Irreversible Investment under L\'evy Uncertainty: an Equation for the Optimal Boundary

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential L\'evy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank-El Karoui representation problem. Such a relation and the Wiener Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying L\'evy process hits any real point with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of (i) Cobb-Douglas type and (ii) CES type. In the first case the function is separable and in the second case non-separable.Comment: 19 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

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

Measuring the Temperature of a Mesoscopic Quantum Electron System by means of Single Electron Statistics

We measure the temperature of a mesoscopic system consisting of an ultra-dilute two dimensional electron gas at the $Si/SiO_2$ interface in a metal-oxide-semiconductor field effect transistor (MOSFET) quantum dot by means of the capture and emission of an electron in a point defect close to the interface. Contrarily to previous reports, we show that the capture and emission by point defects in Si n-MOSFETs can be temperature dependent down to 800 mK. As the finite quantum grand canonical ensemble model applies, the time domain charge fluctuation in the defect is used to determine the temperature of the few electron gas in the channel.Comment: 4 Figures (color

CD40 signalling in ileal Peyer's patch B cells: implications for T cell-dependent antigen selection

The ileal Peyer's patch (PP) plays a central role in B cell development in young sheep and it is hypothesized that this B cell development occurs independent of extrinsic antigen and T cells. Therefore, it was of interest to examine ileal PP folllcular (iPf) B cell responses to CD40 ligand, a molecule integral to T cell-dependent B cell development. A variable level of CD40 expression was detected on a subpopulation of iPfB cells and J558L cells, expressing a membrane form of mouse CD40 ligand (mCD40L), interacted specifically with the CD40 molecule on iPfB cells. In response to mCD40L the non-S phase iPfB cells were rescued from apoptotic cell death and there was a marked proliferative response but viable cell number remained relatively constant. The mCD40L also induced decreased cytoplasmic cAMP levels, blocked anti-Ig-induced iPfB cell death and induced functional IL-2 receptor expression on a subpopulation of iPfB cells. Many of the mCD40L-induced responses of iPfB cells were similar to those reported for germinal centre and immature B cells, and indicated that a cognate T cell-B cell interaction could influence iPfB cell proliferation and differentiation. Finally, that mCD40L induced iPfB cell activation and differentiation was evident as increased expression of CD5, the BAQ44A molecule, the CACT65A molecule and the expansion of surface IgG1+ B cells. These mCD40L-induced phenotypic changes were also observed on subpopulations of freshly isolated iPfB cells and jejunal PP follicular B cells. However, few iPfB cells had a phenotype similar to that observed in co-culture with mCD40L and this suggested that T cell-dependent B cell development may play a minor role in ileal PP B cell development. The possible significance of CD40 signalling is discussed in terms of the selection of iPfB cells during developmen

Taming the Spread of an Epidemic by Lockdown Policies

Federico S, Ferrari G. Taming the Spread of an Epidemic by Lockdown Policies. Center for Mathematical Economics Working Papers. Vol 639. Bielefeld: Center for Mathematical Economics; 2020.We study the problem of a policymaker who aims at taming the spread of an epidemic while minimizing its associated social costs. The main feature of our model lies in the fact that the disease's transmission rate is a diffusive stochastic process whose trend can be adjusted via costly confinement policies. We provide a complete theoretical analysis, as well as numerical experiments illustrating the structure of the optimal lockdown policy. In all our experiments the latter is characterized by three distinct periods: the epidemic is first let freely evolve, then vigorously tamed, and finally a less stringent containment should be adopted. Moreover, the optimal containment policy is such that the product "reproduction number x percentage of susceptible" is kept after a certain date strictly below the critical level of one, although the reproduction number is let oscillate above one in the last more relaxed phase of lockdown.MSC2010 subject classification: 93E20, 49N90, 92D30, 97M4