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RUIN PROBABILITY ON A FINITE TIME HORIZON (Prawdopodobienstwo ruiny

100%

Michna Z.

Mathematical Economics

2010

nr 6(13)

65-74

In this article we investigate the classical risk process. We derive a formula for the ruin probability on a finite time horizon for zero initial capital that is Cramer's formula and for an arbitrary initial capital that is Seal's formula. Applying these formulas and the approximation of a gamma process by compound Poisson processes we obtain a formula for the supremum distribution of a gamma process with a linear drift.

Ruin Probabilities for Two Collaborating Insurance Companies

100%

Michna Z.

Probability and Mathematical Statistics

2020

tom Vol. 40, Fasc. 2

360--386

We find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities if two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions or if the premium rate for a given insurance portfolio is changed at a certain time. As an example we consider a gamma Lévy process, an -stable Lévy process and Brownian motion. Moreover we obtain identities for the Laplace transform of the distribution of the supremum of Lévy processes with a randomly broken drift (random time of the premium rate change) and on random intervals (random time when the insurance portfolio is closed).

The bullwhip effect in the AR(1) demand structure

100%

Michna Z.

Mathematical Economics

2014

nr 10(17)

53-60

In this note we derive the famous formula of F. Chen, Z. Drezner, J.K. Ryan and D. Simchi-Levi [2000a] for the bullwhip effect measure in a simple two-stage supply chain under the assumption that demands constitute autoregressive structure of order 1. Our approach is a little different than in Chen et. al [2000a] and therefore we obtain the formula as an equality unlike Chen et. al [2000a], where they have it as a lower bound. Moreover, we analyze the bullwhip effect measure formula and in some cases we have different conclusions than in Chen et. al [2000a].

Stochastic simulation of storage and inventory systems

63%

Bombała W. , Michna Z.

Mathematical Economics

2012

nr 8 (15)

17-28

This article aims to present the applications of Lévy processes for the stochastic modeling of storage resources. Two cases were considered. In the first one, the volume of supplies to the storehouse is described by a random process (Lévy process), while issuing the products is described by a deterministic and linear function. The second case is reversed: the delivery to the storehouse is described by a linear function (variable: time), while issuing the goods is described by a Lévy process. For both cases the form of the stock level process and examples of its trajectories, when the net supply is a Lévy process, are given. We investigated the following net supply processes: gamma process, α-stable Lévy process with α = 0.5, Cauchy process, Wiener process.

The bullwhip effect in supply chains with stochastic lead times

51%

Michna Z. , Nielsen I. E. , Nielsen P.

Mathematical Economics

2013

nr 9 (16)

71-88

In this article we consider a simple two stage supply chain. We quantify the variance amplification of orders – the bullwhip effect in a model with stochastic lead times. Employing the moving average forecasting method for lead time demands we obtain an exact value of the bullwhip effect measure. We analyze the formula using numerical examples.

On the supremum from gaussian processes over infinite horizon

51%

Dębicki K. , Michna Z. , Rolski T.

Probability and Mathematical Statistics

1998

tom Vol. 18, Fasc. 1

83--100

In the paper we study the asymptotic of the tail of distribution function P(A(X,c) > x) for x→∞, where A(X, c) is the supremum of X (t)—ct over [0, ∞). In particular, X(t) is the fractional Brownian motion, a nonlinearly scaled Brownian motion or some integrated stationary Gaussian processes. For the fractional Brownian motion we give a stronger result than a recent one of Duffield and O’Connell [5].