Help. me to answer the following attachments.

(i) Consider two probability measures, P and Q. (a) State the conditions under which the two measures are equivalent. (b) State the necessary and sufficient conditions for a continuous process I, to be a Brownian motion under measure P (called a P-Brownian motion). [3] (ii) State the Cameron-Martin-Girsanov theorem for a P-Brownian motion process W, and an F-previsible (and bounded) process 7, where F, is a filtration (history) of events up to time t. [2] (iii) Let W, be a continuous P-Brownian motion process. Let X, = uf+owl, be a drifting Brownian motion process, where u and o are both constant. (a) Show that there exists a measure Q, equivalent to P, under which X, is a driftless Q-Brownian motion. [Hint: Consider the ratio Y =~ (b) Find the volatility of X, under both P and Q. (c) Evaluate the Radon-Nikodym derivative aQ at time t. [6] dPProblem 1: motivating workforce (7 Pts.) An individual has preferences over leisure hours during the day (21 ) and dollars spent on consumption of goods (12) given by u(71, 12) = 1412 Her endowment is (24,0), and her hourly wage is $ 10. a) Find the individual's choice of leisure and consumption (r;, r;), and her supply of labor s" b) Her employer, who needs to motivate her to work some more hours, is contemplating two possible solutions: either raise the salary to $14, or pay her overtime: keep the salary to $10 for the first s' hours of work (the s' you found in item a)), but raise it to $16 for any hour worked above s'. Find the individual's supply of labor under the two schemes to figure out which of the two approaches is more succesful. (Hint: for overtime, draw carefully the budget set, and the indifference curve going through the choice (ar;. r;) from a), to understand what is going on). c) Even if you had not found the exact labor supplies in part b), could you make a general argument, independent of the particular utility and the wages at play, for why one of the two payment schemes would be more succsessful than the other?Math 81 Final Exam Spring 2004.doc Directions: Show all work in a neat and organized manner. Name: Part I (Computational, Show all steps on 4-7 and only verify with calculators): 1. The solution set of Ax - b. Solve the system and write in the form x = p +h , where p is the I - 21, +1 -1, -4 2x, - 3x, + 4x, - 3x, = -1 particular solution and h is the homogeneous solution. 3.x, - 5x, + 5x, -4x, - 3 -X, + X -3x, + 2x, - 5 2. Determine whether S = { - x,2-3x, x + 2x} is a basis for the vector space P, of polynomials of degree at most 2 together with the zero polynomial. 3. Let F be the vector space of all functions f: R - R, and let cER . Show that the evaluation function T : F -- R defined by 7 (f)- f (c), which maps each function fin F into its value at c. is a linear transformation. [5x, - 21, + 1 -1] 4. Solve the system using Cramer's rule. 3x, +2x, = 3 . (Show all steps in calculating the determinants. Check it on your calculators.) 2 1 5. Find the eigenvalues and eigenvectors of the matrix / - -1 0 1 . State the algebraic and 1 3 geometric multiplicity. -3 31 6. Diagonalize the matrix A = 0 6 10 -3 " Find a unitary matrix C that diagonalizes the Hermitian matrix A - - 1 6Exercise 2. Load the beersales dataset in the TSA package by running data(beersales). You will model monthly beersales in millions of barrels and perform the following tasks. (a) Create plots using tsdisplayo and comment on them. [5] (b) Estimate a trend by smoothing the series and plot it. What order do you choose for the moving average smoother and why? [5] (c) Hence decompose the time series as an additive model with trend, seasonal, and random components. Find the seasonal and random components and plot them against time. Centre the seasonal component on zero. [5]