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Kunita–Watanabe Decomposition of a Martingale

Article Quant Q&A · Author: glork
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Summary

The question asks why a martingale of conditional expectations can be represented as a stochastic integral against another martingale plus a residual martingale orthogonal to it. The answer identifies this representation as the Kunita–Watanabe decomposition and points to external presentations for its derivation.

Conceptually, the decomposition separates the component of a martingale that can be expressed through stochastic integration against a chosen martingale from the remaining orthogonal component. This is useful in stochastic analysis and can support reasoning about hedging and projection onto traded sources of risk. The document itself does not state the theorem’s assumptions, define the precise meaning of orthogonality, or provide a proof. In practice, those details matter: existence and uniqueness depend on conditions on the martingales and integrands, so the brief identification should be treated as a pointer to the underlying result rather than a complete explanation.

Key ideas

  • The representation is identified as the Kunita–Watanabe decomposition.
  • It splits a martingale into a stochastic integral against a selected martingale and an orthogonal residual martingale.
  • The decomposition provides a way to separate the component associated with one source of martingale risk.
  • The brief answer points to references but does not state the theorem’s assumptions or prove it.

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Full text
# martingale decomposition problem


# martingale decomposition problem












Let $G_{t}$ be a filtration and $M_{t}$ a $G_{t}$-martingale. Why do we have this decomposition: $H_{t}=\mathbb{E}[H|G_t]=\int_{0}^{t}h_{s}dM_{s}+R_{t}$ where $R_{t}$ is a martingale orthogonal with M

Thank you.

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/17671

As per your comments, this is the Kunita Watanabe decomposition. See the post at https://math.stackexchange.com/questions/413103/kunita-watanabe-decomposition and the presentation http://www.eurandom.nl/events/workshops/2011/ISI_MRM/Presentation/Vanmaele.pdf

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