English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

IIn this paper we provide predictable and chaotic representations forIt\^{o}-Markov additive processes $X$. Such a process is governed by afinite-state CTMC $J$ which allows one to modify the parameters of theIt\^{o}-jump process (in so-called regime switching manner). In addition, thetransition of $J$ triggers the jump of $X$ distributed depending on the statesof $J$ just prior to the transition. This family of processes includes Markovmodulated It\^{o}-L\'evy processes and Markov additive processes. The derivedchaotic representation of a square-integrable random variable is given as a sumof stochastic integrals with respect to some explicitly constructed orthogonalmartingales. We identify the predictable representation of a square-integrablemartingale as a sum of stochastic integrals of predictable processes withrespect to Brownian motion and power-jumps martingales related to all the jumpsappearing in the model. This result generalizes the seminal result of Jacod-Yorand is of importance in financial mathematics. The derived representation thenallows one to enlarge the incomplete market by a series of power-jump assetsand to price all market-derivatives.

A note on chaotic and predictable representations for It\^o-Markov additive processes