Numerical solution of the time-fractional Fokker-Planck equation with general forcing

We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^\alpha)$ for a uniform time step $k$, where $\alpha\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.Comment: 3 Figure

Finite element approximation of non-Fickian polymer diffusion

The problem of nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with an adjoined spatially local evolution equation for a viscoelastic stress is considered (see, for example, Cohen, White & Witelski, SIAM J. Appl. Math. 55, pp. 348–368, 1995). We present numerical schemes based, spatially, on the Galerkin finite element method and, temporally, on the Crank-Nicolson method. Special attention is paid to linearising the discrete equations by extrapolating the value of the nonlinear term from previous time steps. Optimal a priori error estimates are given, based on the assumption that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates

An Optimal Execution Problem with Market Impact

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales, otherwise the value function is continuous. Moreover, we show the semi-group property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of the optimal strategies change completely, depending on the amount of the trader's security holdings and where optimal strategies in the Black-Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.Comment: 36 pages, 8 figures, a modified version of the article "An optimal execution problem with market impact" in Finance and Stochastics (2014