It is shown that for a parabolic problem with maximal Lp-regularity (for 1 < p < ∞), the time discretization by a linear multistep method or Runge–Kutta method has maximal `p-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank–Nicolson method, the second-order backward difference formula ...

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocationmethod based on gradedmeshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing probl...

We consider a backward parabolic partial differential equation with variable coefficient a(x, t) in time. A new modification is used on the logarithmic convexity method to obtain a conditional stability estimate. Based on a formal solution, we reveal the essence of the ill-posedness and propose a simple regularization method. Moreover, we apply the regularization method to two representative ca...

The paper investigates a contingent claim replicating problem for a diiusion market model. It is proposed an approach which does not call to solve the backward parabolic equation, unlike for the classical method, and this approach is applied for a case when the volatility coeecient is random and depends on time. The replicating strategy is decomposed to series of explicit strategies which does ...

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the backward Euler method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for cor...

We show non-existence of solutions of the Cauchy problem in R for the nonlinear parabolic equation involving fractional diffusion ∂tu + (-∆) s φ(u) = 0, with 0 < s < 1 and very singular nonlinearities φ . More precisely, we prove that when φ(u) = −1/u with n > 0, or φ(u) = log u, and we take nonnegative L initial data, there is no (nonnegative) solution of the problem in any dimension N ≥ 2. We...

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solut...

In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in t...

A semilinear parabolic problem of second order with an unknown solely time-dependent convolution kernel is considered. An additional given global measurement (a space integral of the solution) ensures the existence of a unique weak solution. The unknown kernel function can be approximated by a time-discrete numerical scheme based on Backward Euler’s method (Rothe’s method). In this contribution...