What makes for a beautiful problem in Science?

In a paper with the title, What makes for a beautiful problem in Science? Paul Samuelson (Journal of Political Economy, Vol. 78, No. 6, 1970), writes:

What is the shortest distance between two points? A straight line. Can you prove it?

What is the longest distance between two points? Surely, no answer is possible. But consider two points a mile apart on a horizontal line in the plane. Suppose that the admissible paths cannot have an absolute slope exceeding one million. Then it is not hard to envisage that the longest distance is a bit over a million miles. But it takes the methods of Pontryagin and not those of the classical theory to handle such a problem in which the control variables are constrained within a closed set.

Following Session 9 you might like to see if you can use Pontryagin's maximum principle to prove what Samuelson claims. You must start deciding what he means by an admissible path. I expect he is thinking about a curve between $(0,0)$ and $(1,0)$ that is a graph $y(x)$ determined by $y'(x)=u(x)$, $0\leq x \leq 1$. I have added (sic) (for sic erat scriptum) because I think the word "not" should be deleted. He  is trying to say that the maximum distance is a bit over a million miles, in fact $\sqrt{10^{12}+1}$. Samuelson goes on:

A scientific problem gains in aesthetic interest if you can explain it to your neighbor at a dinner party. 

That no map requires more than four colors is such a problem. Can the turnpikes of Professor Chakravarty pass this test? In a system in which everything produces everything, a critical set of proportions at which balanced growth is most rapid seems reasonable. So the general notion of the von Neumann turnpike is conveyed. But suppose we begin away from this highway, and also are to end up away from it. Still, if the journey is to be a sufficiently long one, as seen we shall do well to move in the beginning toward that fast highway; catch a fast ride along it, staying so long there as to leave behind in all directions any rival who sticks to any slower path; then, toward the end of our journey, we can swing off the turnpike to our final rendezvous.

We will see this in Session 9 (the turnpike of economic growth, Section 15.6).

I cannot resist one more excerpt from Samuelson's paper, in which he muses on the way in which an optimal path may sometimes seem strange.

Begin with durable hammers and robots that can make new hammers or robots with the same symmetrical input requirements for the one as for the other. Each input is subject to familiar diminishing returns; but increasing both inputs proportionally leads to constant returns to scale. I begin with 10 robots and 5 hammers and must end up in minimum time with (at least) 1,000 robots and 5,000 hammers. How should I proceed? Surely, I begin by producing only hammers, since the redundant robots are subject to diminishing returns. When I reach 10 of each input, I shift gears to produce equal amounts of both, proceeding along what is known as a von Neumann-DOSSO turnpike. You might think I go along it until 1,000 of each are attained, thereafter producing only the 4,000 more hammers. And you might be right. But you might be wrong. Depending upon the exact technology, it might pay to produce more of both along the turnpike, ceasing to produce more robots only after they are somewhere between 1,000 and 5,000 in number. We end up with more than we needed of robots. Irrational, since we don't want them and they cost something? Not necessarily, since they may have been worthwhile for their help in producing the extra hammers we do want.

Kalman Filter in economics and finance

To follow up a question this morning from Yiangos. I asked my colleague Lord Eatwell, a famous economist and President of my college, "Is the Kalman filter much used in economics". He immediately replied, "Yes, all the time".

Eatwell is one of the compliers of the Palgrave Dictionary of Economics. I searched in Palgrave for the Kalman Filter, and read:

The Kalman filter

The Kalman filter deals with state-space representations where the transition and measurement equations are linear and where the shocks to the system are Gaussian. The procedure was developed by Kalman (1960) to transform (‘filter’) some original observables $y_t$ into Wold innovations at and estimates of the state $x_t$. With the innovations, we can build the likelihood function of the dynamic model. With the estimates of the states, we can forecast and smooth the stochastic process.

The use of unobserved components opens up a new range of possibilities for economic modelling. Furthermore, it provides insights and a unified approach to many other problems. The examples below give a flavour.

The local linear trend model generalizes (1) by the introduction of a stochastic slope, βt, which itself follows a random walk. Thus ...

I recommend the New Palgrave, as a good place to go if you need to find a short article on any economics topic. My short answer to Yiangos's question is that econometricians are always engaged in building linear models in which some variables are explained by other variables (in some sort of regression model). As the values of variables become known over time one wants to update estimates of other variables. The machinery for doing this is provided by the Kalman filter. Notice that the Kalman filter does not have anything to do with the Q assumptions of our LQG model. It is only the Linear Gaussian parts that are relevant.

I tried a Google search for "Kalman Filter and finance". Many things turn up. For example, here is a talk, Kalman Filtering in Mathematical Finance. 

Session 8

I have added to the notes, on page 77, the argument by which we can verify that in the fishing problem the value function $F(x)$ is concave.

The homework for next week is just three questions: the first three on Problems Sheet 4. See hints below.

The name "Kalman filter" refers to the estimation equation (16.1) and takes its name from Rudolf Kalman (1930 –), who developed it in the years 1958-64. He also coined the terms controllable and observable, and gave the criteria that we have seen in previous lectures. The fact that a system is controllable iff the matrix $[B\ AB\ \cdots\ A^{n-1}B]$ is of full rank is sometimes called Kalman's criteria. In the IEEE biography of Kalman it is stated

The Kalman filter, and its later extensions to nonlinear problems, represents perhaps the most widely applied by-product of modern control theory. It has been used in space vehicle navigation and control (e.g. the Apollo vehicle), radar tracking algorithms for ABM applications, process control, and socioeconomic systems.

The theory in this lecture is admittedly quite tricky - partly because the notation. As a test of memory, can you say what roles in the theory are taken by each of these?

 $x_t$, $\hat x_t$, $y_t$, $u_t$, $\epsilon_t$, $\eta_t$, $\Delta_t$, $\xi_t$, $\zeta_t$, $A$, $B$, $C$, $R$, $S$, $Q$, $K_t$, $\Pi_t$, $N$, $L$, $M$, $H_t$,  $V_t$. 

 You will understand the ideas better once you have worked through the details of a scalar example (in which $n=m=p=1$). You do this in Problems Sheet 4 Question 1. When you do this question, start by supposing that $\hat x_t=\hat x_{t-1}+u_{t-1}+h_t(y_t-\hat x_{t-1})$, and then find the value of $h_t$ that minimizes the variance of $\hat x_t$. You can start by subtracting $x_t=x_{t-1}+u_{t-1}+3\epsilon_t$ and using $y_t=x_{t-1}+2\eta_t$. I also think that Question 2 is helpful in gaining an appreciation of the duality between control and estimation problems. One of the things you will learn from this question is a sufficient condition for the variance of $\hat x_t$ (that is, $V_t$) to tend to a finite limit. Question 3 is about solving a HJB equation (easier than the fishing example).

Notice that the Riccati equation for $V_t$, i.e. $V_t = g\, V_{t-1}$ runs in the opposite time direction to the one we had for $\Pi_t$ in Chapter 13, where $\Pi_{t-1} = f\, \Pi_t$. We are given $V_0$ and $\Pi_h$.