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Optimal Investment: Utility Functions, Market Completeness, and HJB Equation, Exams of Statistics

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

2012/2013

Uploaded on 02/26/2013

dharmaraaj
dharmaraaj 🇮🇳

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M. PHIL. IN STATISTICAL SCIENCE
Friday 8 June 2007 9.00 to 11.00
OPTIMAL INVESTMENT
Attempt THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Cover sheet None
Treasury Tag
Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
pf3
pf4

Partial preview of the text

Download Optimal Investment: Utility Functions, Market Completeness, and HJB Equation and more Exams Statistics in PDF only on Docsity!

M. PHIL. IN STATISTICAL SCIENCE

Friday 8 June 2007 9.00 to 11.

OPTIMAL INVESTMENT

Attempt THREE questions. There are FOUR questions in total.

The questions carry equal weight.

STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS

Cover sheet None Treasury Tag Script paper

You may not start to read the questions

printed on the subsequent pages until

instructed to do so by the Invigilator.

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

E

[

U

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.

END OF PAPER

Optimal Investment