Non-robustness of impulse control Non-robustness of some impulse control problems with respect to intervention costs Bernt Øksendal 1,2 , Jan Ubøe 3 , and Tusheng Zhang 3 1 Department of Mathematics, University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway 2 Norwegian School of Economics and Business Administration, Helleveien 30, 5035 Bergen- Sandviken, Norway 3 Stord/Haugesund College, Sk˚ aregaten 103, N-5 500, Haugesund, Norway ABSTRACT. We study how the value function (minimal cost function) V c of certain impulse control problems depends on the intervention cost c. We consider the case when the cost of interfering with an impulse control of size ζ∈R is given by c+|ζ| with c≥0,λ>0 constants, and we show (under some assumptions) that V c is very sensitive (non-robust) to an increase in c near c=0 in the sense that dV c dc   c=0 =+∞ 1. Introduction A mathematical model is often a tradeoff between i) mathematical simplicity and tractability on one hand and ii) accuracy in the description of the real life situation that the model claims to represent, on the other. In view of this, a natural requirement for a model to be good is robustness with respect to the parameters involved. For example, if some of the values of the parameters change slightly, this should not cause a too dramatic change in the conclusions from the model. The purpose of this paper is to study one such robustness question in connection with a class of impulse control problems. More precisely, we study a class of impulse control problems of 1-dimensional jump diffusion processes where the cost of interfering with an impulse of size ζ ∈ R is given by c + λ|ζ | where c ≥ 0,λ > 0 are constants. The constant λ is called the proportional cost coefficient and the constant c is called the intervention cost. The value function/minimal cost function corresponding to c when the jump diffusion starts at y is denoted by V c (y). (See precise definitions below.) Several authors have adressed impulse control problems with a similar type of cost functional, see, e.g., [BL], [BØ2], [F], [HST], [JS], [LØ], [MØ], [MR1], [MR2], and [V]. For the particular impulse control problem to be studied below, it is well known that the mapping c → V c (y) is continuous at c = 0, see [MR1]. Continuity alone, however, is not sufficient for robustness of the construction. Consider f [x]=  − 1000 ln[x] if x> 0 0 if x =0 Certainly, x → f [x] is continuous at x = 0. Changing x from x = 0 to x = 1 10 000 , we change the value of f [x] from 0 to more than 100. This change is in no proportion to the change in x. In fact, from a practical point of view it may be difficult to distinguish such a behaviour from 1