Symmetries of stochastic differential equations using Girsanov transformations

Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models

Variational principles for involutive systems of vector fields

In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar way by means of an higher order variational principle, and how this extends to involutive systems of vector fields.Comment: 31 pages. To appear in International Journal of Geometric Methods in Modern Physics (IJGMMP

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

A note on symmetries of diffusions within a martingale problem approach

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and It\uf4 integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes

A priori estimates for 3D incompressible current-vortex sheets

We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics

NHS commissioning practice and health system governance: a mixed-methods realistic evaluation

Background By 2010 English health policy-makers had concluded that the main NHS commissioners [primary care trusts (PCTs)] did not sufficiently control provider costs and performance. After the 2010 general election, they decided to replace PCTs with general practitioner (GP)-controlled Clinical Commissioning Groups (CCGs). Health-care commissioners have six main media of power for exercising control over providers, which can be used in different combinations (‘modes of commissioning’). Objectives To: elicit the programme theory of NHS commissioning policy and empirically test its assumptions; explain what shaped NHS commissioning structures; examine how far current commissioning practice allowed commissioners to exercise governance over providers; examine how commissioning practices differ in different types of commissioning organisation and for specific care groups; and explain what factors influenced commissioning practice and the relationships between commissioners and providers. Design Mixed-methods realistic evaluation, comprising: Leximancer and cognitive frame analyses of policy statements to elicit the programme theory of NHS commissioning policy; exploratory cross-sectional analysis of publicly available managerial data about PCTs; systematic comparison of case studies of commissioning in four English sites – including commissioning for older people at risk of unplanned hospital admission; mental health; public health; and planned orthopaedic surgery – and of English NHS commissioning practice with that of a German sick-fund and an Italian region (Lombardy); action learning sets, to validate the findings and draw out practical implications; and two framework analyses synthesising the findings and testing the programme theory empirically. Results In the four English case study sites, CCGs were formed by recycling former commissioning structures, relying on and maintaining the existing GP commissioning leaderships. The stability of distributed commissioning depended on the convergence of commissioners’ interests. Joint NHS and local government commissioning was more co-ordinated at strategic than operational level. NHS providers’ responsiveness to commissioners reflected how far their interests converged, but also providers’ own internal ability to implement agreements. Commissioning for mental health services and to prevent recurrent unplanned hospital readmissions relied more on local ‘micro-commissioning’ (collaborative care pathway design) than on competition. Service commissioning was irrelevant to intersectoral health promotion, but not clinical prevention work. On balance, the possibility of competition did not affect service outcomes in the ways that English NHS commissioning policies assumed. ‘Commodified’ planned orthopaedic surgery most lent itself to provider competition. In all three countries, tariff payments increased provider activity and commissioners’ costs. To contain costs, commissioners bundled tariff payments into blocks, agreed prospective case loads with providers and paid below-tariff rates for additional cases. Managerial performance, negotiated order and discursive control were the predominant media of power used by English, German and Italian commissioners. Conclusions Commissioning practice worked in certain respects differently from what NHS commissioning policy assumed. It was often laborious and uncertain. In the four English case study sites financial and ‘real-side’ contract negotiations were partly decoupled, clinician involvement being least on the financial side. Tariff systems weakened commissioners’ capacity to choose providers and control costs. Commissioners adapted the systems to solve this problem. Our findings suggest a need for further research into whether or not differently owned providers (corporate, third sector, public, professional partnership, etc.) respond differently to health-care commissioners and, if so, what specific implications for commissioning practice follow. They also suggest that further work is needed to assess how commissioning practices impact on health system integration when care pathways have to be constructed across multiple providers that must tender competitively for work, perhaps against each other. Funding The National Institute for Health Research Health Services and Delivery Research programme