AMA539 Assignment Three
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AMA539 Assignment Three (100 marks in total)
Let {Bt . t > 0} denote a standard one-dimensional Brownian motion.
1. Determine E[Xt] and Var(Xt ), where [10 marks]
4t
Xt = |Bu | dBu r
t
2. Determine
E ┌ / 尸 T 尸 t Bs(1) dBs dBt、1 ┐ r
(Hint: E[Bs(3)] = 3s1 ) [10 marks] 3. Suppose X尸 , μ, and u are positive constants, and
dXt = μXt dt + u dBt r
Determine E[Xt] and E[Xt(1)]. [10 marks]
t
Xt = e −t Bs dBs
尸
is an Itô’s process and determine its quadratic variation process. [10 marks]
5. Let g = inf{t > 0 | Bt(1) + t = 4}. Determine E[Bτ(1)]. [5 marks]
6. Let g = inf{t > 0 | Bt - t (-1. 1)}. Determine E[g]. (Hint: both eaBt − a2 t and Bt are martingales.) [10 marks]
7. Determine if the following utility functions are risk aversion or not.
(a) u(z) = log(3r25(2 - e −1x)3.2上 ), z > 0. [5 marks]
(b) u(z) = inf (Ize −y + y2 - 2y), z > 0. [5 marks]
8. An investor faces two investment alternatives for the coming year. Her first alter- native is to buy Treasury bonds, which will give her a wealth of 10M (M for one million of dollars) for sure at the end of next year. The second alternative has three possible outcomes: 20M , 10M and 5M with corresponding probabilities of 20%, 40%, 40%. She decides to use the utility function (b) in the previous question to evaluate these alternatives (where z is in one million of dollars). Which is preferred? Give your reason. [10 marks]
9. Let
3t 十t
Xt = Bs dBs . yt = eB5 dBs r
尸 1t
Find E[Xt yt]. (Hint: use the moment generating function of normal distribution) [10 marks]
10. Suppose
dXt = Xt (1 - Xt ) dBt . X尸 = 1(〇);
dyt = yt (1 - yt ) dBt . y尸 = 2(〇);
Vt = exp / 尸 t cs(1) ds - 尸 t cs dBs 、.
where ct = 1 - Xt - yt , for all t > 0.
(a) Show that dVt = Vt ct(1) dt - Vt ct dBt r
(b) Show that Xt > yt for all t > 0. (Hint: study (Xt - yt )Vt )
[5 marks] [10 marks]
2022-04-01