## Stochastic Systems

- Stoch. Syst.
- Volume 3, Number 2 (2013), 500-573.

### Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls

J. G. Dai and Dacheng Yao

#### Abstract

We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy.

#### Article information

**Source**

Stoch. Syst., Volume 3, Number 2 (2013), 500-573.

**Dates**

First available in Project Euclid: 11 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ssy/1392131423

**Digital Object Identifier**

doi:10.1214/11-SSY046

**Mathematical Reviews number (MathSciNet)**

MR3353210

**Zentralblatt MATH identifier**

1298.60084

**Subjects**

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 90B05: Inventory, storage, reservoirs 93E20: Optimal stochastic control

**Keywords**

Impulse control singular control control band verification theorem free boundary problem smooth pasting quasi-variational inequality

#### Citation

Dai, J. G.; Yao, Dacheng. Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls. Stoch. Syst. 3 (2013), no. 2, 500--573. doi:10.1214/11-SSY046. https://projecteuclid.org/euclid.ssy/1392131423