Convergence of switching diffusions

This paper studies the asymptotic behavior of processes with switching. More precisely, the stability under fast switching for diffusion processes and discrete state space Markovian processes is considered. The proofs are based on semimartingale techniques, so that no Markovian assumption for the modulating process is needed

Runs in superpositions of renewal processes with applications to discrimination

AbstractWald and Wolfowitz [Ann. Math. Statist. 11 (1940) 147–162] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let (Sn)n⩾1 and (Tn)n⩾1 be the arrival processes for channel 1 and channel 2, respectively, and (Wn)n⩾1 their be superposition with counting process N(t)=defsup{n⩾1:Wn⩽t}. Let further Rn* be the number of runs in W1,…,Wn and Rt=RN(t)* the number of runs observed up to time t. We study the asymptotic behavior of Rn* and Rt, first for the case where (Sn)n⩾1 and (Tn)n⩾1 have exponentially distributed increments with parameters λ1 and λ2, and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level α tests for testing λ1=λ2 against λ1≠λ2 in the first case, and for testing for equal scale parameters in the second

A statistical equilibrium model of competitive firms

We find that the empirical density of firm profit rates, measured as returns on assets, is markedly non-Gaussian and reasonably well described by an exponential power (or Subbotin) distribution. We start from a statistical equilibrium model that leads to a stationary Subbotin density in the presence of complex interactions among competitive heterogeneous firms. To investigate the dynamics of firm profitability, we construct a diffusion process that has the Subbotin distribution as its stationary probability density. This leads to a phenomenologically inspired interpretation of variations in the shape parameter of the Subbotin distribution, which essentially measures the competitive pressure in and across industries. Our findings have profound implications both for the previous literature on the ‘persistence of profits’ as well as for understanding competition as a dynamic process. Our main formal finding is that firms' idiosyncratic efforts and the tendency for competition to equalize profit rates are two sides of the same coin, and that a ratio of these two effects ultimately determines the dispersion of the equilibrium distribution

MÜNSTER Runs in Superpositions of Renewal Processes with Applications to Discrimination

Wald and Wolfowitz [11] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let (Sn)n≥1 and (Tn)n≥1 be the arrival processes for channel 1 and channel 2, respectively, and (Wn)n≥1 their be superposition with counting process N(t) def = sup{n ≥ 1:Wn≤t}. Let further R ∗ n be the number of runs in W1,..., Wn and Rt = R ∗ the number of runs N(t) observed up to time t. We study the asymptotic behavior of R ∗ n and Rt, first for the case where (Sn)n≥1 and (Tn)n≥1 have exponentially distributed increments with parameters λ1 and λ2, and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level α tests for testing λ1 = λ2 against λ1 = λ2 in the first case, and for testing for equal scale parameters in the second

American Options with guarantee – A class of two-sided stopping problems

We introduce a class of optimal stopping problems in which the gain is at least a fraction of the initial value. From a financial point of view this structure can be seen as a guarantee for the holder of an American option. It turns out that the optimal strategies are of two-sided type under weak conditions. If the driving process is a diffusion we use harmonic-functions techniques to obtain general results. For an explicit solution we derive two differential equations that characterize the optimal strategies. Furthermore we study the case of L\ue9vy processes. An explicit solution is obtained for spectrally negative processes using scale function