Problem 5: Let (∆(0),∆) be a portfolio with the associated wealth process X. Assume that ∆(0) and ∆ are adapted processes and that X satisfies the self-financing condition: X =∆ S +∆(0) (1+r)n =∆ S +∆(0)(1+r)n, n=1,...,N. n n−1 n n−1 n n n

(a). Is X an adapted process?

(b). Forn=1,...,N, show thatXn−Xn−1 =∆n−1(Sn−Sn−1).

WriteXn := Xn n andSn := Sn n forn=0,...,N. (1+r) (1+r)

(c). Prove that X is a P-martingale.
(It is an intro to math finance course problem)

Use set builder notation to represent the following set: { -3, -2, -1, 0}

How to do this , and can some one explain

In new york city at the spring equinox there are 12 hours 8 minutes of daylight. the longest and shortest days of the year very by two hours and 53 minutes from the equinox in this year the equinox falls on march 21 in this task you use trigonometric function to model the hours of daylight hours on certain days of the year in new york city a. what is the independent and dependent variables? b. find the amplitude and the period of the function. c. create a trigonometric function that describes the hours of sunlight for each day of the year. d. graph the function you build in part c. e. use the function you build in part c to find out how many fewer daylight hours february 10 will have than march 21. you may look at the calendar.