Solve the following problems;

Problem 1 Let de [0, 1] and c = (1 - d)/4. A Markov chain has transition matrix 0 1 2 3 4 5 6 d C C C 0 1 0 1 2 .3 4 0 0 2 0 2 3 4 1 0 0 P = 3 0 4 1 2 0 0 4 0 .4 .1 2 30 0 5 C C C cod 6 0 C C C d 0 (a). Determine, for each value of de [0, 1] the positive recurrent state(s) (if any), and the null recurrent states (if any). (b). Determine, for each value of de [0, 1], the communication classes. (c). For what - if any - value(s) of d will the Markov chain be regular? (d). Let * = (*0, #1, . .., #6) = (0, 7. 7. 7, 7, 0,0). For what - if any - value(s) of d will * be i) a stationary distribution? ii) a limiting distribution? (e). Assume that the Markov chain starts in state 6. Determine, for each value of de [0, 1] for which state 6 is transient, the expected time until the Markov chain leaves the communication class state 6 belongs to.Problem 2 Throughout this problem, Y will be a birth/death-process with state space {0. 1, 2}. The intensity of an upward jump from state i ( {0, 1} will be 2 -i. The inten- sity of a downward jump from state ic {1, 2} will be }?. (a). Write down the generator.Problem 4 Throughout this problem, let 7 > 0, r > 0 and * 2 1 be given constants. Consider for time t e [0, 7] a frictionless arbitrage-free market consisting of one investment opportunity whose price at time t is deterministic and equal to ert, and furthermore an in- vestment opportunity with stochastic price S(t) at time t, with S(0) = 1. Consider three Arrow-Debreu securities with arbitrage-free prices as follows: . security #1 pays 1 if S(T) 2 k (and 0 otherwise), and has price c, at time 0, . security #2 pays 1 if S(T) S(T) 2 1/k (and 0 otherwise), and has price c; at time 0. (a). Find c for the case when S(t) = exp ((u - 202)1 + OB(t)) (gBm) where / 2 r and o * 0 are given constants, and where B(t) is standard Brownian motion starting at B(0) = 0. (b). It is a fact that c +oz + c3 = e " when S is given by formula (gBm) above. However, there are other cases where there is inequality. Let & = 1 and find a suitable condition on the process S for which a + c2 + c3 * e ", and decide whether the sum is then >el or condition, and consider instead for the rest of the problem, the system of stochastic differential equations for X = (X1, X2, X3) T: dXi(t) = m(X2(t) - Xi(t))dt + o, dB(t) dX2(t) = ra(X3(t) - X2(t))dt + 0, dB(t) (3) dX3(t) = ra(m - Xa(t))dt + 0; dB(t). where all three r, are > 0, and where X starts at X(0) = x.(g). Assume that E is the identity matrix. Use your diagonalisation (4) to simplify the solution X of (3). Your formula should not include A explicitely. (Again, you are not supposed to calculate U-1.) (h). Again, consider equation system (3) written on the form (1), but assume now that 71 = = r. Is the matrix A now diagonalisable