Renewal equations for option pricing

In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these equations for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.

Langevin equations with multiplicative noise: Application to domain growth

Langevin Equations of Ginzburg-Landau form, with multiplicative noise, are proposed to study the effects of fluctuations in domain growth. These equations are derived from a coarse-grained methodology. The Cahn-Hiliard-Cook linear stability analysis predicts some effects in the transitory regime. We also derive numerical algorithms for the computer simulation of these equations. The numerical results corroborate the analytical predictions of the linear analysis. We also present simulation results for spinodal decomposition at large times.

Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions

The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of state has the form of the ideal gas equation of state, but with the density depending on the potential characterizing long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable that modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation that relates the potential energy to the temperature, pressure, and chemical potential. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of the application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.

Anticipating linear stochastic differential equations driven by a Lévy process

In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a Lévy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we extend the method based on Girsanov transformations on Wiener space and developped by Buckdahn [7] to the canonical Lévy space, which is introduced in [25].

Risk Prediction Scores for Recurrence and Progression of Non-Muscle Invasive Bladder Cancer: An International Validation in Primary Tumours

Abstract Objective: We aimed to determine the validity of two risk scores for patients with non-muscle invasive bladder cancer in different European settings, in patients with primary tumours. Methods: We included 1,892 patients with primary stage Ta or T1 non-muscle invasive bladder cancer who underwent a transurethral resection in Spain (n = 973), the Netherlands (n = 639), or Denmark (n = 280). We evaluated recurrence-free survival and progression-free survival according to the European Organisation for Research and Treatment of Cancer (EORTC) and the Spanish Urological Club for Oncological Treatment (CUETO) risk scores for each patient and used the concordance index (c-index) to indicate discriminative ability. Results: The 3 cohorts were comparable according to age and sex, but patients from Denmark had a larger proportion of patients with the high stage and grade at diagnosis (p,0.01). At least one recurrence occurred in 839 (44%) patients and 258 (14%) patients had a progression during a median follow-up of 74 months. Patients from Denmark had the highest 10- year recurrence and progression rates (75% and 24%, respectively), whereas patients from Spain had the lowest rates (34% and 10%, respectively). The EORTC and CUETO risk scores both predicted progression better than recurrence with c-indices ranging from 0.72 to 0.82 while for recurrence, those ranged from 0.55 to 0.61. Conclusion: The EORTC and CUETO risk scores can reasonably predict progression, while prediction of recurrence is more difficult. New prognostic markers are needed to better predict recurrence of tumours in primary non-muscle invasive bladder cancer patients.

Efficacy of methotrexate to prevent postoperative recurrence of Crohn's disease

BACKGROUND: endoscopic postoperative recurrence (POR) of Crohn’s disease (CD) is the presence of lesions in previously unaffected intestinal segments and occurs in up to 85% of patients one year after bowel resection. Patients at low risk for POR can either remain untreated until lesions recur or receive immediate prevention after surgery with mesalazine, azathioprine (AZA) and/or metronidazole, although with moderate benefit. Out of the postoperative setting, methotrexate (MTX) has been shown to be efficacious for induction and maintenance of remission and has been established as the second-line immunosuppressant for patients with CD unresponsive or intolerant to AZA.AIMS: to determine the efficacy and safety of MTX to prevent endoscopic and clinical POR at 24 weeks after surgery in low risk patientsMETHODS: the study consists on a multicenter, randomized, double-blind and placebo-controlled clinical trial that will enroll 132 patients at low risk for POR (non-smokers, first intestinal resection, non-penetrating behavior). Patients will be randomized to receive subcutaneous MTX at doses of 25 mg/week or an identical placebo, for 24 weeks. Endoscopic and clinical assessment of POR will be performed after 24 weeks (6 months) of treatment. The main outcome is endoscopic POR, defined as a Rutgeerts score of >i2, and secondary outcomes include clinical POR, defined as >i2 lesions plus a Crohn’s Disease Activity Index (CDAI) >150, and description of adverse events