1973

An Intertemporal Capital Asset Pricing Model

Robert C. Merton

Econometrica, Vol. 41, No. 5. (Sep., 1973), pp. 867-887.

This is one of the series of papers that made a revolution in Finance by introducing methods of stochastic analysis and partial differential equations. Finance applications were one of the inspirations of probability theory from the times of Bachelier but this was the first application that really changed the practice of doing financial transactions.

The basic model of stock returns that Merton uses in this paper is

where , , and are correlated Wiener processes.

Consumer number k maximizes the following inter-temporal utility function

,

where is the static utility function, is the random time of death, is the bequest function , and is wealth at the time of death.

The wealth dynamics is described by the following equation:

,

where y is the wage income.

Finally we have budget constraint

,

or

.

Merton then writes the system in a more consistent way by introducing notation for the state vector of factors, , and writing the evolution equations in a vector form as

.

After that Merton introduces the value function that depends on the wealth, time and factors. Then he refers to his previous papers and writes down the Bellman equation for this system, which is quite complicated. From this equation he derives first order conditions, that are simple but depend on an unknown value function. Finally he speculates about the changes in the demand functions that are caused by the correlation of the stock price and factor Wiener processes.

As an application, Merton derives a theorem that says that if there is only one changing and correlated factor which can be represented by a portfolio tradable assets then all investors should have the portfolios that can be decomposed into riskless asset, market portfolio and the portfolio that represents the factor.

Subadditive Ergodic Theory

J. F. C. Kingman

The Annals of Probability, Vol. 1, No. 6. (Dec., 1973), pp. 883-899.

This is a theory that unified several beautiful results.

The concept of sub-additive process was introduced by Hammersley and Welch. A subadditive process is such a family of random variables , , with nonnegative integer , that the following properties are satisfied:

S1. Whenever s<t<u,

.

S2. The expectation exists, and satisfies

for some constant A and all t>1.

S3. The joint distributions of are the same as of

Theorem1. If x is a subadditive process, then the finite limit

exists with probability one and in mean, and

Remark: If L denotes the field of events defined in terms of x and invariant under the shift , then is L-measurable. So if we are able to prove that L is trivial then is degenerate and

.

For continuous time processes we need to add additional conditions.

The following problems were set:

1) What is a good method to calculate ?

2) How fast is the convergence of the subadditive process?

3) Given a one-parameter process , when can we represent it as a the following difference of two subadditive processes: ?

4) If is a family of additive processes, then is subadditive. Is every subadditive process is represented in this way?

5) A subadditive process has independent increments if for any sequence of , the random variables are independent. What are the consequences of this assumption?

The theory is connected with several beautiful theories. First, it is connected with percolation theory. Let each edge of a graph is associated with a number, then each pair of vertices corresponds to a minimal sum, U, of the edge numbers that connect these two vertices. If there is an automorphism of the graph , then we can define the following subadditive process , where A is an arbitrarily chosen vertex of the graph.

Then there is a connection with the theory of products of random matrices. Namely, let be a stationary sequence of matrices with positive random entries. Then the process is subadditive.

Next is a theorem about norm of product of random Banach algebras. Let be a stationary random element of a Banach algebra. Then is subadditive

Then there is a curious application to the theory of random permutations. The theory of subadditive processes helps to proof the existence of a limit for a maximal increasing sequence in a permutation. We generate a 2-dimensional process and restrict it to a square with coordinates , , , and . The direction on x-axis is initial order of points and the direction on y axis is the after-permutation order of points. (Note that number of points in the permuation is random). Look at the convex hull of the points from above. It defines the longest increasing sequence. Let the number of points in the sequence is . Then is subadditive. From this it follows that limit of exist, where is the length of largest increasing sequence in the permutation of order .

On the Existence of Invariant Measures for Piecewise Monotonic Transformations

A. Lasota; James A. Yorke

Transactions of the American Mathematical Society, Vol. 186. (Dec., 1973), pp. 481-488.

This is a paper from ergodic theory. The power of this paper is that it provides a general class of transformations, for which the absolutely continuous invariant measure exist. Let be a measurable non-singular transformation. The Frobenius-Perron operator is defined by the formula

The main property of this operator is that if and only if the measure is invariant under , that is if for any measurable A.

The following theorem is proved.

Theorem 1. Let be a piecewise function such that . Then for any the sequence

is convergent in norm to a function . The limit function has the following properties:

(1) .

(2)

(3) and consequently the measure is invariant under .

(4) The function is of bounded variation; moreover, there exists a constant independet of the choice of initial such that the variation of the limiting satisfies the inequality

.

The method of proof is by explicitly computing bounds on variation and proving inequality in (4) for . This means relative compactness of the sequence. After that a general theorem (Kakutani-Yosida) of functional analysis is used to prove convergence.

The authors also constructed an example that have the derivative of transformation equal to 1 at only 1 point and still has no invariant absolutely continuous measure.