Filtrations and Probability Measures in Martingale Models

Summary

The document asks which filtration is intended in a financial martingale model and whether the filtration’s probability measure differs from the measure used to define the martingale. The accepted response identifies the filtration as the one generated by the original or equivalent Brownian process, rather than the alternative process shown in the question. It also emphasizes that a martingale is defined relative to both a filtration and a probability measure, and that the measure associated with the filtration need not be the same as the measure used for the martingale.

The explanation is brief and tentative: the answer cites a reference and reports an instructor’s recollection, but provides no derivation or detailed definitions. It does not fully resolve how risk-neutral and forward measures relate in a specific model. Readers should therefore treat it as a conceptual pointer and consult the referenced material or a formal treatment of measure changes and filtrations for precise conditions.

Key ideas

• A martingale is defined relative to a filtration and a probability measure.
• The filtration in the question is identified as the one generated by the original or equivalent Brownian process.
• The probability measure associated with a filtration need not be the measure under which a process is a martingale.
• The brief answer does not establish the details of the risk-neutral and forward measure relationship.

Tags

# What is the filtration described?


# What is the filtration described?












What is the filtration $(\mathfrak{F}_t)$ encircled below?

Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$?

Or is it $(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in [0,T]$?

The reference (p. 271, 275, 336) and suggests that it is in fact the $(\sigma(W_t)) = (\sigma(\tilde{W_t}))$, but I am not really sure I am reading this right.

If so, does that mean there are 2 probability measures being considered in the martingale? Risk neutral measure for filtration and Forward measure for probability measure>?

## Answer by BCLC (score 0, accepted)

https://quant.stackexchange.com/a/16812

It is the former. Martingales are defined by filtration and probability space*. The probability space* for the filtration need not be the same as the probability space* for the martingale.

I think?

That's what my prof said (iirc).

*specifically the probability measure

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