Applications of the Banach fixed-point theorem to analyze insolvency problems of an insurance company

Abstract We explore new areas of applications of the Banach fixed-point theorem to solve some problems arising in mathematical economics with a special attention paid to the insurance mathematics and the solvency challenges faced by insurance companies in their daily business activities. We use Banach’s contraction principle to construct sequences of new monotone, attainable upper and lower bounds for the deficit distribution at ruin $$\Psi (u,\,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the insurer, considered as a function of the initial surplus (capital) u and the severity of ruin y . We investigate a monotone risk operator L on a properly defined complete metric space $$\langle {\mathcal {R}}, {d_r} \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> in which $$\Psi (u,\,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is shown to be the unique fixed point. What is more, L is proven to be a contraction on $$\langle {\mathcal {R}}, {d_r} \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> . The resulting procedure allows the insurer to approximate $$\Psi (u,\,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> (and effectively control the error of approximation) by iterating L on any point from $$\langle {\mathcal {R}}, {d_r} \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> . The approach presented in this paper enables to treat in a unified way some discrete and continuous-time insolvency risk models. All the lower and upper bounds for $$\Psi (u,\,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> introduced in this paper are shown to be attainable for arbitrary values of u and y under both discrete and continuous-time setting.