Applications to Optimal Stopping Problems

Abstract

The aim of this chapter is to show how some of the established fluctuation identities for (reflected) Lévy processes can be used to solve quite specific, but nonetheless exemplary, optimal stopping problems. To some extent, this will be done in an unsatisfactory way, without first giving a thorough account of the general theory of optimal stopping. However, we shall give rigorous proofs relying on the method of “guess and verify”. That is to say, our proofs will start with a candidate solution, the choice of which is inspired by intuition, and then we shall prove that this candidate verifies sufficient conditions in order to confirm its status as the actual solution.

Exercises

11.1

Suppose that X is a spectrally negative Lévy process. Consider the process $Y^{x} = \{Y^{x}_{t} : t\geq0\}$, where $Y^{x}_{t} = (x\vee\overline {X}_{t}) - X_{t}$. Recall the definition

$$\overline{\sigma}^x_a = \inf\bigl\{ t> 0 : Y^x_t >a\bigr\} , $$

for 0≤x≤a. Use the Poisson point process of excursions, described in Theorem 6.14, to show that

$$\mathbb{P}\bigl(\overline{\sigma}^x_a = \infty\bigr) = \lim_{t\uparrow\infty}\frac {W(a-x)}{W(a)} \exp\bigl\{ -tW'(a)/W(a) \bigr\} , $$

and hence $\mathbb{P}(\overline{\sigma}^{x}_{a}<\infty) = 1$, for all 0≤x≤a.

11.2

The following exercise is based on Novikov and Shiryaev (2004). Suppose that X is a Lévy process and either:

$$q>0\quad\text{or}\quad q=0\quad\text{and}\quad\lim_{t\uparrow \infty}X_t = -\infty. $$

Consider the optimal stopping problem

$$ v(x) =\sup_{\tau\in\mathcal{T}}\mathbb{E}_x \bigl( \mathrm{e}^{-q\tau}\bigl(1- \mathrm{e}^{-(X_{\tau})^+}\bigr)\bigr),\quad x\in\mathbb{R}, $$

(11.33)

where $\mathcal{T}$ is the set of $\mathbb{F}$-stopping times.

• (i) For a>0, prove the identity

  $$\mathbb{E}_x \bigl(\mathrm{e}^{-qT^+_a} \bigl(1 - \mathrm{e}^{ - X_{T^+_a}} \bigr)\mathbf{1}_{(T^+_a <\infty)} \bigr) = \mathbb{E}_x \biggl( \biggl(1- \frac{\mathrm{e}^{- \overline{X}_{\mathbf{e}_q}}}{\mathbb{E}(\mathrm{e}^{ - \overline{X}_{\mathbf{e}_q}})} \biggr) \mathbf{1}_{(\overline{X}_{\mathbf{e}_q} \geq a)} \biggr), $$

  where $T^{+}_{a} = \inf\{t\geq0 : X_{t} \geq a\}$ and $x\in\mathbb{R}$.

• (ii) Show that a solution to (11.33) is given by the pair $(v_{x^{*}}, T^{+}_{x^{*}})$, where $v_{x^{*}}(x)$ is equal to the left-hand side of the identity in part (i) with

  $$a = x^*: = -\log\mathbb{E}\bigl(\mathrm{e}^{- \overline{X}_{\mathbf{e}_q}}\bigr). $$

• (iii) Show that there is smooth fit at x ^∗ if and only if 0 is regular for (0,∞) for X, and otherwise there is continuous fit.

11.3

This exercise is taken from Baurdoux (2007) and is based on a method of Beibel and Lerche (1997). Suppose that X is a spectrally negative α-stable Lévy process, with α∈(1,2) and probabilities $\{\mathbb{P}_{x} : x\in\mathbb{R}\}$. Let η>0 and define for, $x\in\mathbb{R}$,

$$H(x) = \int_0^\infty\mathrm{e}^{ux - u^\alpha} u^{\alpha\eta -1}\mathrm{d}u. $$

Now suppose that h is a function on $\mathbb{R}$ such that there exists some x ^∗ satisfying

$$x^* = \mathrm{argmax}_{x\in\mathbb{R}} \frac{h(x)}{H(x)}. $$

• (i) Show that, for all $x\in\mathbb{R}$,

  $$\frac{H((t+1)^{-1/\alpha} X_t)}{H(x)(t+1)^\eta}, \quad t\geq0 $$

  is a martingale under $\mathbb{P}_{x}$.

• (ii) Use the martingale in part (i) to deduce that, for any stopping time τ,

  $$\mathbb{E}_x \biggl[ \frac{h((\tau+1)^{-1/\alpha} X_\tau)}{(\tau+ 1)^\eta}\mathbf{1}_{(\tau <\infty)} \biggr]\leq H(x)\frac{h(x^*)}{H(x^*)}. $$

• (iii) Define

  $$\tau^* = \inf\bigl\{ t>0 : (1+t)^{-1/\alpha}X_t = x^* \bigr\} . $$

  Assuming that $\mathbb{P}(\tau^{*} <\infty) = 1$ for x<x ^∗,⁵ show that τ ^∗ is an optimal stopping time for

  $$V(x) =\sup_{\tau}\mathbb{E}_x \biggl[ \frac{h((\tau +1)^{-1/\alpha}X_\tau )}{(\tau+ 1)^{\eta}}\mathbf{1}_{(\tau<\infty)} \biggr], $$

  for x<x ^∗, where the supremum is taken over all stopping times for X. Moreover deduce that V(x)=h(x ^∗)H(x)/H(x ^∗).