Martingale Approach to Derive Lundberg-Type Inequalities
In this paper, we find the upper bound for the tail probability <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><mfenced separators="" open="(" close=")"><msub...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/10/1742 |
| _version_ | 1797551329058488320 |
|---|---|
| author | Tautvydas Kuras Jonas Sprindys Jonas Šiaulys |
| author_facet | Tautvydas Kuras Jonas Sprindys Jonas Šiaulys |
| author_sort | Tautvydas Kuras |
| collection | DOAJ |
| description | In this paper, we find the upper bound for the tail probability <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><mfenced separators="" open="(" close=")"><msub><mo movablelimits="true" form="prefix">sup</mo><mrow><mi>n</mi><mo>⩾</mo><mn>0</mn></mrow></msub><msubsup><mo>∑</mo><mrow><mi mathvariant="normal">I</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>ξ</mi><mi mathvariant="normal">I</mi></msub><mo>></mo><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> with random summands <inline-formula><math display="inline"><semantics><mrow><msub><mi>ξ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ξ</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity <inline-formula><math display="inline"><semantics><mrow><msub><mi>ϱ</mi><mn>1</mn></msub><mo form="prefix">exp</mo><mrow><mo>{</mo><mo>−</mo><msub><mi>ϱ</mi><mn>2</mn></msub><mi>x</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula> with some positive constants <inline-formula><math display="inline"><semantics><msub><mi>ϱ</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>ϱ</mi><mn>2</mn></msub></semantics></math></inline-formula>. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model. |
| first_indexed | 2024-03-10T15:43:08Z |
| format | Article |
| id | doaj.art-15f9a85e07874d1ea7c520579771a4a4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:43:08Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-15f9a85e07874d1ea7c520579771a4a42023-11-20T16:37:32ZengMDPI AGMathematics2227-73902020-10-01810174210.3390/math8101742Martingale Approach to Derive Lundberg-Type InequalitiesTautvydas Kuras0Jonas Sprindys1Jonas Šiaulys2Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaInstitute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaInstitute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaIn this paper, we find the upper bound for the tail probability <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><mfenced separators="" open="(" close=")"><msub><mo movablelimits="true" form="prefix">sup</mo><mrow><mi>n</mi><mo>⩾</mo><mn>0</mn></mrow></msub><msubsup><mo>∑</mo><mrow><mi mathvariant="normal">I</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>ξ</mi><mi mathvariant="normal">I</mi></msub><mo>></mo><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> with random summands <inline-formula><math display="inline"><semantics><mrow><msub><mi>ξ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ξ</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity <inline-formula><math display="inline"><semantics><mrow><msub><mi>ϱ</mi><mn>1</mn></msub><mo form="prefix">exp</mo><mrow><mo>{</mo><mo>−</mo><msub><mi>ϱ</mi><mn>2</mn></msub><mi>x</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula> with some positive constants <inline-formula><math display="inline"><semantics><msub><mi>ϱ</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>ϱ</mi><mn>2</mn></msub></semantics></math></inline-formula>. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.https://www.mdpi.com/2227-7390/8/10/1742exponential estimatesupremum of sumstail probabilityrisk modelinhomogeneityruin probability |
| spellingShingle | Tautvydas Kuras Jonas Sprindys Jonas Šiaulys Martingale Approach to Derive Lundberg-Type Inequalities Mathematics exponential estimate supremum of sums tail probability risk model inhomogeneity ruin probability |
| title | Martingale Approach to Derive Lundberg-Type Inequalities |
| title_full | Martingale Approach to Derive Lundberg-Type Inequalities |
| title_fullStr | Martingale Approach to Derive Lundberg-Type Inequalities |
| title_full_unstemmed | Martingale Approach to Derive Lundberg-Type Inequalities |
| title_short | Martingale Approach to Derive Lundberg-Type Inequalities |
| title_sort | martingale approach to derive lundberg type inequalities |
| topic | exponential estimate supremum of sums tail probability risk model inhomogeneity ruin probability |
| url | https://www.mdpi.com/2227-7390/8/10/1742 |
| work_keys_str_mv | AT tautvydaskuras martingaleapproachtoderivelundbergtypeinequalities AT jonassprindys martingaleapproachtoderivelundbergtypeinequalities AT jonassiaulys martingaleapproachtoderivelundbergtypeinequalities |