Finite difference scheme: Monotonicity violation of a class of interpolations

The Barles–Souganidis (1991) convergence theorem states that a finite difference scheme for a fully nonlinear second-order PDE converges uniformly to the unique viscosity solution provided the following conditions hold:

• Monotonicity
• Stability
• Consistency
• PDE that satisfies a comparison principle

While most conditions are automatically satisfied in many economic models, economists must be very careful with the monotonicity condition, which may fail in some cases and hard to detect at first glance.

Before diving into technical details, let’s setup a generic problem and formalize notations.

1. Notations

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1.1 Boundary problem

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Consider HJB equation, which is a 2nd-order non-linear PDE, looking like:

$$ \begin{aligned} & \rho v(x) = u + \mathcal{L}_x[v] \\ & \mathcal{L}_x[v] := \left<\nabla v(x), \mu \right> + \frac{1}{2}\text{tr}\left\{ \sigma^T \cdot \nabla^2 v(x) \cdot \sigma \right\} \\ & x \in \mathcal{X} \subset \mathbb{R}^n \\ & v\in\mathbb{R} \\ & \nabla v(x) \in\mathbb{R}^n, \nabla^2 v(x)\in\mathbb{R}^{n\times n} \\ & u \in \mathbb{R} \\ & \mu \in \mathbb{R}^{n}, \sigma \in\mathbb{R}^{n\times m} \\ & \text{d}x = \mu \cdot \text{d}t + \sigma \cdot \text{d}W, W\in\mathbb{R}^{m} \end{aligned} $$

where

• \(\rho\) is discounting rate
• \(n\) is not necessary to be equal to \(m\)
• \(u = u(x,v(x),\nabla v(x),\nabla^2 v(x))\) is instantaneous payoff term that can be a function of \(x\), function value at \(x\), derivatives and Hessian at \(x\)
• \(\mu = \mu(x,v(x),\nabla v(x),\nabla^2 v(x))\) is drift term that can be a function
• \(\sigma = \sigma(x,v(x),\nabla v(x),\nabla^2 v(x))\) is diffusion/volatility matrix that can be a function

Meanwhile, also consider boundary conditions up to 1st order:

$$ b(x,v(x),\nabla v(x)) = \mathbf{0}_{2^n}, x \in \partial\mathcal{X} $$

1.2 Grid space & index algebra

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TBD

1.3 Monotonic numerical scheme

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1.3.1 Monotonic scheme

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The Barles–Souganidis (1991) theorem requires monotonic scheme for local convergence.

1.3.1 Explicit (Euler forward) scheme

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1.3.2 Fully implicit (Euler backward) scheme

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Solution

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Reference

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Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic analysis, 4(3):271–. 283, 1991.