollar with probability one. The winnings process \( (Z_n) \) is a martingale starting at zero and ending at one. Speeding up the time scale to fit infinitely many tosses into a unit interval produces a process \( (X_t) \) that is a local martingale, yet \( E[X_1] = 1 \ne 0 = E[X_0] \), so it is not a martingale<sup>[4](https://almostsuremath.com/2009/12/24/local-martingales/)</sup>.

**Jumps and filtrations.** Explicit examples of naturally occurring strict local martingales with jumps are scarce in the literature; one construction method produces them via filtration shrinkage, a phenomenon held to be common in applications such as filtering, control, and mathematical finance<sup>[3](https://www.sciencedirect.com/science/article/pii/S030441491400266X)</sup>. Conversely, filtration enlargement can also act in this direction: an initial expansion of the filtration can turn a true martingale into a strict local martingale in stochastic volatility models<sup>[10](https://math.ethz.ch/imsf/courses/mf-seminar.html/1000?s=fs16)</sup>.

## A small-moment criterion

Xue-Mei Li, a mathematician then working on stochastic analysis, gave a criterion based on the small moments of the running supremum: a continuous local martingale is a strict local martingale if its supremum process is not in \( L_\alpha \) for any positive number \( \alpha \) smaller than 1. Using this she constructed a family of strict local martingales without employing Bessel processes or the speed measures of one-dimensional diffusions<sup>[11](https://arxiv.org/abs/1609.00935)</sup>. The same source observes that the sample paths of a strict local martingale oscillate faster than those of a martingale, a wild oscillation that may explain why strict local martingales are useful for bubble modeling<sup>[12](https://www.sciencedirect.com/science/article/pii/S0167715217301645)</sup>.

For one-dimensional diffusions the classification is complete. The solution of a one-dimensional SDE without drift is a strict local martingale if and only if an explicit condition holds, a clean solution given by Delbaen and Shirakawa (2002, Theorem 1.6) and Kotani (2006, Theorem 1); Alexander Mijatović and Mikhail Urusov solved the corresponding question for the exponential martingale of a stochastic integral in 2012<sup>[12](https://www.sciencedirect.com/science/article/pii/S0167715217301645)</sup>.

## Role in financial mathematics

The popularity of strict local martingales is largely due to the 1995 paper of Freddy Delbaen and Walter Schachermayer on [incomplete markets](https://www.edgepedia.org/incomplete-markets), and to the use of strict local martingales in bubble modeling by Cox and Hobson (2005) and Jarrow, Protter, and Shimbo (2010)<sup>[12](https://www.sciencedirect.com/science/article/pii/S0167715217301645)</sup>. Within the local martingale theory of bubbles, a bubble exists when the price process, under some equivalent local martingale measure, is a strict local martingale rather than a true martingale.<sup>[5](https://onlinelibrary.wiley.com/doi/abs/10.1111/irfi.12404)</sup> Because a local martingale bounded below by zero is a supermartingale, the asset's price bubble is always nonnegative<sup>[5](https://onlinelibrary.wiley.com/doi/abs/10.1111/irfi.12404)</sup>.

The connection to market completeness is structural: under the no-free-lunch-with-vanishing-risk conditions, the existence of bubbles implies that markets must be incomplete, and the birth of bubbles is impossible in a complete market but possible in incomplete markets<sup>[12](https://www.sciencedirect.com/science/article/pii/S0167715217301645)</sup><sup> • </sup><sup>[13](https://link.springer.com/article/10.1007/s11147-023-09200-x)</sup>. Working in a numéraire-independent setting, Herdegeen and Schweizer established—assuming only a mild no-arbitrage condition, essentially NUPBR—that a market exhibits a strong bubble exactly when asset prices fail to be true martingales, being strict local martingales, under every equivalent local martingale measure; they additionally constructed a case where prices are strict local martingales with respect to one such measure yet genuine martingales with respect to another.<sup>[14](https://doi.org/10.1142/S0219024916500229)</sup>

**Detecting bubbles from option prices.** Because the underlying fundamental value is unknown, direct bubble tests face a joint hypothesis problem; Jarrow (2015) instead suggested specifying an option pricing model, validating it separately, and checking whether it implies an underlying price process that is a true martingale rather than a strict local martingale<sup>[13](https://link.springer.com/article/10.1007/s11147-023-09200-x)</sup>. Piiroinen et al. (2018) demonstrated that, under a parameterization in which stock and option prices are positively correlated, the SABR model becomes a strict local martingale, which makes bubble detection a matter of calibration<sup>[13](https://link.springer.com/article/10.1007/s11147-023-09200-x)</sup>. In a related line of work, Biagini et al. propose a model-free method, computationally expensive, that uses a neural network to decide whether a call option price smile stems from a strict local martingale or a genuine one.<sup>[13](https://link.springer.com/article/10.1007/s11147-023-09200-x)</sup>

Strict local martingale behavior also changes option prices qualitatively. For a positive continuous strict local martingale modeling a discounted stock price, the price of a European call priced by \( E[(S_T - K)^+] \) can decrease monotonically as maturity increases after some point, whereas for true martingales the price curve strictly increases by [Jensen's inequality](https://www.edgepedia.org/jensens-inequality); consequently Merton's no-early-exercise theorem for American calls need not hold when the underlying is a strict local martingale<sup>[9](https://pi.math.cornell.edu/~soumik/strictlocmgle4.pdf)</sup>.

## References

1. [Lecture 20. Local martingales, Semimartingales and Integrators, Fabrice Baudoin](https://fabricebaudoin.blog/2012/09/17/lecture-20-local-martingales-semimartingales-and-integrators/)
2. [Introduction to stochastic calculus, Djalil Chafaï (course notes)](https://djalil.chafai.net/docs/M2/m2-stochastic-calculus-course-2020-2021.pdf)
3. [Strict local martingales with jumps, Stochastic Processes and their Applications 125 (2015)](https://www.sciencedirect.com/science/article/pii/S030441491400266X)
4. [Local Martingales, Almost Sure (George Lowther)](https://almostsuremath.com/2009/12/24/local-martingales/)
5. [An explosion time characterization of asset price bubbles, Jarrow, International Review of Finance (2023)](https://onlinelibrary.wiley.com/doi/abs/10.1111/irfi.12404)
6. [Lecture notes on continuous-time stochastic processes, A. Thévenin](https://www.normalesup.org/~thevenin/LectureNotes/Lecture_fin.pdf)
7. [Local Martingale, an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/local-martingale)
8. [Advanced Financial Models, lecture notes, Cambridge](https://www.statslab.cam.ac.uk/~mrt31/AFM/notes.pdf)
9. [Analysis of continuous strict local martingales via h-transforms (Cornell copy)](https://pi.math.cornell.edu/~soumik/strictlocmgle4.pdf)
10. [Post/Doctoral Seminar in Mathematical Finance, ETH Zurich, Spring 2016](https://math.ethz.ch/imsf/courses/mf-seminar.html/1000?s=fs16)
11. [Xue-Mei Li, Strict local martingales: examples, arXiv:1609.00935](https://arxiv.org/abs/1609.00935)
12. [Strict local martingales: Examples, Statistics and Probability Letters (2017)](https://www.sciencedirect.com/science/article/pii/S0167715217301645)
13. [Martingale defects in the volatility surface and bubble conditions in the underlying, Review of Derivatives Research 27 (2024)](https://link.springer.com/article/10.1007/s11147-023-09200-x)
14. [Strong bubbles and strict local martingales, Herdegen and Schweizer, International Journal of Theoretical and Applied Finance](https://doi.org/10.1142/S0219024916500229)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Martingales and filtrations › Continuous-time martingales*

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