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This paper presents an explicit finite-difference method for nonlinear partial differential equation appearing as a transformed Black-Scholes equation for American put option under logarithmic front fixing transformation. Numerical analysis of the method is provided. The method preserves positivity and monotonicity of the numerical solution. Consistency and stability properties of the scheme are studied. Explicit calculations avoid iterative algorithms for solving nonlinear systems. Theoretical results are confirmed by numerical experiments. Comparison with other approaches shows that the proposed method is accurate and competitive.

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Abstract. Fire spotting is often responsible for dangerous flare-ups in wildfires and causes secondary ignitions isolated from the primary fire zone, which lead to perilous situations. The main aim of the present research is to provide a versatile probabilistic model for fire spotting that is suitable for implementation as a post-processing scheme at each time step in any of the existing operational large-scale wildfire propagation models, without calling for any major changes in the original framework. In particular, a complete physical parameterisation of fire spotting is presented and the corresponding updated model RandomFront 2.3 is implemented in a coupled fire–atmosphere model: WRF-SFIRE. A test case is simulated and discussed. Moreover, the results from different simulations with a simple model based on the level set method, namely LSFire+, highlight the response of the parameterisation to varying fire intensities, wind conditions and different firebrand radii. The contribution of the firebrands to increasing the fire perimeter varies according to different concurrent conditions, and the simulations show results in agreement with the physical processes. Among the many rigorous approaches available in the literature to model firebrand transport and distribution, the approach presented here proves to be simple yet versatile for application to operational large-scale fire spread models.

The challenge of removing the mixed derivative terms of a second order multidimensional partial differential equation is addressed in this paper. The proposed method, which is based on proper algebraic factorization of the so-called diffusion matrix, depends on the semidefinite or indefinite character of this matrix. Computational cost of the transformed equation is considerably reduced and well-known numerical drawbacks are avoided.

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In this work, we apply the local Wendland radial basis function (RBF) for solving the time-dependent multi dimensional option pricing nonlinear PDEs. Firstly, cross derivative terms of the PDE are removed with a change of spatial variables based in LDLT factorization of the diffusion matrix. Then, it is discussed that the valuation of a multi-asset option up to 4D can be computed using a modified shape parameter algorithm. In fact, several experiments containing of three and four assets are worked out showing that the results of the presented method are in good agreement with the literature and could be much more accurate once the shape parameter is chosen carefully.

American put option pricing under regime switching is modelled by a system of coupled partial differential equations. The proposed model combines better the reality of the market by incorporating the regime switching jointly with the emotional behaviour of traders using the rationality parameter approach recently introduced by Tågholt Gad and Lund Petersen to cope with possible irrational exercise policy. The classical rational exercise is recovered as a limit