29510776
Acquisition of Project-Specific Assets with Bayesian Updating
We study the impact of learning on the optimal policy and the time-to-decision in an infinite-horizon Bayesian sequential decision model with two irreversible alternatives: exit and expansion. In our model, a firm undertakes a small-scale pilot project to learn, via Bayesian updating, about the project's profitability, which is known to be in one of two possible states. The firm continuously observes the project's cumulative profit, but the true state of the profitability is not immediately revealed because of the inherent noise in the profit stream. The firm bases its exit or expansion decision on the posterior probability distribution of the profitability. The optimal policy is characterized by a pair of thresholds for the posterior probability. We find that the time-to-decision does not necessarily have a monotonic relation with the arrival rate of new information.
118683853
Optimal Stopping and Free-Boundary Problems
Optimal stopping: General facts.- Stochastic processes: A brief review.- Optimal stopping and free-boundary problems.- Methods of solution.- Optimal stopping in stochastic analysis.- Optimal stopping in mathematical statistics.- Optimal stopping in mathematical finance.- Optimal stopping in financial engineering.
154386984
Investment and Hysteresis
It seems that firms behave contrary to the standard economic theory of investment. We observe that firms do not invest as soon a price rises above long-run average cost; instead firms wait until price rises substantially above long-run average cost. On the downside, firms stay in business for lengthy periods while absorbing operating losses, and price can fall substantially below average variable cost without inducing disinvestment or exit. Recent developments in the theory of investment under uncertainty offer an interesting new explanation. The new approach builds on an interesting analogy between real investments and options in financial markets: In the timing of investment, waiting has positive value because time brings more information about the future prospects of a project. This new approach suggests that textbook pictures of the dynamics of a competitive industry need to be substantially redrawn. More generally, it says that a great deal of inertia is optimal when dynamic decisions are being made in an uncertain environment.
154490860
Experimentation in Markets
We present a model of entry and exit with Bayesian learning and price competition. A new product of initially unknown quality is introduced in the market, and purchases of the product yield information on its true quality. We assume that the performance of the new product is publicly observable. As agents learn from the experiments of others, informational externalities arise. We determine the Markov Perfect Equilibrium prices and allocations. In a single market, the combination of the informational externalities among the buyers and the strategic pricing by the sellers results in excessive experimentation. If the new product is launched in many distinct markets, the path of sales converges to the efficient path in the limit as the number of markets grows.
3212528
Resources, real options, and corporate strategy
The types of investments a ﬁrm undertakes will depend in part on what it expects the outcome of those investments to reveal about its skills, capabilities, and assets (i.e., its resources). We predict that a ﬁrm will specialize when young, then experiment in a new line of business for some time, and then either expand into a large, multisegment business or focus and scale up its specialized business. We derive several empirical implications for ﬁrm valuations and the reaction of stock prices to news about ﬁrm prospects. We also offer a novel explanation for the well-documented ‘‘diversiﬁcation’’ discount.
154753537
Information Acquisition and the Adoption of New Technology
The profitability of a new technology is rarely known with certainty at its announcement date. Consequently, prior to making an adoption decision it behooves the firm considering the adoption of this innovation to reduce the level of uncertainty associated with its profitability. The firm accomplishes this by sequentially gathering information, updating its prior estimate of profitability in a Bayesian manner. Quantifying the uncertainty regarding the innovation permits application of dynamic programming techniques: criteria are derived which tell the firm when to stop collecting information and make the adoption decision. It will be shown that it is optimal for the firm to continue to collect information until its estimate of profitability crosses one of two thresholds: upon crossing the upper threshold the firm adopts the technology, whereas the firm rejects the technology if the lower threshold is crossed. The model predicts that even the manager who behaves optimally will occasionally adopt unprofitable technologies and reject profitable ones.
14309644
Uncertainty, Information Acquisition, and Technology Adoption
Consumers or firms contemplating purchasing a new product or adopting a new technology are often plagued by uncertainty: Will the benefits outweigh the costs? Should we buy now or wait and gather more information? In this paper, we study a dynamic programming model of this technology adoption problem. In each period, the consumer decides whether to adopt the technology, reject it, or wait and gather additional information by observing a signal about the technology's benefit. The technology's actual benefit may be constant or changing stochastically over time. The dynamic programming state variable is a probability distribution that describes the consumer's beliefs about the benefits of the technology. We allow general probability distributions on benefits and general signal processes and assume that the consumer updates her beliefs over time using Bayes' rule. We are interested in structural properties of this model. We show that improving the technology's benefit need not make the consumer better off and that first-order stochastic dominance improvements in the consumer's distribution on benefits need not increase the consumer's value function. Nevertheless, the model possesses a great deal of structure. For example, we obtain monotonic value functions and policies if we order distributions using likelihood-ratio dominance rather than first-order stochastic dominance. We also examine convexity properties and provide many comparative statics results.
122901079
SEQUENTIAL ANALYSIS: SOME CLASSICAL PROBLEMS AND NEW CHALLENGES
We give a brief review of the developments in several classical problems of sequential analysis and their applications to biomedicine, economics and engi- neering. Even though it can only focus on a limited number of topics, the review shows that sequential analysis is still a vibrant subject after six decades of contin- ual development, with fresh ideas brought in from various fields of application and through interactions with other branches of statistics and probability. We conclude with some remarks on the opportunities and challenges ahead.
123624130
Stochastic differential equations : an introduction with applications
Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.
35449775
Reward functionals, salvage values, and optimal stopping
Abstract. We consider the optimal stopping of a linear diffusion in a problem subject to both a cumulative term measuring the expected cumulative present value of a continuous and potentially state-dependent profit flow and an instantaneous payoff measuring the salvage or terminal value received at the optimally chosen stopping date. We derive an explicit representation of the value function in terms of the minimal r-excessive mappings for the considered diffusion, and state a set of necessary conditions for optimal stopping by applying the classical theory of linear diffusions and ordinary non-linear programming techniques. We also state a set of conditions under which our necessary conditions are also sufficient and prove that the smooth pasting principle follows directly from our approach, while the contrary is not necessarily true.
6710763
A Sequential Entry Problem with Forced Exits
We consider a continuous time optimal stopping problem with multiple entries and forced exits. The value for such an optimization problem with a general payoff function is solved in closed form under the assumption that the state process is a geometric Brownian motion and the forced exits come in according to a Poisson process. The effect due to the forced exits is analyzed. It is shown that the presence of the forced exits is a true risk (meaning that it will reduce the value and enlarge the "continuation" region) if and only if the entry cost is large enough compared to the running cost.
153168206
Optimal Experimentation in a Changing Environment
This paper studies optimal experimentation by a monopolist who faces an unknown demand curve subject to random changes, and who maximises profits over an infinite horizon in continuous time. We show that there are two qualitatively very different regimes, determined by the discount rate and the intensities of demand curve switching, and the dependence of the optimal policy on these parameters is discontinuous. One regime is characterised by extreme experimentation and good tracking of the prevailing demand curve, the other by moderate experimentation and poor tracking. Moreover, in the latter regime the agent eventually becomes `trapped' into taking actions in a strict subset of the feasible set.
152920650
The Optimal Level of Experimentation
We assume that an impatient decision maker (DM) runs variable-size experiments at an increasing, strictly convex cost before choosing an irreversible action. We introduce and solve a tractable continuous time version of this problem --- a control of variance of a diffusion with uncertain mean. Assuming two states and two actions, we prove: (a) the optimal experimentation level rises in the Bellman value; and deduce testable implications, like (b) experimentation costs drift up; and (c) a more impatient decision maker may experiment more, given lump-sum final payoffs. We show that (a) and (b) are robust to finitely many states and actions, and we also extend an R&D interpretation of the model, where experimentation is monotonic not only in the value, but also in beliefs. Our intuition for our key monotonicity finding (a) is very economic. There are two decisions at each instant: stop or experiment, and then at what level n. The second choice equates the marginal costs and benefits of information: c'(n) = MB(n). In our diffusion setting, the marginal benefit of experimentation is constant, and so the total benefit TB is linear in the level: TB = n*MB = nc'(n). So the DM acts like a neoclassic competitive firm, producing information at an increasing marginal cost and selling it to himself at the fixed price c'(n). Since postponing the final decision entails a discounting cost, optimal stopping demands that the DM equate his producer surplus from experimentation nc'(n)-c(n) to the delay cost rV (given the interest rate r). Intuitively, the DM closes down his information firm (i.e. acts) when he cannot generate profits (producer surplus) to justify his capital rental (his deferred action). Since this surplus rises in quantity with convex costs, greater experimentation is needed to generate the higher surplus for a higher value V.
122556484
OPTIMAL EXIT FROM A PROJECT WITH NOISY RETURNS
We consider the problem of selecting a stopping time τ which determines when to exit an investment project when the project's cumulative profit up to time t is Xt, where {Xt : t ≥ 0} is a Brownian motion with drift μ and variance σ2. The profit rate μ never changes over time, but μ is not directly observable. Specifically, μ takes the value μH > 0 when in the high state and μL < 0 when in the low state, and the initial probability p0 that the project is in the high state is known. The decision-maker seeks to maximize the expected discounted profit up to time τ. Using the theory of stochastic differential equations, we show that it is optimal to exit only when the posterior probability Pt of being in the high state falls below a critical number p*, and we produce a simple, closed form for p*. Our most surprising comparative-statics result is that the expected discounted profit increases with |μL|, provided |μL| is large.
1864820
Investment Timing Under Incomplete Information
We study the decision of when to invest in a project whose value is perfectly observable but driven by a parameter that is unknown to the decision maker ex ante. This problem is equivalent to an optimal stopping problem for a bivariate Markov process. Using filtering and martingale techniques, we show that the optimal investment region is characterized by a continuous and nondecreasing boundary in the value-belief state space. This generates path-dependency in the optimal investment strategy. We further show that the decision maker always benefits from an uncertain drift relative to an average drift situation and that the value of the option to invest is not globally increasing with respect to the volatility of the value process.
11741767
Irreversible investment in alternative projects
Summary.We study the problem of a risk-neutral decision-maker who has to choose among two alternative investment projects of different scales under output price uncertainty. We provide parameter restrictions under which the optimal investment strategy is not a trigger strategy and the optimal investment region is dichotomous. Whenever the decision-maker has the opportunity to switch from the smaller scale to the larger scale project, the dichotomy of the investment region can persist even when the volatility of the output price process becomes large.
41924562
Invest or Exit? Optimal Decisions in the Face of a Declining Profit Stream
Even in the face of deteriorating and highly volatile demand, firms often invest in, rather than discard, aging technologies. To study this phenomenon, we model the firm's profit stream as a Brownian motion with negative drift. At each point in time, the firm can continue operations, or it can stop and exit the project. In addition, there is a one-time option to make an investment that boosts the project's profit rate. Using stochastic analysis, we show that the optimal policy always exists and that it is characterized by three thresholds. There are investment and exit thresholds before investment, and there is a threshold for exit after investment. We also effect a comparative statics analysis of the thresholds with respect to the drift and the volatility of the Brownian motion. When the profit boost upon investment is sufficiently large, we find a novel result: the investment threshold decreases in volatility.
122158202
On optimal timing of investment when cost components are additive and follow geometric diffusions
Abstract The paper gives a partial characterization of the optimal stopping rule for a problem with reward function g ( t, x ) = e − rt ( x 1 − x 2 − ··· − x n ), where x 1 ,…, x n follow geometric Brownian motions. The characterization takes the form of a simple and operational sufficient condition for when it is optimal to stop (invest). It is further shown that the optimal value function for the problem at hand is convex in the state variables, and that this fact has some interesting implications for comparative static results.
29785127
Optimal time to invest when the price processes are geometric Brownian motions
Abstract. Let 
$X_1(t)$, 
$\cdots$, 
$X_n(t)$ be 
$n$ geometric Brownian motions, possibly correlated. We study the optimal stopping problem: Find a stopping time 
$\tau^*<\infty$ such that 
 \[ \sup_{\tau}{\Bbb E}^x\Big\{ X_1(\tau)-X_2(\tau)-\cdots -X_n(\tau)\Big\}={\Bbb E}^x \Big\{ X_1(\tau^*)-X_2(\tau^*)-\cdots -X_n(\tau^*)\Big\} , \] the 
$\sup$ being taken all over all finite stopping times 
$\tau$, and 
${\Bbb E}^x$ denotes the expectation when 
$(X_1(0), \cdots, X_n(0))=x=(x_1,\cdots, x_n)$. For 
$n=2$ this problem was solved by McDonald and Siegel, but they did not state the precise conditions for their result. We give a new proof of their solution for 
$n=2$ using variational inequalities and we solve the 
$n$-dimensional case when the parameters satisfy certain (additional) conditions.

121011814
Handbook of Brownian Motion - Facts and Formulae
I: Theory.- I. Stochastic processes in general.- II. Linear diffusions.- III. Stochastic calculus.- IV. Brownian motion.- V. Local time as a Markov process.- VI. Differential systems associated to Brownian motion.- Appendix 1. Briefly on some diffusions.- II: TABLES OF DISTRIBUTIONS OF FUNCTIONALS OF BROWNIAN MOTION AND RELATED PROCESSES.- 1. Brownian motion.- 2. Brownian motion with drift.- 3. Reflecting Brownian motion.- 4. Bessel process of order ?.- 5. Bessel process of order 1/2.- 6. Bessel process of order zero.- 7. Ornstein-Uhlenbeck process.- 8. Radial Ornstein-Uhlenbeck process.- 9. Geometric Brownian motion.- Appendix 2. Special functions.- Appendix 3. Inverse Laplace transforms.- Appendix 4. Differential equations and their solutions.- Appendix 5. Formulae for n-fold differentiation.