447 research outputs found
Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources
In this paper we study a continuous time, optimal stochastic investment
problem under limited resources in a market with N firms. The investment
processes are subject to a time-dependent stochastic constraint. Rather than
using a dynamic programming approach, we exploit the concavity of the profit
functional to derive some necessary and sufficient first order conditions for
the corresponding Social Planner optimal policy. Our conditions are a
stochastic infinite-dimensional generalization of the Kuhn-Tucker Theorem. The
Lagrange multiplier takes the form of a nonnegative optional random measure on
[0,T] which is flat off the set of times for which the constraint is binding,
i.e. when all the fuel is spent. As a subproduct we obtain an enlightening
interpretation of the first order conditions for a single firm in Bank (2005).
In the infinite-horizon case, with operating profit functions of Cobb-Douglas
type, our method allows the explicit calculation of the optimal policy in terms
of the `base capacity' process, i.e. the unique solution of the Bank and El
Karoui representation problem (2004).Comment: 25 page
Optimal Dynamic Procurement Policies for a Storable Commodity with L\'evy Prices and Convex Holding Costs
In this paper we study a continuous time stochastic inventory model for a
commodity traded in the spot market and whose supply purchase is affected by
price and demand uncertainty. A firm aims at meeting a random demand of the
commodity at a random time by maximizing total expected profits. We model the
firm's optimal procurement problem as a singular stochastic control problem in
which controls are nondecreasing processes and represent the cumulative
investment made by the firm in the spot market (a so-called stochastic
"monotone follower problem"). We assume a general exponential L\'evy process
for the commodity's spot price, rather than the commonly used geometric
Brownian motion, and general convex holding costs.
We obtain necessary and sufficient first order conditions for optimality and
we provide the optimal procurement policy in terms of a "base inventory"
process; that is, a minimal time-dependent desirable inventory level that the
firm's manager must reach at any time. In particular, in the case of linear
holding costs and exponentially distributed demand, we are also able to obtain
the explicit analytic form of the optimal policy and a probabilistic
representation of the optimal revenue. The paper is completed by some computer
drawings of the optimal inventory when spot prices are given by a geometric
Brownian motion and by an exponential jump-diffusion process. In the first case
we also make a numerical comparison between the value function and the revenue
associated to the classical static "newsvendor" strategy.Comment: 28 pages, 3 figures; improved presentation, added new results and
section
Continuous-Time Public Good Contribution under Uncertainty
Ferrari G, Riedel F, Steg J-H. Continuous-Time Public Good Contribution under Uncertainty. Center for Mathematical Economics Working Papers. Vol 485 Version February 2015. Bielefeld: Center for Mathematical Economics; 2015.We study a continuous-time problem of public good contribution under uncertainty
for an economy with a finite number of agents. Each agent aims to maximize his expected
utility allocating his initial wealth over a given time period between private consumption and
repeated but irreversible contributions to increase the stock of some public good. We study the
corresponding social planner problem and the case of strategic interaction between the agents.
These problems are set up as stochastic control problems with both monotone and classical controls
representing the cumulative contribution into the public good and the consumption of the
private good, respectively. We characterize the optimal investment policies by a set of necessary
and sufficient stochastic Kuhn-Tucker conditions, which in turn allow to identify a universal signal
process that triggers the public good investments. Further we show that our model exhibits
a dynamic free rider effect. We explicitly evaluate it in a symmetric Black-Scholes setting with
Cobb-Douglas utilities and we show that uncertainty and irreversibility of public good provisions
need not affect the degree of free-riding
Continuous-Time Public Good Contribution under Uncertainty
Ferrari G, Riedel F, Steg J-H. Continuous-Time Public Good Contribution under Uncertainty. Center for Mathematical Economics Working Papers. Vol 485. Bielefeld: Center for Mathematical Economics; 2013.We study a continuous-time problem of optimal public good contribution under uncertainty for an economy with an finite number of agents. Each agent can allocate his wealth between private consumption and repeated but irreversible contributions to increase the stock of some public good. We study the corresponding social planner problem and the case of strategic interaction between the agents and we characterize the optimal investment policies by a set of necessary and sufficient stochastic Kuhn-Tucker conditions. Suitably combining arguments from Duality Theory and the General Theory of Stochastic Processes, we prove an abstract existence result for a Nash equilibrium of our public good contribution game. Also, we show that our model exhibits a dynamic free rider effect. We explicitly evaluate it in a symmetric Black-Scholes setting with Cobb-Douglas utilities and we show that uncertainty and irreversibility of public good provisions do not affect free-riding
Investment, irreversibility, and financing constraints in transition economies
Using a panel of 4223 Bulgarian, Czech, Polish, and Romanian firms, over the period 1998-2005, we show that financially constrained firms likely to face irreversibility constraints exhibit low and insignificant sensitivities of investment to cash flow. These firms typically use their cash flow to accumulate cash instead of investing. Our findings provide a new explanation for why some financially constrained firms may exhibit low investment-cash flow sensitivities. Specifically, controlling for investment irreversibility may matter for the interpretation of these sensitivities.Investment; Irreversibility; Cash flow; Cash accumulation; Capital market imperfections
On a Class of Infinite-Dimensional Singular Stochastic Control Problems
Federico S, Ferrari G, Riedel F, Röckner M. On a Class of Infinite-Dimensional Singular Stochastic Control Problems. Center for Mathematical Economics Working Papers. Vol 614. Bielefeld: Center for Mathematical Economics; 2019.We study a class of infinite-dimensional singular stochastic control problems with
applications in economic theory and finance. The control process linearly affects an abstract
evolution equation on a suitable partially-ordered infinite-dimensional space X, it takes values in
the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a
rigorous formulation of the problem by properly defining the controlled dynamics and integrals
with respect to the control process. We then exploit the concave structure of our problem
and derive necessary and sufficient first-order conditions for optimality. The latter are finally
exploited in a specification of the model where we find an explicit expression of the optimal
control. The techniques used are those of semigroup theory, vector-valued integration, convex
analysis, and general theory of stochastic processes
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