Thus it is possible to reduce a given impulse control problem to a sequence of iterated optimal stopping problems.

This is useful both for theoretical purposes and numerical applications. See 6. In other words, Vn is the set of all admissible controls with at most n interventions.

### Description

Moreover, we have Lemma 7. Hence by 7. The main result of this chapter is the following : Theorem 7. To prove this we need a dynamic programming principle or Bellman principle. This principle is due to Krylov [K], Theorem 9 and Theorem 11, p. The proof of the dynamic programming principle for jump processes can be found in Ishikawa [Ish], Section 4.

Lemma 7. Then by 7. This proves that U t is a supermartingale. Then by Corollary 7. Summing 7. Now apply the argument 7. Combined with 7. Remark 7. Note that the proof of Theorem 7. Corollary 7. Let be as in 7. Then by Lemma 7. A similar argument, based on Corollary 7. See Figure 7. In the limiting case when there is no bound on the number of interventions the corresponding value function will be 7.

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Using the notation of Exercise 6. Use Theorem 7. This is an addition to Exercise 6. Thus in this case we achieve the optimal result with just one intervention. The proof is similar to the proof of Theorem 6. Example 8. Suppose there are two investment possibilities, say a bank account and a stock. Let X1 t , X2 t denote the amount of money invested in these two assets, respectively, at time t.

In the absence of consumption and transactions suppose that 8. This consumption is automatically drawn from the bank account holding with no extra costs. In addition the investor may at any time transfer money from the bank to the stock and conversely.

## Applied Stochastic Control of Jump Diffusions (2nd ed.)

The set of admissible controls is denoted by W. See also Shreve and Soner [SS]. See Figure 8. See Example 5. A similar result holds for combined control problems. More precisely, a combined stochastic control and impulse control problem can be regarded as a limit of iterated combined stochastic control and optimal stopping problems. We now describe this in more detail. The presentation is similar to the approach in Chapter 7. Moreover, we have Lemma 8. The proof is similar to the proof of Lemma 7.

Then we have, as in Chapter 7, Theorem 8. The proof is basically the same as the proof of Theorem 7. Use Theorem 8. Exercise 8. Compare with Exercise 3. This models the situation where one is trying to keep X t close to 0 with a minimum of cost of the two controls, represented by the rate u2 t and the intervention cost c. These requirements are made as weak as possible in order to include as many cases as possible.

In fact, it need not even be continuous everywhere. More precisely, they should be interpreted in the sense of viscosity solutions. See [Is2]. However, the nice feature of the viscosity solution is that it also applies to the nonlinear equations appearing in control theory. We will do this in two steps: First we consider the viscosity solutions of the variational inequalities appearing in the optimal stopping problems of Chapter 2.

Then we proceed to discuss more general equations. Theorem 9. First note that 9. So it remains to consider 9. Then by the dynamic programming principle Lemma 7. Hence 9. Therefore the question of uniqueness is crucial. In general we need not have uniqueness.

The following simple example illustrates this: Example 9. Suppose that a. In this case Theorem 2.

## Applied Stochastic Control of Jump Diffusions (2nd ed.)

However, we can use Theorem 9. The corresponding variational inequality is see 9. Let us guess that the continuation region D has the form 9. Then 9. And if 0 9. We have proved: Suppose 9. Theorem 8. However, it turns out that if we interpret 9. This result is an important supplement to Theorem 8. Lemma 9.

### Reward Yourself

This completes the proof of Theorem 9. Many types of uniqueness results can be found in the literature. See the references in the end of this section. Here we give a proof in the case when the process Y t has no jumps, i. We have now ready for the second main theorem of this section: Theorem 9. Let u be a viscosity subsolution and v a viscosity supersolution of 9. Sketch We argue by contradiction. Then by 9. Since v is a supersolution of 9.