Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model

• Main Authors: Andrius Grigutis, Jonas Šiaulys
• Format: Article
• Language: English
• Published: MDPI AG 2020-01-01
• Series: Mathematics
• Subjects: multi-risk model; discrete-time risk model; ruin probability; survival probability; ultimate time; net profit condition
• Online Access: https://www.mdpi.com/2227-7390/8/2/147

In this paper, we prove recursive formulas for ultimate time survival probability when three random claims <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> </mrow> </semantics> </math> </inline-formula> in the discrete time risk model occur in a special way. Namely, we suppose that claim <i>X</i> occurs at each moment of time <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>&#8712;</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>&#8230;</mo> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>, claim <i>Y</i> additionally occurs at even moments of time <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>&#8712;</mo> <mo>{</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>&#8230;</mo> <mo>}</mo> </mrow> </semantics> </math> </inline-formula> and claim <i>Z</i> additionally occurs at every moment of time, which is a multiple of three <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>&#8712;</mo> <mo>{</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mo>&#8230;</mo> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>. Under such assumptions, the model that is obtained is called the three-risk discrete time model. Such a model is a particular case of a nonhomogeneous risk renewal model. The sequence of claims has the form <inline-formula> <math display="inline"> <semantics> <mrow> <mo>{</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo>+</mo> <mi>Z</mi> <mo>,</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>+</mo> <mi>Z</mi> <mo>,</mo> <mo>&#8230;</mo> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>. Using the recursive formulas, algorithms were developed to calculate the exact values of survival probabilities for the three-risk discrete time model. The running of algorithms is illustrated via numerical examples.