PDE Approach Research Articles

Stochastic maximal regularity for rough time-dependent problems

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain L^{p}(L^{q}) estimates for all p>2 and qge 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain L^{p}(L^{p}) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces T^{p,2}_{sigma } of Coifman–Meyer–Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.

On Calibration and Simulation of Local Volatility Model with Stochastic Interest Rate

Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.

SDE-driven modeling of phenotypically heterogeneous tumors: The influence of cancer cell stemness

We deduce cell population models describing the evolution of a tumor (possibly interacting with its environment of healthy cells) with the aid of differential equations. Thereby, different subpopulations of cancer cells allow accounting for the tumor heterogeneity. In our settings these include cancer stem cells known to be less sensitive to treatment and differentiated cancer cells having a higher sensitivity towards chemo- and radiotherapy. Our approach relies on stochastic differential equations in order to account for randomness in the system, arising e.g., due to the therapy-induced decreasing number of clonogens, which renders a pure deterministic model arguable. The equations are deduced relying on transition probabilities characterizing innovations of the two cancer cell subpopulations, and similarly extended to also account for the evolution of normal tissue. Several therapy approaches are introduced and compared by way of tumor control probability (TCP) and uncomplicated tumor control probability (UTCP). A PDE approach allows to assess the evolution of tumor and normal tissue with respect to time and to cell population densities which can vary continuously in a given set of states. Analytical approximations of solutions to the obtained PDE system are provided as well.

Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE Approach

Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

Stochastic hybrid models of gene regulatory networks – A PDE approach

A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation approach based on a system of partial differential equations, where we assume a continuous-deterministic evolution for the protein counts. We discuss efficient analysis methods for both modeling approaches and compare their performance. We show that the hybrid approach yields accurate results for sufficiently large molecule counts, while reducing the computational effort from one ordinary differential equation for each state to one partial differential equation for each mode of the system. Furthermore, we give an analytical steady-state solution of the hybrid model for the case of a self-regulatory gene.

On uniqueness sets of additive eigenvalue problems and applications

In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton--Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton--Jacobi equations is also obtained.

Numerical methods for fractional diffusion

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

A PDE approach to a 2-dimensional matching problem

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by Caracciolo et al. (Phys Rev E 90:012118, 2014) that linearizes the Monge–Ampère equation.

Pricing and hedging barrier options

European options are a significant financial product. Barrier options, in turn, are European options with a barrier constraint. The investor may pay less buying the barrier option obtaining the same result as that of the European option whenever the barrier is not breached. Otherwise, the option's payoff cancels. In this paper, we obtain closed‐form expressions of the exact no‐arbitrage prices, delta hedges, and gammas of a call option with a moving barrier that tracks the prices of the risk‐free asset. Besides the interest in its own right, this class of options constitutes the core element to obtain, via an original and simple technique, the closed‐form expressions for the estimates of the prices of call options with barriers of arbitrary shape. Equally important is the fact that a bound for the worst associated error is provided, so the investor can evaluate beforehand if the accuracy provided is according to his/her needs or not. Discrete monitored barrier provisions are also allowed in the estimates. Simulations are performed illustrating the accuracy of the estimates. A quality of the aforementioned procedures is that the time consumed in computations is very small. In turn, we observe that the approximate prices, delta hedges, and gammas of the barrier option associated to the risk‐free asset, obtained via a PDE approach in conjunction with a good finite difference method, converge to the closed‐form expressions of the prices, hedges, and gammas of the option. This attests the correctness of the analytical results.

Reaching goals under ambiguity: Continuous-time optimal portfolio selection

In this paper, the problem of reaching goals for an investor in the financial market is studied. We follow the context of Jin and Zhou (2015) who studied the continuous-time optimal portfolio selection in which the appreciation rates are only known to be in a certain convex closed set. The cost functional in our problem is of type I{x≥1} which is not concave or convex, even not continuous, we prove that the min–max theorem is still applicable. Via PDE approach, we construct the optimal portfolio strategy. At last, we obtain the saddle point for our problem explicitly.

Speckle removal and change preservation by distance-driven anisotropic diffusion of synthetic aperture radar temporal stacks

We propose a method that smoothes spatially and uses, but also preserves, temporal information of synthetic aperture radar image stacks. To provide smoothing of such imagery without effacing temporal changes in the scene, we put forth an anisotropic diffusion technique, using a PDE approach. This approach smoothes uniform areas and preserves and enhances edges, such as roads or other features. In order to use this anisotropic diffusion technique, a homogeneous region must first be selected in the image to calculate statistics of the speckle noise. We propose a method that uses the temporal information to automatically select a homogeneous region prior to smoothing. Our proposed smoothing method calculates the distance between either pixel time-series or pixel CDFs as a measure of similarity, or uniformity. Results demonstrate the efficacy of the approach on real and synthetic data, showing lower mean squared ratio and higher structural similarity index than leading methods, as well as temporal change preservation.

Beneficial mutation-selection dynamics in finite asexual populations: a free boundary approach

Using a free boundary approach based on an analogy with ice melting models, we propose a deterministic PDE framework to describe the dynamics of fitness distributions in the presence of beneficial mutations with non-epistatic effects on fitness. Contrarily to most approaches based on deterministic models, our framework does not rely on an infinite population size assumption, and successfully captures the transient as well as the long time dynamics of fitness distributions. In particular, consistently with stochastic individual-based approaches or stochastic PDE approaches, it leads to a constant asymptotic rate of adaptation at large times, that most deterministic approaches failed to describe. We derive analytic formulas for the asymptotic rate of adaptation and the full asymptotic distribution of fitness. These formulas depend explicitly on the population size, and are shown to be accurate for a wide range of population sizes and mutation rates, compared to individual-based simulations. Although we were not able to derive an analytic description for the transient dynamics, numerical computations lead to accurate predictions and are computationally efficient compared to stochastic simulations. These computations show that the fitness distribution converges towards a travelling wave with constant speed, and whose profile can be computed analytically.

Simulations of flow-like landslides invading urban areas: a cellular automata approach with SCIDDICA

Different methodologies are used for modelling flow-like landslides. A common critical point concerns the flooding into town areas, which cannot be assimilated straight to a morphology, especially, when the urban tissue is very irregular with narrow streets and odd setting of buildings, so that discretization processes of approximating numerical methods have to be carefully examined in their limits. A semi-empirical approach by the computational paradigm of cellular automata is here considered with SCIDDICA, a competitive (related to PDE approach) cellular automata model for 3-dimensions simulation of flow-like landslides. This paper presents innovations to the transition function of SCIDDICA-SS2, which manage opportunely building data in the cells corresponding to the urban tissue. The novelties of the transition function need a theorem, here demonstrated which regards the Algorithm of Minimization of Differences in the new context of inhomogeneous cells. This progress permits to simulate the complete evolution of landslides, from the detachment area to its exhaustion almost on the same precision level. This is an advantage for hazard and risk analyses in threatened zones. Improved SCIDDICA-SS2 was applied successfully to all the well-known 2009 debris flows of Giampilieri Superiore (Sicily) also in comparison with simulation results of the previous versions.

Pricing European Call Options Under a Hard-to-Borrow Stock

In 2009, Avellaneda and Lipkin (A&L) proposed a dynamic model for hard-to-borrow stocks, in which the stock price and the buy-in rate, an additional factor introduced by them, are full coupled. In order to obtain a semi-explicit pricing formula for European call options, A&L had to make an independence assumption which has facilitated the derivation of their formula. However, it has also limited its application to more general cases. When such an assumption is removed, a more general approach is needed. A PDE approach is presented in this paper, taking the pricing of European call options as an example to demonstrate that it is a broader approach than A&L’s semi-explicit pricing formula. The PDE system governing the value of European call options together with a set of appropriate boundary conditions is first established, followed by numerical results obtained with two carefully chosen methods, the difference of which lies in the treatment of the jump term. Through this work, it is verified that the formula presented by A&L is a good approximate solution, provided that the independence assumption is indeed a reasonable one. However, in the event that this is not the case, the full model needs to be solved with the PDE approach as demonstrated in this paper.

Computing cross fields A PDE approach based on the Ginzburg-Landau theory

Cross fields are auxiliary in the generation of quadrangular meshes. A method to generate cross fields on surface manifolds is presented in this paper. Algebraic topology constraints on quadrangular meshes are first discussed. The duality between quadrangular meshes and cross fields is then outlined, and a generalization to cross fields of the Poincaré-Hopf theorem is proposed, which highlights some fundamental and important topological constraints on cross fields. A finite element formulation for the computation of cross fields is then presented, which is based on Ginzburg-Landau equations and makes use of edge-based Crouzeix-Raviart interpolation functions. It is first presented in the planar case, and then extended to a general surface manifold. Finally, application examples are solved and discussed.

Nonlinear PDE Approach to Time-Inconsistent Optimal Stopping

We present a novel method for solving a class of time-inconsistent optimal stopping problems by reducing them to a family of standard stochastic optimal control problems. In particular, we convert an optimal stopping problem with a nonlinear function of the expected stopping time in the objective into optimization over an auxiliary value function for a standard stochastic control problem with an additional state variable. This approach differs from those in the previous literature which primarily employ Lagrange multiplier methods or rely on exact solutions. In contrast, we characterize the auxiliary value function as the unique viscosity solution of a nonlinear elliptic PDE which satisfies certain growth constraints, and investigate basic regularity properties. We demonstrate a connection between optimal stopping times for the original problem and optimal controls of the auxiliary control problem. Finally, we discuss extensions to more general dynamics and time-inconsistencies, as well as potential conne...

A superconvergent partial differential equation approach to price variance swaps under regime switching models

We present a superconvergent finite difference algorithm to price discretely sampled variance swaps. We consider the Black–Scholes model, the Merton’s jump–diffusion model, stochastic volatility models that use constant-elasticity of variance for the instantaneous variance and corresponding regime switching models. PDE approach provides a universal and efficient framework for pricing under these models. To obtain extremely accurate results, we solve PDEs whose associated terminal conditions can be represented as second-order polynomials based on the two popular definitions of realised variance and for which the spatial derivatives greater than second-order are all zero. We then apply second-order finite difference discretisations in space with an exponential time integration. We also derive analytical solutions under the Merton’s model and some regime switching models to validate our superconvergent results.

Existence of weak quasi-periodic solutions for a second order Hamiltonian system with damped term via a PDE approach

Existence of weak quasi-periodic solutions for a second order Hamiltonian system with damped term via a PDE approach

A PDE Approach to Space-Time Fractional Parabolic Problems

We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition. We propose and analyze an implicit fully discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time. We prove stability and error estimates for this scheme.

A potential theory for the k-curvature equation

In this paper, we introduce a potential theory for the k-curvature equation, which can also be seen as a PDE approach to curvature measures. We assign a measure to a bounded, upper semicontinuous function which is strictly subharmonic with respect to the k-curvature operator, and establish the weak continuity of the measure.