symmetry reductions exact solutions and conservation laws of the generalized zakharov equations in this paper the generalized zakharov equations which describe interactions between high and low frequency waves in plasma physics are studied from the perspective of lie symmetry analysis and conservation laws based on some subalgebras of symmetries several reductions and numerous new exact solutions are obtained all of these solutions represent modified traveling waves the obtained solutions include expressions involving airy functions bessel functions whittaker functions and generalized hypergeometric functions previously unknown conservation laws are constructed for the generalized zakharov equations using the direct method profiles are presented for some of these new solutions

a coupled korteweg de vries equation with dispersion a dispersive system describing a vector multiplet interacting with the korteweg de vries field is shown to be a number of a bi hamiltonian integrable hierarchy

group analysis and exact solutions of the time fractional fokker planck equation abstract in this paper the lie symmetry analysis method is extended to deal with the nonlinear time fractional fokker planck fp equation with riemann liouville derivative the erdelyi kober fractional derivative which is depending on a parameter is used for the reduction of fp equation symmetry reduction is provided and some exact analytic solutions to the time fractional fp equation are investigated by virtue of the reduction method introduced by m c nucci

option pricing stochastic volatility singular dynamics and constrained path integrals abstract stochastic volatility models have been widely studied and used in the financial world the heston model heston 1993 7 is one of the best known models to deal with this issue these stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter which relates the two brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset solutions to the heston model in the context of option pricing using a path integral approach are found in lemmens et al 2008 21 while in baaquie 2007 1997 12 13 propagators for different stochastic volatility models are constructed in all previous cases the propagator is not defined for extreme cases 1 it is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator in this paper we study in detail a general class of stochastic volatility models for extreme values 1 and show that in these two cases the associated classical dynamics corresponds to a system with second class constraints which must be dealt with using dirac s method for constrained systems dirac 1958 1967 22 23 in order to properly obtain the propagator in the form of a euclidean hamiltonian path integral henneaux and teitelboim 1992 25 after integrating over momenta one gets an euclidean lagrangian path integral without constraints which in the case of the heston model corresponds to a path integral of a repulsive radial harmonic oscillator in all the cases studied the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral

the mobius function and distal flows we prove that the mobius function is linearly disjoint from an analytic skew product on the 2 torus these flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points the previous cases for which such disjointness has been proved are all regular we also establish the linear disjointness of the mobius function from various distal homogeneous flows

numerical method for solving fractional coupled burgers equations in this paper we use the fractional variational iteration method fvim to solve a time and space fractional coupled burgers equations some numerical examples are presented to show the efficiency of considered method a comparison of the proposed method is made with the exact solution adomain decomposition method adm generalized differential transformation method gdtm and homotopy perturbation method hpm

on invariant analysis of some time fractional nonlinear systems of partial differential equations i an investigation of lie point symmetries for systems of time fractional partial differential equations including ito system coupled burgers equations coupled korteweg de vries equations hirota satsuma coupled kdv equations and coupled nonlinear hirota equations has been done using the obtained symmetries each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving erdelyi kober fractional differential operator depending on a parameter

the quantum dark side of the optimal control theory abstract in a recent article a generic optimal control problem was studied from a physicist s point of view contreras et al 2017 through this optic the pontryagin equations are equivalent to the hamilton equations of a classical constrained system by quantizing this constrained system using the right ordering of the operators the corresponding quantum dynamics given by the schrodinger equation is equivalent to that given by the hamilton jacobi bellman equation of bellman s theory the conclusion drawn there were based on certain analogies between the equations of motion of both theories in this paper a closer and more detailed examination of the quantization problem is carried out by considering three possible quantization procedures right quantization left quantization and feynman s path integral approach the bellman theory turns out to be the classical limit 0 of these three different quantum theories also the exact relation of the phase s x t of the wave function x t e i s x t of the quantum theory with bellman s cost function j x t is obtained in fact s x t satisfies a conjugate form of the hamilton jacobi bellman equation which implies that the cost functional j x t must necessarily satisfy the usual hamilton jacobi bellman equation thus the bellman theory effectively corresponds to a quantum view of the optimal control problem

lie symmetry analysis and exact solutions of the time fractional biological population model abstract we first perform a complete lie symmetry classification for the time fractional biological population model with riemann liouville fractional derivative and then construct the corresponding reduced equations with the obtained lie symmetries moreover with the f expansion method and the reduced equations we obtain several new exact solutions for the equation and show the propagation pattern via the evolutional figures by means of the power series theory an exact power series solution of the equation is also constructed

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

multi asset black scholes model as a variable second class constrained dynamical system abstract in this paper we study the multi asset black scholes model from a structural point of view for this we interpret the multi asset black scholes equation as a multidimensional schrodinger one particle equation the analysis of the classical hamiltonian and lagrangian mechanics associated with this quantum model implies that in this system the canonical momentums cannot always be written in terms of the velocities this feature is a typical characteristic of the constrained system that appears in the high energy physics to study this model in the proper form one must apply dirac s method for constrained systems the results of the dirac s analysis indicate that in the correlation parameters space of the multi assets model there exists a surface called the kummer surface k where the determinant of the correlation matrix is null on which the constraint number can vary we study in detail the cases with n 2 and n 3 assets for these cases we calculate the propagator of the multi asset black scholes equation and show that inside the kummer k surface the propagator is well defined but outside k the propagator diverges and the option price is not well defined on k the propagator is obtained as a constrained path integral and their form depends on which region of the kummer surface the correlation parameters lie thus the multi asset black scholes model is an example of a variable constrained dynamical system and it is a new and beautiful property that had not been previously observed

lie symmetry analysis and exact explicit solutions for general burgers equation in this paper the lie symmetry analysis is performed for the general burgers equation the exact solutions and similarity reductions generated from the symmetry transformations are provided furthermore the all exact explicit solutions and similarity reductions based on the lie group method are obtained some new method and techniques are employed simultaneously such exact explicit solutions and similarity reductions are important in both applications and the theory of nonlinear science

time fractional 2 1 dimensional wu zhang system dispersion analysis similarity reductions conservation laws and exact solutions abstract this paper examines the time fractional 2 1 dimensional wu zhang system describing dispersive long waves the dispersion relation of waves has been obtained from linear analysis and used to find the phase and group velocity the dispersion relation shows an anomalous dispersion of waves lie symmetries and the corresponding similarity reductions are carried out using riemann liouville fractional derivative and the order of the system is reduced to lower dimension furthermore the system is analysed for nonlinear self adjointness and the conservation laws are obtained associated with infinitesimal symmetries by applying new conservation theorem the adjoint variables are used to convert conservation laws into original dependent variables the exponential rational function method is employed to find exact solutions revealing kink and bell type profiles the effect of fractional order on the wave profile of solutions is discussed graphically

invariant analysis of time fractional generalized burgers and korteweg de vries equations abstract a systematic investigation to derive lie point symmetries to time fractional generalized burgers as well as korteweg de vries equations is presented using the obtained lie point symmetries we have shown that each of them has been transformed into a nonlinear ordinary differential equation of fractional order with a new independent variable the derivative corresponding to time fractional in the reduced equation is usually known as the erdelyi kober fractional derivative

lie symmetry analysis explicit solutions and conservation laws for the space time fractional nonlinear evolution equations abstract this paper studies the symmetry analysis explicit solutions convergence analysis and conservation laws cls for two different space time fractional nonlinear evolution equations with riemann liouville rl derivative the governing equations are reduced to nonlinear ordinary differential equation ode of fractional order using their lie point symmetries in the reduced equations the derivative is in erdelyi kober ek sense power series technique is applied to derive an explicit solutions for the reduced fractional odes the convergence of the obtained power series solutions is also presented moreover the new conservation theorem and the generalization of the noether operators are developed to construct the nonlocal cls for the equations some interesting figures for the obtained explicit solutions are presented

path integral approach to closed form option pricing formulas with applications to stochastic volatility and interest rate models we present a path integral method to derive closed form solutions for option prices in a stochastic volatility model the method is explained in detail for the pricing of a plain vanilla option the flexibility of our approach is demonstrated by extending the realm of closed form option price formulas to the case where both the volatility and interest rates are stochastic this flexibility is promising for the treatment of exotic options our analytical formulas are tested with numerical monte carlo simulations

exact solutions to a class of time fractional evolution systems with variable coefficients we explicitly give new group invariant solutions to a class of riemann liouville time fractional evolution systems with variable coefficients these solutions are derived from every element in an optimal system of lie algebras generated by infinitesimal symmetries of evolution systems in the class we express the solutions in terms of mittag leffler functions generalized wright functions and fox h functions and show that these solutions solve diffusion wave equations with variable coefficients these solutions contain previously known solutions as particular cases some plots of solutions subject to the order of the fractional derivative are illustrated we explicitly give new group invariant solutions to a class of riemann liouville time fractional evolution systems with variable coefficients these solutions are derived from every element in an optimal system of lie algebras generated by infinitesimal symmetries of evolution systems in the class we express the solutions in terms of mittag leffler functions generalized wright functions and fox h functions and show that these solutions solve diffusion wave equations with variable coefficients these solutions contain previously known solutions as particular cases some plots of solutions subject to the order of the fractional derivative are illustrated

invariance of a partial differential equation of fractional order under the lie group of scaling transformations abstract in this article a symmetry group of scaling transformations is determined for a partial differential equation of fractional order containing among particular cases the diffusion equation the wave equation and the fractional diffusion wave equation for its group invariant solutions an ordinary differential equation of fractional order with the new independent variable z xt 2 is derived the derivative then is an erdelyi kober derivative depending on a parameter its complete solution is given in terms of the wright and the generalized wright functions

group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

a comparison between several correlated stochastic volatility models abstract we compare the most common stochastic volatility models such as the ornstein uhlenbeck ou the heston and the exponential ou models we try to decide which is the most appropriate one by studying their volatility autocorrelation and leverage effect and thus outline the limitations of each model we add empirical research on market indices confirming the universality of the leverage and volatility correlations

generalized lie symmetry approach for fractional order systems of differential equations iii the generalized lie symmetry technique is proposed for the derivation of point symmetries for systems of fractional differential equations with an arbitrary number of independent as well as dependent variables the efficiency of the method is illustrated by its application to three higher dimensional nonlinear systems of fractional order partial differential equations consisting of the 2 1 dimensional asymmetric nizhnik novikov veselov system 3 1 dimensional burgers system and 3 1 dimensional navier stokes equations with the help of derived lie point symmetries the corresponding invariant solutions transform each of the considered systems into a system of lower dimensional fractional partial differential equations

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

comment on lie symmetries and group classification of a class of time fractional evolution systems j math phys 56 123504 2015 in this article the prolongation formulae proposed in huang and shen j math phys 56 123504 2015 for the lie symmetry analysis of time fractional systems of partial differential equations are proved incomplete and the required correct prolongation operators are suggested with the help of some examples the efficiency of the operators introduced in this study is illustrated

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

mobius disjointness for homogeneous dynamics we prove sarnak s mobius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected lie groups namely we show that if g is any such group g a lattice and u g an ad unipotent element then for every x g and every function f continuous on the 1 point compactification of g the sequence f xun cannot correlate with the mobius function on average

minimal model of financial stylized facts in this work we propose a statistical characterization of a linear stochastic volatility model featuring inverse gamma stationary distribution for the instantaneous volatility we detail the derivation of the moments of the return distribution revealing the role of the inverse gamma law in the emergence of fat tails and of the relevant correlation functions we also propose a systematic methodology for estimating the parameters and we describe the empirical analysis of the standard poor s 500 index daily returns confirming the ability of the model to capture many of the established stylized facts as well as the scaling properties of empirical distributions over different time horizons

the first integral method for the time fractional kaup boussinesq system with time dependent coefficient the first integral method fim is applied to reach the different type solutions of the time fractional kaup boussinesq fkb system with time dependent coefficient it is obtained the envelope bell shaped triangular and kink soliton solutions of this equation the method is an effective one to reach the different types of solutions of fractional nonlinear partial differential equations fnpde

improved fractional sub equation method for 3 1 dimensional generalized fractional kdv zakharov kuznetsov equations in the present paper we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics namely the time fractional kdv zakharov kuznetsov kdv zk and space time fractional modified kdv zakharov kuznetsov mkdv zk equations by using improved fractional sub equation method as a result new types of exact analytical solutions are obtained the obtained results are shown graphically here the fractional derivative is described in the jumarie s modified riemann liouville sense

lie symmetry analysis of a class of time fractional nonlinear evolution systems abstract we study a class of nonlinear evolution systems of time fractional partial differential equations using lie symmetry analysis we obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems we find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain in each case the dimension of the lie algebra generated by the infinitesimal symmetries is greater than 2 and for this reason we present the structures and one dimensional optimal systems of these lie algebras the reduced systems corresponding to the optimal systems are also obtained explicit group invariant solutions are found for particular cases

numerical solutions of coupled burgers equations with time and space fractional derivatives in this paper by introducing the fractional derivative in the sense of caputo the adomian decomposition method is directly extended to study the coupled burgers equations with time and space fractional derivatives as a result the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components the figures show the effectiveness and good accuracy of the proposed method

stochastic volatility models at ρ 1 as second class constrained hamiltonian systems abstract the stochastic volatility models used in the financial world are characterized in the continuous time case by a set of two coupled stochastic differential equations for the underlying asset price s and volatility in addition the correlations of the two brownian movements that drive the stochastic dynamics are measured by the correlation parameter 1 1 this stochastic system is equivalent to the fokker planck equation for the transition probability density of the random variables s and solutions for the transition probability density of the heston stochastic volatility model heston 1993 were explored in dragulescu and yakovenko 2002 where the fundamental quantities such as the transition density itself depend on in such a manner that these are divergent for the extreme limit 1 the same divergent behavior appears in hagan et al 2002 where the probability density of the sabr model was analyzed in an option pricing context the propagator of the bi dimensional black scholes equation was obtained in lemmens et al 2008 in terms of the path integrals and in this case the propagator diverges again for the extreme values 1 this paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum all of them are second class constrained systems for the most extreme correlated limit 1 in this way the stochastic dynamics of the 1 cases are different of the 1 1 case and it cannot be obtained as a continuous limit from the 1 regimen this conclusion is achieved by considering the fokker planck equation or the bi dimensional black scholes equation as a euclidean quantum schrodinger equation then the analysis of the underlying classical mechanics of the quantum model implies that stochastic volatility models at 1 correspond to a constrained system to study the dynamics in an appropriate form dirac s method for constrained systems dirac 1958 1967 must be employed and dirac s analysis reveals that the constraints are second class in order to obtain the transition probability density or the option price correctly one must evaluate the propagator as a constrained hamiltonian path integral henneaux and teitelboim 1992 in a similar way to the high energy gauge theory models in fact for all stochastic volatility models after integrating over momentum variables one obtains an effective euclidean lagrangian path integral over the volatility alone the role of the second class constraints is determining the underlying asset price s completely in terms of volatility so it plays no role in the path integral in order to examine the effect of the constraints on the dynamics for both extreme limits the probability density function is evaluated by using semi classical arguments in an analogous manner to that developed in hagan et al 2002 for the sabr model

on spectral disjointness of powers for rank one transformations and mobius orthogonality abstract we study the spectral disjointness of the powers of a rank one transformation for a large class of rank one constructions including those for which the cutting and stacking parameters are bounded and other examples such as rigid generalized chacon s maps and katok s map we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum our proof involves the existence in the weak closure of u t k k z of sufficiently many analytic functions of the operator u t then we apply these disjointness results to prove sarnak s conjecture for the possibly non uniquely ergodic symbolic models associated to these rank one constructions all sequences realized in these models are orthogonal to the mobius function

dynamic optimization and its relation to classical and quantum constrained systems abstract we study the structure of a simple dynamic optimization problem consisting of one state and one control variable from a physicist s point of view by using an analogy to a physical model we study this system in the classical and quantum frameworks classically the dynamic optimization problem is equivalent to a classical mechanics constrained system so we must use the dirac method to analyze it in a correct way we find that there are two second class constraints in the model one fix the momenta associated with the control variables and the other is a reminder of the optimal control law the dynamic evolution of this constrained system is given by the dirac s bracket of the canonical variables with the hamiltonian this dynamic results to be identical to the unconstrained one given by the pontryagin equations which are the correct classical equations of motion for our physical optimization problem in the same pontryagin scheme by imposing a closed loop strategy the optimality condition for the action gives a consistency relation which is associated to the hamilton jacobi bellman equation of the dynamic programming method a similar result is achieved by quantizing the classical model by setting the wave function x t e i s x t in the quantum schrodinger equation a non linear partial equation is obtained for the s function for the right hand side quantization this is the hamilton jacobi bellman equation when s x t is identified with the optimal value function thus the hamilton jacobi bellman equation in bellman s maximum principle can be interpreted as the quantum approach of the optimization problem

numerical solutions for fractional kdv burgers equation by adomian decomposition method abstract based on the symbolic computation system maple adomian decomposition method developed for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional kdv burgers equation the fractional derivatives are described in the caputo sense according to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional kdv burgers equation with time and space fractional derivatives finally the solutions of our model equation are calculated in the form of convergent series with easily computable components

symmetries and exact solutions of the time fractional harry dym equation with riemann liouville derivative abstract in this paper group analysis of the time fractional harry dym equation with riemann liouville derivative is performed its maximal symmetry group in lie s sense and the corresponding optimal system of subgroups are determined similarity reductions of the equation under study are performed as a result the reduced fractional ordinary differential equations are deduced and some group invariant solutions in explicit form are obtained as well

fractional variational problems from extended exponentially fractional integral in this letter we study the fractional variational problems from extended exponentially fractional integral both the lagrangian and hamiltonian formulations are explored and discussed in some details some interesting consequences are revealed

mixing of all orders of lie groups actions we show that for lie groups whose adjoint representation reflects their topology mixing implies mixing of all orders in particular we prove that mixing actions of a semisimple lie group are mixing of all orders answering a conjecture of b marcus