A Dynamic Programming Approach to Impulse Control of Brownian Motions

This thesis considers an impulse control problem of a standard Brownian motion under a discounted criterion, in which every intervention incurs a strictly positive cost. The value function and an optimal $(\tau_{*}, Y_{*})$ policy are found using the dynamic programming principle together with the smooth pasting technique. The thesis also performs a sensitivity analysis by analyzing the limiting behaviors of the value function and the $(\tau_{*}, Y_{*})$ policy when the fixed intervention cost converges to zero. It is demonstrated that the limits agree with the classic fuel follower problem. The thesis next formulates and analyzes an $N$-player stochastic game of an impulse control problem under a discounted criterion. In the $N$-player stochastic game, each player controls an object. The objects are molded by an $N$-dimensional Brownian motion. A key aspect of the formulation is that each player aims to minimize her total impulse control cost and the total distance of her object to the moving center of the $N$ objects. The interaction mandates the players to closely follow each other\u27s movements. The Nash equilibrium is characterized and analyzed by a system of Hamilton-Jacobi-Bellman equations. The case when $N=2$ is studied in detail

The Impact of Stochastic Convenience Yield on Long-term Forestry Investment Decisions

This paper investigates whether convenience yield is an important factor in determining optimal decisions for a forestry investment. The Kalman filter method is used to estimate three different models of lumber prices: a mean reverting model, a simple geometric Brownian motion and the two-factor price model due to Schwartz (1997). In the latter model there are two correlated stochastic factors: spot price and convenience yield. The two-factor model is shown to provide a reasonable fit of the term structure of lumber futures prices. The impact of convenience yield on a forestry investment decision is examined using the Schwartz (1997) long-term model which transforms the two-factor price model into a single factor model with a composite price. Using the long-term model an optimal harvesting problem is analyzed, which requires the numerical solution of an impulse control problem formulated as a Hamilton-Jacobi-Bellman Variational Inequality. We compare the results for the long-term model to those from single-factor mean reverting and geometric Brownian motion models. The inclusion of convenience yield through the long-term model is found to have a significant impact on land value and optimal harvesting decisions.

Partially reversible capital investment with both fixed and proportional costs under demand risk (Financial Modeling and Analysis)

This study investigates a firm's capital expansion and reduction policy with both fixed and proportional costs when the output demand follows the geometric Brownian motion. We formulate the firm's problem as an impulse control problem and solve it by using quasi-variational inequalities. Through numerical analysis, we find that the output demand risk delays the capital expansion and reduction. Furthermore, the output demand risk decreases the magnitude of capital expansion, but it increases that of capital reduction