An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise

We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.Comment: 16 pages. Many misprints have been correcte

An Hilbert space approach for a class of arbitrage free implied volatilities models

We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.Comment: 21 page

Second order parabolic Hamilton–Jacobi–Bellman equations in Hilbert spaces and stochastic control: Lμ2 approach

AbstractWe study a Hamilton–Jacobi–Bellman equation related to the optimal control of a stochastic semilinear equation on a Hilbert space X. We show the existence and uniqueness of solutions to the HJB equation and prove the existence and uniqueness of feedback controls for the associated control problem via dynamic programming. The main novelty is that we look for solutions in the space L2(X,μ), where μ is an invariant measure for an associated uncontrolled process. This allows us to treat controlled systems with degenerate diffusion term that are not covered by the existing literature. In particular, we prove the existence and uniqueness of solutions and obtain the optimal feedbacks for controlled stochastic delay equations and for the first order stochastic PDE’s arising in economic and financial models

Time irregularity of generalized Ornstein--Uhlenbeck processes

The paper is concerned with the properties of solutions to linear evolution equation perturbed by cylindrical L\'evy processes. It turns out that solutions, under rather weak requirements, do not have c\`adl\`ag modification. Some natural open questions are also stated

Lognormality of Rates and Term Structure Models

A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting rates are well defined (they do not explode) and remain positive. They are bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann (1993), (1994) for Markovian lognormal short rates to (non-Markovian) lognormal forward rates.Term structure of interest rates, lognormal volatility structure, Heath, Jarrow and Morton models.

Increasing trans-cleavage catalytic efficiency of Cas12a and Cas13a with chemical enhancers: Application to amplified nucleic acid detection

The exceptional programable trans-cleavage ability of type V and VI CRISPR/Cas nucleases paved the way for ultrasensitive CRISPR/Cas based sensing of nucleic acid and alternative targets. However, the enhancement of the trans-cleavage activity of Cas effector with organic chemical agents has not been explored thus far. We report here chemically enhanced trans-cleavage activity of Cas12a and Cas13a nucleases which improves sensor performance in CRISPR/Cas biosensing. Improved trans-ssDNA cleavage of Cas12a and trans-ssRNA cleavage of Cas13a were demonstrated by using sulfhydryl reductants and non-ionic surfactants. DTT and PVA were demonstrated to be the most effective chemical enhancers in both cases. By using a fluorescence resonance energy transfer (FRET)-based intramolecular distance measurements, we identified the mechanism of this enhancement to be the conformation change of the ribonucleoprotein and quantified it to be major (about 50% increase of a relevant intramolecular distance). These chemical enhancers have been integrated into the established CRISPR/Cas biosensing protocols without additional modifications. For the detection of Helicobacter Pylori DNA and SARS-CoV-2 RNA, we found a decreased reaction time by 75–83% and 4–6-fold increased sensitivity. These results indicate that chemical enhancers provide a versatile and broadly applicable approach to break through the barriers of long reaction time and sensitivity in CRISPR/Cas sensors

A pricing formula for delayed claims: appreciating the past to value the future

We consider the valuation of contingent claims with delayed dynamics in a Samuelson complete market model. We find a pricing formula that can be decomposed into terms reflecting the current market values of the past and the future, showing how the valuation of prospective cashflows cannot abstract away from the contribution of the past. As a practical application, we provide an explicit expression for the market value of human capital in a setting with wage rigidity. The formula we derive has successfully been used to explicitly solve the infinite dimensional stochastic control problems addressed in [7], [6] and [16]

Formation and dissociation of hydrogen-related defect centers in Mg-doped GaN

Moderately and heavily Mg-doped GaN were studied by a combination of post-growth annealing processes and electron beam irradiation techniques during cathodoluminescence (CL) to elucidate the chemical origin of the recombination centers responsible for the main optical emission lines. The shallow donor at 20-30 meV below the conduction band, which is involved in the donor-acceptor-pair (DAP) emission at 3.27 eV, was attributed to a hydrogen-related center, presumably a (VN-H) complex. Due to the small dissociation energy (<2 eV) of the (VNH) complex, this emission line was strongly reduced by low-energy electron irradiation. CL investigations of the DAP at a similar energetic position in Si-doped (n-type) GaN indicated that this emission line is of different chemical origin than the 3.27 eV DAP in Mg-doped GaN. A slightly deeper DAP emission centered at 3.14 eV was observed following low-energy electron irradiation, indicating the appearance of an additional donor level with a binding energy of 100-200 meV, which was tentatively attributed to a VN-related center. The blue band (2.8-3.0 eV) in heavily Mg-doped GaN was found to consist of at least two different deep donor levels at 350±30 meV and 440±40 meV. The donor level at 350±30 meV was strongly affected by electron irradiation and attributed to a H-related defect