Let W be a two-dimensional Brownian motion defined on the probability space the augmented natural filtration generated by W. Consider an economy defined for the finite time interval and composed of...

Let W be a two-dimensional Brownian motion defined on the probability space

the augmented natural filtration generated by W. Consider an economy defined for the finite time interval

and composed of three assets having price process

satisfying

Where

are continuous

martingales such that

for constants

and

(i) Let E be an economy defined on the probability space

is a numeraire pair for E and let VT be some attainable contingent claim. Starting from the martingale valuation formula, equation (7.15), show that

for some pricing kernel Z.

(ii) Find a pricing kernel for the economy described in Exercise 10. Is this economy complete? Write down the Radon–Nikod´ym derivative

Where

denotes an EMM corresponding to the numeraire D.