A Filtered Version of the Bipolar Theorem of Brannath and Schachermayer

By Gordan Zitkovic. 2007

46 citations recorded by the source corpus.

We extend the Bipolar Theorem of Kramkov and Schachermayer(12) to the space of nonnegative càdlàg supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer(12) and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market.

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/ff1be2e23262de97dbbbc6c6412f68c992aff55b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

In Finance

In Management Science and Operations Research

Citations in the corpus

Duality theory for portfolio optimisation under transaction costs (2014, methodology): “The existence of a local martingale deflator Z = (Zt)0≤t≤T now allows us to apply part (a) of Theorem 4 in [47] to obtain that Y (φ + φŜ) is a nonnegative local supermartingale and hence a true supermartingale for all (φ, φ) ∈ A(x) and all Y ∈ Y(y; Ŝ).”
Finitely Additive Probabilities and the Fundamental Theorem of Asset Pricing (2009, unknown): “4The notation “X ××(1)” is borrowed from [27] since it is suggestive of the fact that X ××(1) is the process-bipolar of X (1), as is defined in [27].”
Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment (2005, background): “…details, we note that it is the consequence of the continuity of local martingales on Brownian filtrations, the filtered bipolar theorem [Žitković (2002), Theorem 2], and Lemma 2.5, Theorem 2.10 and Proposition 4.1 in Karatzas and Žitković (2003) that (Y Qt )t∈[0,∞) possesses a Pκ…”