A Characterization of the Gamma Distribution

Lukács, Eugene

doi:10.1214/aoms/1177728549

Cited by 226 publications

(143 citation statements)

References 0 publications

Supporting

Mentioning

139

Contrasting

Order By: Relevance

“…As follows from a remarkable result of Lukacs [11] for gamma variables, this property is characteristic of the gamma processes. That is, if η is a Lévy process such thatη and η(X) are independent, then η is a gamma process (possibly with some scale parameter).…”

Section: Lemma 1 the Total Charge γ(X) Of The Gamma Process And The mentioning

confidence: 84%

On the Markov–Krein Identity and Quasi-Invariance of the Gamma Process

Vershik¹,

Yor²,

Цилевич³

2004

Journal of Mathematical Sciences

View full text Add to dashboard Cite

Section: Lemma 1 the Total Charge γ(X) Of The Gamma Process And The mentioning

confidence: 84%

On the Markov–Krein Identity and Quasi-Invariance of the Gamma Process

Vershik¹,

Yor²,

Цилевич³

2004

Journal of Mathematical Sciences

View full text Add to dashboard Cite

“…are independent of g t /g T and g s /g T . The following lemma is a classical result (Lukacs 1955) which can be used as the basis of an alternative proof of proposition 3.2.…”

Section: ð3:6þmentioning

confidence: 99%

Dam rain and cumulative gain

2008

View full text Add to dashboard Cite

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process. An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow-Debreu security on the cumulative gains process is determined, and is used to obtain a closedform expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions and rainfall.

show abstract

“…Notice that S * is no longer independent from S t , for, according to Lukacs [13], this independence essentially characterizes Gamma subordinators. The next proposition shows that S * = S when S is close enough to a stable subordinator.…”

Section: It Remains To See That Dmentioning

confidence: 99%

“…., γ t n ). A (much less elementary) converse is also known: Lukacs has proved in [13] that given two independent, strictly positive, nondegenerate random variables X and Y , if…”

mentioning

confidence: 99%