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A portion of a master of philosophy (m.phil.) thesis in statistical science, focusing on optimal investment. It covers topics such as utility functions, market completeness, and the hamilton-jacobi-bellman equation. Proofs and computations related to these concepts, including the properties of a utility function's conjugate, the relationship between an investor's wealth and utility functions, and the derivation of the hamilton-jacobi-bellman equation. It also discusses the concept of a complete market and its implications.
Typology: Exams
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Friday 8 June 2007 9.00 to 11.
Attempt THREE questions. There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Let U : R → R ∪ {−∞} be a utility function that is finite, twice-differentiable, strictly increasing and strictly concave on the interval (0, ∞) and such that the Inada conditions hold. Let the conjugate function V : R → R ∪ {∞} be
V (y) = sup x> 0
[U (x) − xy].
Show that V is finite, twice-differentiable, strictly decreasing and strictly convex on (0, ∞) and satisfies lim y↓ 0
V ′(y) = −∞ and lim y↑∞
V ′(y) = 0.
Now consider a market with cash (that is, zero-interest rate) and d assets whose prices are given by the d-dimensional process (Sn)n> 0. Assume this market is free of arbitrage. Let u(x) = sup π
XNπ
where XNπ is the wealth at time N for an investor using trading strategy π = (πn)N n=0^ −^1 with initial wealth X 0 = x, and let
v(y) = inf ZN
E[V (yZN )]
where the infimum is taken over all state price densities ZN.
Prove that the inequality
u(x) ≤ inf y> 0
[v(y) + xy]
holds for all x > 0.
What does it mean to say the market is complete? Prove that if the market is complete then there exists a unique state price density. Compute u(x) for x > 0 as explicitly as you can in the case when the market is complete and
U (x) =
log(x) if x > 0 −∞ if x ≤ 0.
Optimal Investment
4 Consider a market with cash and d assets whose prices have stochastic dynamics
dSt = diag(St)(μtdt + σtdWt)
for a Rd-valued Wiener process (Wt)t≥ 0 , a bounded Rd-valued process (μt)t≥ 0 , and a uniformly elliptic d × d matrix-valued process (σt)t≥ 0 , all adapted to the filtration (Ft)t≥ 0.
Consider an investor who does not consume. What is an admissible trading strategy for this investor? What is an arbitrage? Prove that this market is free of arbitrage.
Let Zt = e−^
12 ∫^ t 0 |λs|
(^2) ds−^ ∫^ t 0 λs·dWs
where λt = σ t− 1 μt. Prove that the process (ZtSt)t≥ 0 is a local martingale. Prove that (ZtSt)t≥ 0 is a true martingale if (σt)t≥ 0 is bounded.
Optimal Investment
