w symmetries of ito stochastic differential equations we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones

random lie point symmetries of stochastic differential equations we study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random lie point symmetries of stochastic differential equations both in ito and in stratonovich forms we also discuss relations with previous results in the literature

stochastic calculus of variations abstract a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes generalizations of the euler equation and noether s theorem are obtained and several conservation laws are discussed an application to nelson s probabilistic framework of quantum mechanics is also given

a connection between quantum dynamics and approximation of markov diffusions it is observed that the existence of an attracting set in the class of solutions to the stochastic lagrangian variational principle leads to a natural problem of convergence of diffusions in the carlen class it is then shown how dynamical properties enable one to prove some convergence results in the two dimensional gaussian case

variational processes and stochastic versions of mechanics the dynamical structure of any reasonable stochastic version of classical mechanics is investigated including the version created by nelson e nelson quantum fluctuations princeton u p princeton nj 1985 phys rev 150 1079 1966 for the description of quantum phenomena two different theories result from this common structure one of them is the imaginary time version of nelson s theory whose existence was unknown and yields a radically new probabilistic interpretation of the heat equation the existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of yasue k yasue j math phys 22 1010 1981

malliavin calculus and euclidean quantum mechanics i functional calculus abstract we give a rigorous version of the functional calculus developed by r feynman in relation to his path integral formulation of quantum mechanics our approach is euclidean but distinct from the one founded on the feynman kac formula it uses two basic ingredients a new probabilistic interpretation of the classical heat equation introduced recently in the framework of euclidean quantum mechanics and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation

closed form expansion conditional expectation and option valuation enlightened by the theory of watanabe watanabe s 1987 analysis of wiener functionals malliavin calculus and its applications to heat kernels ann probab 15 1 39 for analyzing generalized random variables and its further development in yoshida yoshida n 1992a asymptotic expansions for statistics related to small diffusions j japan statist soc 22 139 159 takahashi takahashi a 1995 essays on the valuation problems of contingent claims ph d thesis haas school of business university of california berkeley takahashi a 1999 an asymptotic expansion approach to pricing contingent claims asia pacific financial markets 6 115 151 as well as kunitomo and takahashi kunitomo n takahashi a 2001 the asymptotic expansion approach to the valuation of interest rate contingent claims math finance 11 1 117 151 kunitomo n takahashi a 2003 on validity of the asymptotic expansion approach in contingent claim analysis ann appl probab 13 3 914 952 etc we focus on a wide range of mult

towards a quantum analog of weak kam theory we discuss a quantum analogue of mather s minimization principle for lagrangian dynamics and provide some formal calculations suggesting the corresponding euler lagrange equation we then rigorously construct from the dual eigenfunctions of a certain non selfadjoint operator a candidate for a minimizer and recover aspects of weak kam theory in the limit as h 0 regarding our state as a quasimode we furthermore derive some error estimates although it remains an open problem to improve these bounds

on the behaviour near expiry for multi dimensional american options in this paper we analyse the behaviour near expiry of the free boundary appearing in the pricing of multi dimensional american options in a financial market driven by a general multi dimensional ito diffusion in particular we prove regularity for the pricing function up to the terminal state and we establish a sufficient criteria for the conclusion that the optimal exercise boundary approaches the terminal state faster than parabolically

the density manifold and configuration space quantization on developpe la structure geometrique differentielle d une variete de frechet de densites ceci fournit un cadre geometrique a la quantification reliee a la mecanique stochastique de nelson on etudie les structures riemanniennes et symplectiques de la variete densite et on deduit l equation de schrodinger a partir d un principe variationnel

hamilton s principle in stochastic mechanics in this paper we establish three variational principles that provide new foundations for nelson s stochastic mechanics in the case of nonrelativistic particles without spin the resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature we first develop two stochastic variational principles whose hamilton jacobi like equations are precisely the two coupled partial differential equations that are obtained from the schrodinger equation madelung equations the two problems are zero sum noncooperative stochastic differential games that are familiar in the control theory literature they are solved here by means of a new absolutely elementary method based on lagrange functionals for both games the saddle point equilibrium solution is given by the nelson s process and the optimal controls for the two competing players are precisely nelson s current velocity v and osmotic velocity u respectively the first variational principle includes as special cases both the guerra morato variational principle phys rev d 27 1774 1983 and schrodinger original variational derivation of the time independent equation it also reduces to the classical least action principle when the intensity of the underlying noise tends to zero it appears as a saddle point action principle in the second variational principle the action is simply the difference between the initial and final configurational entropy it is therefore a saddle point entropy production principle from the variational principles it follows in particular that both v x t and u x t are gradients of appropriate principal functions in the variational principles the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second combining the two variational principles we get the quantum hamilton principle i e a variational characterization of the logarithm of the wave function the lagrangian is the lagrangian of classical mechanics with the complex valued velocity v iu replacing the classical velocity the dynamics is given by a stochastic differential equation for real valued diffusions with complex valued drift and driving noise processes from the variational principle we derive a newton type law we finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator

geometrical stochastic control and quantization a class of diffusion processes controlled by the geometry of the manifold on which they evolve is considered the kinetic energy of such diffusions is shown to be a geometric object expressed in terms of the curvature and torsion tensors this gives rise to an action functional leading to a variational principle from which a nontrivial critical geometry with nonvanishing torsion emerges the resulting criticality condition is related to the schrodinger equation in a manner that reproduces the features of nelson s stochastic approach to quantum mechanics

managing non homogeneous information in group decision making within the frame of decision aid literature decision making problems with multiple sources of information have drawn the attention of researchers from a wide spectrum of disciplines in decision situations with multiple individuals each one has his own knowledge on the alternatives of the decision problem the use of information assessed in different domains is not a seldom situation this non homogeneous information can be represented as values belonging to domains with different nature as linguistic numerical and interval valued or can be values assessed in label sets with different granularity multi granular linguistic information decision processes for solving these problems are composed by two phases aggregation and exploitation the main problem to deal with non homogeneous contexts is how to aggregate the information assessed in these contexts in this paper taking as base the 2 tuple fuzzy linguistic representation model we shall develop an aggregation process for dealing with non homogeneous contexts in first place we shall develop an aggregation process for combining numerical interval valued and linguistic information afterwards we shall propose different extensions of this process to deal with contexts in which can appear other type of information as intuitionistic fuzzy sets or multi granular linguistic information

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

lagrangians of stochastic mechanics the stochastic variational principle which applies to diffusions and extremises the mean value of the classical action is generalised to include non markovian diffusions a whole family of different stochastic actions labelled by a time dependent parameter is obtained and all of them are shown to lead to the shrodinger equation as a result the well known degeneracy of the class of stochastic processes which correspond to a single quantum state is recovered it is also related to an arbitrariness in the choice of the noise which can occur in two places in the definition of the noise through the time dependent parameter and in the auto correlations of the noise which although constant during the variational method remain undetermined the whole scheme is given a covariant form

a variational principle for the navier stokes equation abstract motivated from arnold s variational characterization of the euler equation in terms of geodesic families of diffeomorphisms a variational principle for the motion of incompressible viscous fluids is presented a volume preserving diffusion process with drift velocity field subject to the navier stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations

quantum mechanics and stochastic control theory a time symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics the main idea is based on nelson s probability theoretical approach to quantum mechanics his approach is reformulated as a time symmetric stochastic control problem several different control constraints equivalent to nelson s are obtained one of them has a close connection to the lagrangian formalism of classical mechanics this suggests to us the use of stochastic calculus of variations within the realm of this time symmetric stochastic control theory it is shown why schrodinger s original variational method of quantization was successful several advantageous points of the stochastic control theoretical approach to quantum mechanics including the analysis of the classical limit are also discussed

an introduction to the semiclassical limit of euclidean quantum mechanics a new class of gaussian bernstein diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new euclidean strategy for the quantization of systems of particles in a potential euclidean quantum mechanics all the measures known by other means up to now are included as particular cases of the present framework it is shown that the theory may be regarded as a well defined probabilistic version of feynman s space time approach distinct in particular from the one due to kac and with much closer relations with classical lagrangian dynamics many explicit examples are considered

preservation of convexity of solutions to parabolic equations abstract in the present paper we find necessary and sufficient conditions on the coefficients of a parabolic equation for convexity to be preserved a parabolic equation is said to preserve convexity if given a convex initial condition any solution of moderate growth remains a convex function of the spatial variables for each fixed time

similarity and dissimilarity measures between fuzzy sets a formal relational study the paper deals with the well known notion of dis similarity measures between fuzzy sets we provide three separate lists of axioms that fit with the respective notions of general comparison measure similarity measure and dissimilarity measure then we review some of the most important axiomatic definitions of dis similarity measures in the literature by referring to the axioms in those lists satisfied by each specific definition this common framework will make our study about the formal relationships among different axiomatic definitions much easier some of them which are apparently different do in fact share many commonalities we provide a self contained picture of these relationships by providing formal results and counterexamples that reflect which of the dis similarity definitions in the literature are connected by implication relations and which of them are not we finalize the paper with an in depth study about the notion of duality between similarity and dissimilarity measures as well as with some concluding remarks

noether s theorem in symmetric stochastic calculus of variations quite recently a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics which is an extension of nelson s stochastic mechanics in this article a noether s theorem is formulated within this calculus of variations conservation laws of momentum angular momentum and energy are proved which are related with the same laws in quantum mechanics

geometry of the schrodinger equation and stochastic mass transportation the schrodinger equation is deduced from a geometric principle lagrange hamilton jacobi and hamilton formalisms are defined almost analogously to the deterministic case which can be identified as geometric optic the form of these formalisms is identical to the deterministic formalisms furthermore it will be shown how the deterministic case is superficial to the stochastic one if planck s constant is very small an elementary proof of heisenberg s uncertainty relation finishes the paper

convexity preserving jump diffusion models for option pricing we investigate which jump diffusion models are convexity preserving the study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters we give a necessary condition for convexity to be preserved in several dimensional jump diffusion models this necessary condition is then used to show that within a large class of possible models the only convexity preserving models are the ones with linear coefficients

discrimination power of measures of comparison this paper is based on a framework 3 for a formalization of measures of comparison of fuzzy objects the purpose is to describe the behaviour of measures of comparison within a given family in order to facilitate the choice of a particular measure it can be done owing to the discrimination power of a measure

on a certain class of two sided continuous local semimartingales toward a sample wise characterization of the nelson process working with the extended framework of stochastic integrals recently discovered by ito a complex of stochastic processes inherent in quantum mechanics the nelson process is characterized in terms of sample paths it is shown that the nelson process belongs to a certain class of two sided continuous local semimartingales several basics of stochastic calculus in this class are presented stochastic calculus of variations is applied in this class to construct the nelson process and to further illustrate some details of its sample paths examples are the bound states the two slit interference and the gravity in quantum mechanics

maupertuis principle of least action in stochastic calculus of variations within the framework of stochastic calculus of variations for time symmetric semimartingales x t we consider two different stochastic versions of maupertuis least action principle in lagrangian and hamiltonian terms the general results are applied to classical statistical mechanics where they coincide with those of classical calculus of variations and to nelson s stochastic mechanics an approach to quantum mechanics where a time symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of hamilton s principle of least action some historic examples of old quantum theory are discussed

canonical stochastic dynamical systems a canonical stochastic dynamical system for time symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations a certain class of stochastic dynamical systems gives a hamiltonian formalism of nelson s stochastic mechanics in a manner analogous to classical mechanics the notions of a stochastic poisson bracket and canonical transformation are introduced to the stochastic dynamical systems it is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation

dynamic programming and quantum mechanical motion abstract in this article we show that quantum dynamics is the most natural generalization of classical dynamics from the point of view of optimal control employing the techniques of dynamic programming we derive the schrodinger equation starting from the lagrangian defined in terms of nelson s forward and backward velocities the generalization to the relativistic case is also analyzed and the klein gordon equation is similarly derived

a simplified variational characterization of schrodinger processes the variational problem e 1 2 t0 t a t xt 2 dt t0c t xt dt min where t is the drift process of a diffusion process with unit diffusion coefficient and given initial and final distributions a is a given vector field and c is a given scalar field is considered it is shown that the solution is given by a certain markovian diffusion process which in the case a 0 c 0 first was investigated by schrodinger sitzungsber preuss akad wiss phys math kl 1931 144

stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper

extremal flows in wasserstein space we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics