Traveling waves and homogeneous fragmentation

We formulate the notion of the classical Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu [In Seminar on Stochastic Processes (1988) 223-242 Birkh\"{a}user] and Chauvin [Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump-Mode-Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365-395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553-570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincar\'{e} Probab. Statist. 46 (2010) 119-134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609-631]) showing their mathematical robustness even within the context of fragmentation theory.Comment: Published in at http://dx.doi.org/10.1214/10-AAP733 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The extended hypergeometric class of L\'evy processes

With a view to computing fluctuation identities related to stable processes, we review and extend the class of hypergeometric L\'evy processes explored in Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of a process in the extended class, and characterise its exponential functional. Finally, we give three concrete examples arising from transformations of stable processes.Comment: 22 page

A Wiener--Hopf Monte Carlo simulation technique for L\'{e}vy processes

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general L\'{e}vy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr's so-called "Canadization" technique as well as Doney's method of stochastic bounds for L\'{e}vy processes; see Carr [Rev. Fin. Studies 11 (1998) 597--626] and Doney [Ann. Probab. 32 (2004) 1545-1552]. We rely fundamentally on the Wiener-Hopf decomposition for L\'{e}vy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos L\'{e}vy en titillant la factorization de Wiener-Hopf (2002) Laboratoire de Math\'{e}matiques de L'INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801--1830]. We illustrate our Wiener--Hopf Monte Carlo method on a number of different processes, including a new family of L\'{e}vy processes called hypergeometric L\'{e}vy processes. Moreover, we illustrate the robustness of working with a Wiener--Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given L\'{e}vy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.Comment: Published in at http://dx.doi.org/10.1214/10-AAP746 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cannabis and schizophrenia

BACKGROUND Schizophrenia is a mental illness causing disordered beliefs, ideas and sensations. Many people with schizophrenia smoke cannabis, and it is unclear why a large proportion do so and if the effects are harmful or beneficial. It is also unclear what the best method is to allow people with schizophrenia to alter their cannabis intake. OBJECTIVES To assess the effects of specific psychological treatments for cannabis reduction in people with schizophrenia.To assess the effects of antipsychotics for cannabis reduction in people with schizophrenia.To assess the effects of cannabinoids (cannabis related chemical compounds derived from cannabis or manufactured) for symptom reduction in people with schizophrenia. SEARCH METHODS We searched the Cochrane Schizophrenia Group Trials Register, 12 August 2013, which is based on regular searches of BIOSIS, CINAHL, EMBASE, MEDLINE, PUBMED and PsycINFO.We searched all references of articles selected for inclusion for further relevant trials. We contacted the first author of included studies for unpublished trials or data. SELECTION CRITERIA We included all randomised controlled trials involving cannabinoids and schizophrenia/schizophrenia-like illnesses, which assessed:1) treatments to reduce cannabis use in people with schizophrenia;2) the effects of cannabinoids on people with schizophrenia. DATA COLLECTION AND ANALYSIS We independently inspected citations, selected papers and then re-inspected the studies if there were discrepancies, and extracted data. For dichotomous data we calculated risk ratios (RR) and for continuous data, we calculated mean differences (MD), both with 95% confidence intervals (CI) on an intention-to-treat basis, based on a fixed-effect model. We excluded data if loss to follow-up was greater than 50%. We assessed risk of bias for included studies and used GRADE to rate the quality of the evidence. MAIN RESULTS We identified eight randomised trials, involving 530 participants, which met our selection criteria.For the cannabis reduction studies no one treatment showed superiority for reduction in cannabis use. Overall, data were poorly reported for many outcomes of interest. Our main outcomes of interest were medium-term data for cannabis use, global state, mental state, global functioning, adverse events, leaving the study early and satisfaction with treatment. 1. Reduction in cannabis use: adjunct psychological therapies (specifically about cannabis and psychosis) versus treatment as usualResults from one small study showed people receiving adjunct psychological therapies specifically about cannabis and psychosis were no more likely to reduce their intake than those receiving treatment as usual (n = 54, 1 RCT, MD -0.10, 95% CI -2.44 to 2.24, moderate quality evidence). Results for other main outcomes at medium term were also equivocal. No difference in mental state measured on the PANSS positive were observed between groups (n = 62, 1 RCT, MD -0.30 95% CI -2.55 to 1.95, moderate quality evidence). Nor for the outcome of general functioning measured using the World Health Organization Quality of Life BREF (n = 49, 1 RCT, MD 0.90 95% CI -1.15 to 2.95, moderate quality evidence). No data were reported for the other main outcomes of interest 2. Reduction in cannabis use: adjunct psychological therapy (specifically about cannabis and psychosis) versus adjunct non-specific psychoeducation One study compared specific psychological therapy aimed at cannabis reduction with general psychological therapy. At three-month follow-up, the use of cannabis in the previous four weeks was similar between treatment groups (n = 47, 1 RCT, RR 1.04 95% CI 0.62 to 1.74, moderate quality evidence). Again, at a medium-term follow-up, the average mental state scores from the Brief Pscychiatric Rating Scale-Expanded were similar between groups (n = 47, 1 RCT, MD 3.60 95% CI - 5.61 to 12.81, moderate quality evidence). No data were reported for the other main outcomes of interest: global state, general functioning, adverse events, leaving the study early and satisfaction with treatment. 3. Reduction in cannabis use: antipsychotic versus antipsychotic In a small trial comparing effectiveness of olanzapine versus risperidone for cannabis reduction, there was no difference between groups at medium-term follow-up (n = 16, 1 RCT, RR 1.80 95% CI 0.52 to 6.22, moderate quality evidence). The number of participants leaving the study early at medium term was also similar (n = 28, 1 RCT, RR 0.50 95% CI 0.19 to 1.29, moderate quality evidence). Mental state data were reported, however they were reported within the short term and no difference was observed. No data were reported for global state, general functioning, and satisfaction with treatment.With regards to adverse effects data, no study reported medium-term data. Short-term data were presented but overall, no real differences between treatment groups were observed for adverse effects. 4. Cannabinoid as treatment: cannabidiol versus amisulprideAgain, no data were reported for any of the main outcomes of interest at medium term. There were short-term data reported for mental state using the BPRS and PANSS, no overall differences in mental state were observed between treatment groups. AUTHORS' CONCLUSIONS Results are limited and inconclusive due to the small number and size of randomised controlled trials available and quality of data reporting within these trials. More research is needed to a) explore the effects of adjunct psychological therapy that is specifically about cannabis and psychosis as currently there is no evidence for any novel intervention being better than standard treatment,for those that use cannabis and have schizophrenia b) decide the most effective drug treatment in treating those that use cannabis and have schizophrenia, and c) assess the effectiveness of cannabidiol in treating schizophrenia. Currently evidence is insufficient to show cannabidiol has an antipsychotic effect