Risk Analysis in Finance and Insurance, 2nd Edition

Financial Risk Management and Related Mathematical Tools 31

Further, let



M

n



n=0,1,...

be a fixed square integrable martingale, then one can

consider all square integrable martingales



N

n



n=0,1,...

that are orthogonal to



M

n



n=0,1,...

and introduce a family of square integrable martingales of the

following form

X

n

= M

n

+ N

n

. (1.30)

Conversely, any square integrable martingale



X

n



n=0,1,...

canbewrittenin

form (1.30), where the orthogonal term N has the form of the stochastic inte-

gral (1.29) with the martingale m that is orthogonal to the given martingale

M. This version of decomposition (1.30) is usually referred to as Kunita-

Watanabe decomposition.

Discrete ...