2.考虑一个从0开始以财富wealth0开始的一期投资者。他们必须选择

𝑅=

2，概率𝑝1= 1
2个
;
1，概率𝑝2= 1
4
;
0，概率𝑝3= 1
4

（a）从问题描述中确定时间1𝑉1处的值函数。 [1分]
（b）用𝑤0，𝑎和𝑅记下时间1的财富演化方程。推断

（c）使用您的最优性方程，否则找到𝑉0（𝑤0），即期望的最大混合
log（𝑊）的值，其中𝑊是时间1的财富。[8分]

（d）陈述𝑉𝑛−1（𝑥）的公式并描述𝑉𝑛−𝑖的一般模式
。 [6分]
（e）不用证明，以𝑤0和terms表示𝑉0（𝑤0）的公式。 [2分]
（f）如果𝑝1=𝑝2=𝑝3= 1，则解如何变化
3
？解释你的答案。 [4分]
（g）如果我们固定𝑝2= 1
3

3
−𝑝1描述了在时间− − 1时的最佳投资为
𝑝1在2之间变化
3

2. Consider a one-period investor who begins at time 0 with wealth 𝑤0. They must choose
at time 0 to invest a quantity 𝑎 ∈ [0,𝑤0] which provides multiplicative returns 𝑅, i.e. 𝑎
becomes 𝑎𝑅, where
𝑅 =

2, with probability 𝑝1 = 1
2
;
1, with probability 𝑝2 = 1
4
;
0, with probability 𝑝3 = 1
4
.
Uninvested wealth is just carried forward to time 1 unchanged. The investor’s objective is
to maximize utility of (terminal) wealth at time 1, using utility function 𝑈(𝑥) = log(𝑥).
(a) Identify the value function at time 1, 𝑉1, from the problem description. [1 mark]
(b) Write down the evolution equation for wealth at time 1 in terms of 𝑤0, 𝑎 and 𝑅. Deduce
the optimality equation for this problem, relating 𝑉0(𝑤0) to the function 𝑉1. [5 marks]
(c) Using your optimality equation, or otherwise, find 𝑉0(𝑤0), i.e. the maximixal expected
value of log(𝑊), where 𝑊 is the wealth at time 1. [8 marks]
Now consider an 𝑛-period version of this terminal wealth problem.
(d) State the formula for 𝑉𝑛−1(𝑥) and describe the general pattern for 𝑉𝑛−𝑖
. [6 marks]
(e) State without proof, the formula for 𝑉0(𝑤0) in terms of 𝑤0 and 𝑛. [2 marks]
(f) How does the solution change if 𝑝1 = 𝑝2 = 𝑝3 = 1
3
(g) If we fix 𝑝2 = 1
3
and set 𝑝3 = 2
3
− 𝑝1 describe the optimal investment at time 𝑛 − 1 as
𝑝1 varies between 2
3
and 0. Explain your answer. [4 marks]

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