We present an analytical option pricing formula for the European options, in which the price dynamics of a risky asset follows a mean-reverting process with a time-dependent parameter. The process can be adapted to describe a seasonal variation in price such as in agricultural commodity markets. An analytical solution is derived based on the solution of a partial differential equation, which shows that a European option price can be decomposed into two terms: the payoff of the option at the initial time and the time-integral over the lifetime of the option driven by a time-dependent parameter. Finally, results obtained from the formula have been compared with Monte Carlo simulations and a Black–Scholes-type formula under various kinds of long-run mean functions, and some examples of option price behaviours have been provided.

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$-norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation (Waring and Siegel (2015) Financial Analysts Journal, 71(1), 91–107) to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst 5% of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.

A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $T_{\mathrm {kin}} \gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength $\alpha $ of the nonlinearity goes to $0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ and $\alpha $ is related to the conserved mass $\lambda $ of the solution via $\alpha =\lambda ^2 L^{-d}$ .

In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $(\alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $\alpha $ approaches $0$ like $L^{-\varepsilon +}$ or like $L^{-1-\frac {\varepsilon }{2}+}$ (for arbitrary small $\varepsilon $ ), we exhibit the wave kinetic equation up to time scales $O(T_{\mathrm {kin}}L^{-\varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $T_*\ll T_{\mathrm {kin}}$ and identify specific interactions that become very large for times beyond $T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $T_*$ toward $T_{\mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.

We consider a model for the dynamics of growing cell populations with heterogeneous mobility and proliferation rate. The cell phenotypic state is described by a continuous structuring variable and the evolution of the local cell population density function (i.e. the cell phenotypic distribution at each spatial position) is governed by a non-local advection–reaction–diffusion equation. We report on the results of numerical simulations showing that, in the case where the cell mobility is bounded, compactly supported travelling fronts emerge. More mobile phenotypic variants occupy the front edge, whereas more proliferative phenotypic variants are selected at the back of the front. In order to explain such numerical results, we carry out formal asymptotic analysis of the model equation using a Hamilton–Jacobi approach. In summary, we show that the locally dominant phenotypic trait (i.e. the maximum point of the local cell population density function along the phenotypic dimension) satisfies a generalised Burgers’ equation with source term, we construct travelling-front solutions of such transport equation and characterise the corresponding minimal speed. Moreover, we show that, when the cell mobility is unbounded, front edge acceleration and formation of stretching fronts may occur. We briefly discuss the implications of our results in the context of glioma growth.

Cells and organisms follow aligned structures in their environment, a process that can generate persistent migration paths. Kinetic transport equations are a popular modelling tool for describing biological movements at the mesoscopic level, yet their formulations usually assume a constant turning rate. Here we relax this simplification, extending to include a turning rate that varies according to the anisotropy of a heterogeneous environment. We extend known methods of parabolic and hyperbolic scaling and apply the results to cell movement on micropatterned domains. We show that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment and generate enhanced diffusion in structured domains.

In this paper, we focus on a Kirchhoff-type equation with singular potential and critical exponent. By virtue of the generalized version of Lions-type theorem and the Nehari–Pohožaev manifold, we established the existence of Nehari–Pohožaev-type ground state solutions to the mentioned equation. Some recent results from the literature are generally improved and extended.

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝn. We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system, we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularising terms.

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at times t where these models have validity, namely t = O(h−2), where h is proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

The existence of a symmetric mode in an elastic solid wedge for all admissible values of the Poisson ratio and arbitrary interior angles close to π has been proven.

We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.

This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system

\[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\]

Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β < 1, whereas whenever β > 1 and $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$, infinite-time blow-up occurs at least in the particular case when ρ = 0.

In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/(β − 1)+) one can identify an L∞-neighbourhood of the homogeneous distribution (u, v, w, z) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u∞, 0, 0, 0) with some u∞ > 0.