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We consider a risky asset following a mean-reverting stochastic process of the form We show that the (singular) diffusion equation which gives the value of a European option on S can be represented, upon expanding in Laguerre polynomials, by a tridiagonal infinite matrix. We analyse this matrix to show that the diffusion equation does indeed have a solution and truncate the matrix to give a simple, highly efficient method for the numerical calculation of the solution.

We study the spectral problem $$\left. {\Delta \mathcal{U} + K^2 \mathcal{U} = 0, \frac{{\partial u}}{{\partial n}} - iK\mathcal{O}\mathcal{U}} \right|_{\partial \Omega } = 0, \mathfrak{S} \geqslant 0$$

By the methods of the theory of extensions, one constructs an explicitly solvable resonator model with a small opening. One computes the high-frequency resonances, i.e., the eigenvalues of the dissipative operator, associated with the considered problem of resonance scattering in the framework of the Lax-Phillips theory.