Real-Time Detection of Asset-Price Bubbles

Definitions and mathematical statement

Let $S=(S_t)_{t\in T}$ denote the asset-price process mentioned in the source, with $T$ its time index set. This notation does not impose a time domain, state space, probability law, or observation scheme, none of which is specified in the supplied record.

Source question. Can one determine in real time whether a given asset's price process exhibits a bubble?

A schematic representation of the requested task is \[ \text{information available by time }t \ \longmapsto\ \text{a decision whether the price process exhibits a bubble}. \] Here "in real time" requires that the decision use only information available at the decision time. The source does not specify what information is available, whether observations are continuous or discrete, or whether the bubble property concerns the current time, an observed interval, or the entire process.

Unresolved formulation. The supplied record gives no mathematical definition of a bubble. It does not specify a fundamental-value benchmark, valuation model, probability measure, or other structure that would determine such a definition. It also does not specify the admissible class of price processes, the quantifiers over models and decision times, or a criterion for successful detection, such as exact identification, controlled statistical error, or asymptotic consistency. Consequently, the displayed task is a schema preserving the source question; the supplied evidence does not determine a unique precise mathematical assertion. These missing specifications require mathematical review before a definitive formulation can be stated.

Short English statement

The problem asks whether information available at the present time can reveal whether an asset-price process has a bubble. A precise detection problem requires definitions of a bubble, the observable information, the admissible models, and successful detection; the supplied record leaves these unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001688$, title "Real-Time Detection of Asset-Price Bubbles," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and names no individual proposer.
• Supplied report citation: Stochastic Analysis in Finance and Insurance (2011), Oberwolfach Reports, pp. 241--292, DOI 10.4171/owr/2011/06. Source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2011/06. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it provides no problem-specific page locator.
• Supplied literature-source label "Jarrow--Kchia--Protter bubble analysis": https://arxiv.org/abs/1105.5717.
• Supplied literature-source label "recent real-time monitoring method": https://doi.org/10.1016/j.ijforecast.2024.12.005.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and reports mathematical and econometric methods for important model classes, while describing the general problem of deciding whether an observed price process has a bubble as unresolved. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. The record explicitly calls for independent verification of time-sensitive status and cited claims.
• The supplied record labels the statement "exact" and reports that manual review found the extracted question self-contained, mathematically coherent, and free of LaTeX or report-layout artifacts. These are supplied verification claims; they do not resolve the missing mathematical specifications identified above.