Hi, able to help with qn 2 & 3 with clear steps and explanations?

2) Let (S )telo, be the solution of the stochastic differential equation dS, = - St 1 dB, (1) VT-t with So > 0. a) (10 marks) Show that (Sthejo,T-] is a martingale on [0, T - =] for every & E (0, T). Hint: Solve the stochastic differential equation (1) by the method of Proposition 6.16-a), and use Exercise 5.11-b). b) (5 marks) Find the value of Sy by a simple argument. c) (10 marks) Show that (Sthejo,7 is a strict local martingale on [0, T']. Hint: Consider the stopping times Th := ((1 - 7 ) T) A if (t = [0, 7] : IS| 2n), n21, and use Proposition 8.1. d) (5 marks) Plot a sample graph of (Sthejo,7 with T = 1, and attach or upload it with your submission. 3) Consider the positive strict local martingale (So)rejo,7 solution of dS, = S/dB, with So > 0, where S, has the probability density function pi(x) = - So (1/x -1/So)" - exp _(1/x + 1/50) 13 v2xt (exp - 2t 2t x >0, LE (0, 7]. a) (5 marks) Plot a sample graph of (S,)ejo,7 with T = 1, and attach or upload it with your submission. b) (10 marks) Compute E[Sy] and check that the condition of Question (1c) is satisfied. Hint: Use the change of variable y = 1/r and the standard normal CDF 4. c) (10 marks) Compute the limit of E[Sr] as So tends to infinity