This document was copied from 2020, and all time references are off! ("last year's", "Easter", ... the order of topics was slightly different)
- Nils

6-10. Diff. eq.: $LaTeX: \displaystyle\frac12\sigma^2x^2V''(x)+\mu xV'(x)-\rho V(x)=w-x$ (Euler-type).

This type of differential equation actually shows up in several optimal stopping (i.e. optimize the time to implement a decision) problems. The application will be discussed at the end; it requires stochastic calculus (my field - but fragments show up in ECON4510 Finance Theory, and last year's Math3 seminar leader Sondre Elstad did use it in his Master thesis, it is possible to grasp a bit of it).

Curriculum clarification:

This is a differential equation you can not be asked to "solve from scratch". However you can be asked to verify solutions and apply the general theory that you are supposed to know.

The problem was assigned to give you an upper bound on what you can be asked on the exam.
As far as (b) (and thus (c)) go, they have an element that I don't consider to be curriculum, but which for a seminar problem is close enough: (b) and (c) test whether you can carry out the C₁u₁+C₂u₂+u* routine, which is on topic, but finding that u* technically isn't. I could have made it within curriculum by specifying the form Ax+B.

General theory recap:

In 6-01(b), we followed the generic form $LaTeX: \ddot x+a(t)\dot x+b(t)x=f(t)$ where the unknown is the function x=x(t) and the solution is C₁u₁+C₂u₂+u* where u*=u*(t) is one particular solution, and u₁=u₁(t) and u₂=u₂(t) are non-proportional solutions of the corresponding homogeneous equation.
6-10 has as unknown a function V=V(x). In order to get the homogeneous equation on your typical form, you can divide by x^2sigma^2 / 2, but as you are given a hint of an x^a form, you can just insert that and see what happens.
- If you get two distinct possible a, you will have two solutions $LaTeX: x^{a_1}$ and $LaTeX: x^{a_2}$ . They are non-proportional functions, so the homogeneous equation will have general solution $LaTeX: C_1x^{a_1}+C_2x^{a_2}$ .
- For the inhomogeneous equation, add a particular solution. (That is guesswork.)

Part (a)

Replace the RHS by zero, insert V(x) = x^a and V'(x) = a x^a^-1 and V''(x) = a(a-1) x^a^-2 into the LHS. The LHS then becomes: $LaTeX: \frac12\sigma^2x^2\cdot a(a-1)x^{a-2}+\mu x\cdot ax^{a-1}-\rho \cdot x^a$ , which equals $LaTeX: x^a\cdot\Big[ a(a-1)\frac12\sigma^2+a\mu-\rho \Big]$ . The bracketed term is a quadratic q(a) in a. and we have a solution if a is chosen so that q(a) = 0.

The assumptions on the parameters grant that there are indeed two such values of a, of opposite sign: the second-order coefficient is >0 and q(0) < 0.
^{(Seriously, you gain a lot from having quadratic functions at your fingertips!)}

That completes the "show that" part. To find the general solution, we should find those roots: $LaTeX: q(a)=\tfrac12\sigma^2a^2+(\mu-\tfrac12\sigma^2)a-\rho$ has the zeroes given in the compendium, top of p. 36.

Parts (b) and (c)

(c) will follow from (b) by inserting the values of the constant.

For (b), it works to try a first-degree polynomial for the particular solution V*(x). Try cx+d; it has a double derivative of zero, so the LHS becomes $LaTeX: 0+\mu x(cx+d)'-\rho (cx+d)=c(\mu-\rho)x-\rho d$ . Since we are given that the constants are positive (hence nonzero) with $LaTeX: \mu\neq \rho$ , we can put $LaTeX: c=-1/(\mu-\rho)$ and $LaTeX: d=-w/\rho$ to get the answer page 36.

Applications note: Problem 6-10 as the right time to buy an asset, using stochastic dynamic programming in continuous time

Did you like applications? If so, here is Easter (or quarantine?) reading for you.

The canonical continuous-time stock price model - if you take ECON4510, you will encounter it - is $LaTeX: ``\ \dot X(t)=X(t)\cdot\Big[\mu +\sigma\cdot\text{white noise}\Big]\ "$ . This in quotation marks, as we need a more thorough mathematical construction (the Ito integral). The "white noise" is taken to aggregate to a standard Gaussian random variable over a timespan of 1, and it is memoryless: noise at distinct times are independent.
You can imagine this as a generalization of a bank account; where the bank account gives you a risk-free interest at rate µ, the yield in the stock market is uncertain - but it is (for a small price-taking investor) proportional to the invested amount X.

There are financial applications of this, https://en.wikipedia.org/wiki/Optimal_stopping#Option_trading ; to fit the problem of 6-10 to a specific case, here is one:

Your discount rate is rho, exponentially.
You hold an asset, the price X developing as above.
By holding it, you get a convenience yield X (say, you rent it out) - but incur a fixed cost w.
- You can at any time dispose of it - irreversibly. That voids all future payoffs.
  Your objective is to optimize that time of stopping. If price is too low and you are running at a loss, then even if prices are expected to increase, your impatience still could mean you want to abandon ship.

Now consider models for this, starting with one which is very close to Mathematics 3

Near-Math3 version: Discrete time, time step = 1.
Your value at price x would be the maximum over decision "stop or not": if you stop, you get zero and that is end of story. Otherwise, you get X-w and then the present value of continuing.
This would give the Bellman equation (sometimes called "Wald-Bellman"): $LaTeX: V(X_t)=\max\big\{0,\quad X_t-w+\beta\cdot \mathsf E[V(X_{t+1})]\big\}$ .
If time-step is smaller, say, h: Instead of a fixed discount rate, use $LaTeX: e^{-\rho h}$ in place of beta:
$LaTeX: V(X_t)=\max\big\{0,\quad (X_t-w)\cdot h+ e^{-\rho h} \mathsf E[V(X_{t+h})]\big\}$ .
Note the latter time index is t+h, not t+1, and that the income (X-w) only is aggregated up over time h.
In the limit as h->0 from above: Assume first a deterministic X'=µX differential equation for X. Then
, and subtracting V from both sides of Wald-Bellman:
or equivalently, since h>0, let h-> 0 to get . That means:
- V ≥ 0 always. You can always get zero by stopping right away, and you stop right away iff the optimal value hits zero.
- If you continue, so V>0, then $LaTeX: 0= x-w-\rho V(x)+V'(x)\cdot \mu x$ which is the diff.eq. of 6-10 except with sigma = 0.
Over to the real deal: it so happens that with this white noise, the expansion for needs a second-order term. Yes, you get a new chain rule for differentiation! Fact:
- So throwing in that second-order term in the equation, yields the equation of 6-10.
- The actual solution of the optimization problem: Find the general solution. Now the interesting solution of the homogeneous equation is the increasing one (this is handwaving, but put the A page 36 equal to zero) and you fit the other constant (B in the notation of p. 36) and a trigger value for x such that
  - $LaTeX: V(\xi)=0$ : we stop when value hits zero
  - $LaTeX: V'(\xi)=0.$ Smooth fit. This condition was allegedly known to Kolmogorov, but I think the first time it appeared in the literature was Samuelson in the mid 60s. Yes, the economist: it was a heuristic but instructive reasoning: much like when "max Cobb-Douglas subject to budget" yields a tangency, you can improve if smooth fit does not hold.
  - Two conditions for two constants. If you want the math behind why I can put A=0, ... oh, that's gonna take some time.