finite arbitrage times and the volatility smile abstract extending previous work on non equilibrium option pricing theory eur phys j 14 2000 383 394 a mean field approach is developed to understand the curvature of implied by black scholes bs volatility surfaces curves as a function of moneyness strike price divided by price the previously developed hypothesis of a finite arbitrage time during which fluctuations around the equilibrium state absence of arbitrage are allowed to occur is generalized as follows instead of a unique arbitrage time independant of moneyness a distribution of arbitrage times will be assumed where the mean arbitrage time will be a function of moneyness this hypothesis is motivated by the fact that the trading volume is the largest for at the money options assuming now the arbitrage time to be inversely proportional to trading volume naturally leads to our generalized hypothesis on the mean arbitrage time consequences on plain vanilla option prices will be studied

martingale measures and hedging for discrete time financial markets the price of stocks is modelled by a discrete time square integrable vector valued process x no further boundedness condition on x is imposed contingent claims h are described by square integrable random variables one looks for values v of the initial wealth v that allow for super hedging h by some portfolio plan in several cases the smallest value v is known to coincide with the maximal expectation of h under equivalent martingale measures here within an l2 framework another sufficient condition is provided which can be looked upon as a stronger form of the no arbitrage condition the mathematical tool and one of the main contributions is an optional decomposition theorem for a process which is a supermartingale under any equivalent martingale measure the upper price process for a contingent claim is shown to be a typical example for such a process moreover it is shown that in a markovian model one can restrict attention to markovian portfolio plans and to markovian martingale measures

markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

follmer schweizer decomposition and mean variance hedging for general claims let x be an r d valued special semimartingale on a probability space f f t 0 t t p with decomposition x x 0 m a and the space of all predictable x integrable processes such that dx is in the space s 2 of semimartingales if h is a random variable in l 2 we prove under additional assumptions on the process x that h can be written as the sum of an f 0 measurable random variable h 0 a stochastic integral of x and a martingale part orthogonal to m moreover this decomposition is unique and the function mapping h with its decomposition is continuous with respect to the l 2 norm finally we deduce from this continuity that the subspace of l 2 generated by dx where is closed in l 2 and we give some applications of this result to financial mathematics

harmonic analysis of stochastic equations and backward stochastic differential equations the bmo martingale theory is extensively used to study nonlinear multi dimensional stochastic equations in mathcal r p p in 1 infty and backward stochastic differential equations bsdes in mathcal r p times mathcal h p p in 1 infty and in mathcal r infty times overline l infty rm bmo with the coefficients being allowed to be unbounded in particular the probabilistic version of fefferman s inequality plays a crucial role in the development of our theory which seems to be new several new results are consequently obtained the particular multi dimensional linear cases for stochastic differential equations sdes and bsdes are separately investigated and the existence and uniqueness of a solution is connected to the property that the elementary solutions matrix for the associated homogeneous sde satisfies the reverse holder inequality for some suitable exponent p 1 finally some relations are established between kazamaki s quadratic critical exponent b m of a bmo martingale m and the spectral radius of the stochastic integral operator with respect to m which lead to a characterization of kazamaki s quadratic critical exponent of bmo martingales being infinite

poisson lie group of pseudodifferential symbols we introduce a lie bialgebra structure on the central extension of the lie algebra of differential operators with scalar or matrix coefficients on the line and on the circle this defines a poisson lie structure on the dual group of pseudodifferential symbols of an arbitrary real or complex order we show that the usual second benney gl n kdv or gl n adler gelfand dickey and kp poisson structures are naturally realized as restrictions of this poisson structure to submanifolds of this universal poisson lie group moreover the reduced sl n versions of these manifolds orw n algebras in physical terminology can be viewed as certain subspaces of the quotient of this poisson lie group by the dressing action of the group of functions on the circle or as a result of a poisson reduction finally we define an infinite set of commuting functions on the poisson lie group that give the standard families of hamiltonians when restricted to the submanifolds mentioned above the poisson structure and hamiltonians on the whole group interpolate between the poisson structures and hamiltonians of benney kp and kdv flows we also discuss the geometrical meaning ofw as a limit of poisson algebraswe as e 0

variance optimal hedging in discrete time we solve the problem of approximating in 2 a given random variable h by stochastic integrals gt of a given discrete time process x we interpret h as a contingent claim to be paid out at time t x as the price evolution of some risky asset in a financial market and g as the cumulative gains from trade using the hedging strategy as an application we determine the variance optimal strategy which minimizes the variance of the net loss h gt over all strategies

mean variance hedging under partial information we consider the mean variance hedging problem under partial information the underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available we show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term which is formulated in terms of observable processes we prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation

utility based indifference pricing in regime switching models article deposited according to the policy found on the elsevier website http www elsevier com wps find authorsview authors postingpolicy june 28 2012

random walks on ultrametric spaces low temperature patterns the authors present the numerically determined distribution of the number of sites visited by thermally activated walkers on ultrametric spaces which mimic the energetic disorder found in amorphous materials at high temperatures the results resemble those found for sierpinski type fractals at low temperatures the distributions are highly structured showing discontinuities which increase with the number of steps in contrast to an intuitive picture that would lead to smoothing they apply the results to the energy transfer in disordered systems and study the decay due to trapping

variance of random walks on cayley trees application to the trapping problem the authors investigate the trapping of particles on regularly multifurcating cayley trees by using random walk methods in order to find a good approximation for the survival probability of the walker they use a cumulant expansion and obtain an exact expression for the variance of the range of nearest neighbour random walks the expressions are corroborated to high accuracy by simulation calculations the method may turn out to be useful for more complex problems such as walks on ultrametric spaces

growth optimal investment and pricing of derivatives abstract we introduce a criterion how to price derivatives in incomplete markets based on the theory of growth optimal strategy in repeated multiplicative games we present reasons why these growth optimal strategies should be particularly relevant to the problem of pricing derivatives under the assumptions of no trading costs and no restrictions on lending we find an appropriate equivalent martingale measure that prices the underlying and the derivative security we compare our result with other alternative pricing procedures in the literature and discuss the limits of validity of the lognormal approximation we also generalize the pricing method to a market with correlated stocks the expected estimation error of the optimal investment fraction is derived in a closed form and its validity is checked with a small scale empirical test

the moyal bracket and the dispersionless limit of the kp hierarchy a new lax equation is introduced for the kp hierarchy which avoids the use of pseudo differential operators as used in the sate approach this lax equation is closer to that used in the study of the dispersionless kp hierarchy and is obtained by replacing the poisson bracket with the moyal bracket the dispersionless limit under which the moyal bracket collapses to the poisson bracket is particularly simple

the hamiltonian structures associated with a generalized lax operator it is shown that with every lax operator which is a pseudodifferential operator of nonzero leading order is associated a kp hierarchy for each such operator we construct the second gelfand dikii bracket associated with the lax equation and show that it defines a hamiltonian structure when the leading order is positive the corresponding compatible first hamiltonian structure which turns out in general to be different from the naive first gelfand dikii bracket is derived the corresponding hamiltonian structures for the constrained lax operator where the next to leading order term vanishes or has a constant coefficient is discussed

a semimartingale backward equation and the variance optimal martingale measure under general information flow we consider a financial market model where the dynamics of asset prices are given by an rd valued continuous semimartingale and the information flow is right continuous using the dynamic programming approach we express the variance optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation we consider two extreme cases when this equation admits an explicit solution in particular we give necessary and sufficient conditions in order that the variance optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case

scr e martingales and their applications in mathematical finance after introducing a new concept the notion of e martingale we extend the well known doob inequality for 1 p and the burkholder davis gundy inequalities for p 2 to e martingales by means of these inequalities we give sufficient conditions for the closedness of a space of stochastic integrals with respect to a fixed r d valued semimartingale a question which arises naturally in the applications to financial mathematics we also provide a necessary and sufficient condition for the existence and uniqueness of the follmer schweizer decomposition

utility deviation risk portfolio selection we here provide a comprehensive study of the utility deviation risk portfolio selection problem by considering the first order condition for the corresponding objective function we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem called the nonlinear moment problem we then illustrate the application of this general necessity result by revisiting the nonexistence of the optimal solution for the mean semivariance problem second we establish an alternative version of the verification theorem serving as the sufficient condition that the solution which satisfies another mild condition different from that for necessity of the nonlinear moment problem is the optimal terminal wealth of the original utility deviation risk portfolio selection problem we then apply this general sufficiency result to revisit the various well posed mean risk problems already known in the literature and to also establish the ex

local risk minimization under transaction costs we propose a new approach to the pricing and hedging of contingent claims under transaction costs in a general incomplete market in discrete time under the assumptions of a bounded mean variance tradeoff substantial risk and a nondegeneracy condition on the conditional variances of asset returns we prove the existence of a locally risk minimizing strategy inclusive of transaction costs for every square integrable contingent claim then we show that local risk minimization is robust under the inclusi on of transaction costs the preceding strategy which is locally risk minimizing inclusive of transaction costs in a model with bid ask spreads on the underlying asset is also locally risk minimizing without transaction costs in a fictitious model which i s frictionless and where the fictitious asset price lies between the bid and ask price processes of the original model in particular our results apply to any nondegenerate model with a finite state space if the transaction cost parameter is sufficiently small

robustness of quadratic hedging strategies in finance via fourier transforms in this paper we investigate the consequences of the choice of the model to partial hedging in incomplete markets in finance in fact we consider two models for the stock price process the first model is a geometric levy process in which the small jumps might have infinite activity the second model is a geometric levy process where the small jumps are truncated or replaced by a brownian motion which is appropriately scaled to prove the robustness of the quadratic hedging strategies we use pricing and hedging formulas based on fourier transform techniques we compute convergence rates and motivate the applicability of our results with examples

on diff s1 covariantization of pseudodifferential operator a study of diff s1 covariant properties of a pseudodifferential operator of integer degree is presented first it is shown that the action of diff s1 defines a hamiltonian flow defined by the second gelfand dickey bracket if and only if the pseudodifferential operator transforms covariantly second the covariant form of a pseudodifferential operator of degree n 0 1 is constructed by exploiting the inverse of the covariant derivative this in particular implies the existence of a primary basis for wkp n n 0 1

generalization of the algebra of the matrix pseudodifferential operators we introduce the new nonhomogeneous graded matrix algebra matn r over the ring r of the pseudodifferential operators of the arbitrary orders this algebra forms the generalization of the algebra of the standard matrix pseudodifferential operators we show that the matn r algebra admits the nontrivial central extension the case n 2 is considered in some detail as the example we consider the mat2 w1 mat2 virasoro and mat2 sl 2 c algebras

a stochastic control approach to the pricing of options we construct a stochastic control model of a portfolio in which the investor may invest in stock call options on the stock or risk free bonds he may also borrow funds we assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power 0 1 this includes both the risk averse and risk neutral cases we allow the borrowing interest rate r to be distinct from the lending interest rate r we make two fundamental assumptions i the investor cannot achieve infinite expected utility of wealth in finite time stable markets and ii the market will set the option price to minimize the investor s maximal expected utility of wealth i e options are priced using a minimax strategy the market is stable only when r r we prove in the case r r and 1 that the minimax european call option price ft s is given as the solution of the black scholes equation ft i i 2s2fss i s fs i f 0 where i r when r i 2 1 i r when i 2 1 r and i i 2 1 if r i 2 1 r the parameter is the stock return drift and i is the stock return diffusion if r r then the call option price must by stable markets always be the black scholes price with rate r in the risk neutral 1 case we must have r r for a stable market and the option price given by the black scholes equation but with i explicit solutions are presented and the optimal portfolios and returns are also derived comparable results are obtained for american call options which can be exercised at any time up to and including the expiration date in this case the option pricing function is shown to be the solution of an obstacle problem

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

ultradiffusion the relaxation of hierarchical systems the authors show that the dynamics of relaxation in hierarchical structures leads to an anomalous decay process which they term ultradiffusion using renormalisation group techniques they explicitly calculate the behaviour of the autocorrelation function in a one dimensional hierarchical system and show its universal power law decay as a function of temperature the authors also demonstrate analytically the emergence of a hierarchy of time scales this phenomenon is relevant to systems ranging from macromolecules to computing structures

maximization of nonconcave utility functions in discrete time financial market models this paper investigates the problem of maximizing expected terminal utility in a generically incomplete discrete time financial market model with finite time horizon by contrast to the standard setting a possibly nonconcave utility function u is considered with domain of definition equal to the whole real line simple conditions are presented that guarantee the existence of an optimal strategy for the problem in particular the asymptotic elasticity of u plays a decisive role existence can be shown when it is strictly greater at than at

forward backward semimartingale systems for utility maximization we consider the problem of maximizing the expected utility of terminal wealth with a terminal random liability when the underlying asset price process is a continuous semimartingale the optimal strategy is characterized in terms of a forward backward semimartingale system of equations the results cover the cases of exponential logarithmic and power utilities which we analyze as illustrative examples

approximating random variables by stochastic integrals let x be a semimartingale and the space of all predictable x integrable processes such that dx is in the space 2 of semimartingales we consider the problem of approximating a given random variable h l 2 by a stochastic integral 0 t s dx s with respect to the l 2 norm if x is special and has the form x x 0 m d we construct a solution in feedback form under the assumptions that 2 d is deterministic and that h admits a strong f s decomposition into a constant a stochastic integral of x and a martingale part orthogonal to m we provide sufficient conditions for the existence of such a decomposition and we give several applications to quadratic optimization problems arising in financial mathematics

on the minimal entropy martingale measure let x be a locally bounded semimartingale using the theory of bmo martingales we give a sufficient criterion for a martingale measure for x to minimize relative entropy among all martingale measures this is applied to prove convergence of the q optimal martingale measure to the minimal entropy martingale measure in entropy for q 1 under the assumption that x is continuous and that the density process of some equivalent martingale measure satisfies a reverse llogl inequality

dynamic programming and pricing of contingent claims in an incomplete market the problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market this paper studies the same problem in an incomplete market when the market is incomplete prices cannot be derived from the absence of arbitrage since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities in this situation there is a price range for the actual market price of the contingent claim the maximum and minimum prices are studied using stochastic control methods the main result of this work is the determination that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities a similar result is obtained for the minimum price which corresponds to the purchase price

random walks on ultrametric spaces mean and variance of the range the authors study thermally activated hopping processes on regularly multifurcating ultrametric spaces ums and investigate the first two moments of the range rn where n is the number of steps paralleling the recent findings for cayley trees the knowledge of exact generating functions for these moments permits to draw conclusions about their behaviour for large n for the first moment the mean number of distinct sites visited sn they derive analytically the cross over from a linear behaviour in the transient regime gamma 1 to a sublinear increase sn approximately ngamma in the recurrent regime gamma 1 here as usual for ums the parameter gamma depends linearly on the temperature the second moment of rn the variance vn displays different behaviour corresponding to strongly transient gamma 2 not strongly transient 1 gamma 2 and recurrent gamma 1 random walks all results are verified by comparison to computer data obtained through simulations and through numerical series expansions

optimal consumption from investment and random endowment in incomplete semimartingale markets we consider the problem of maximizing expected utility from consumption in a con strained incomplete semimartingale market with a random endowment process and establish a general existence and uniqueness result using techniques from convex duality the notion of as ymptotic elasticity of kramkov and schachermayer is extended to the time dependent case by imposing no smoothness requirements on the utility function in the temporal argument we can treat both pure consumption and combined consumption terminal wealth problems in a com mon framework to make the duality approach possible we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of l 1 to its topological bidual l 1 a space of finitely additive measures as an application we treat a constrained ito process market model as well as a totally incomplete model

hedged monte carlo low variance derivative pricing with objective probabilities abstract we propose a new hedged monte carlo hmc method to price financial derivatives which allows to determine simultaneously the optimal hedge the inclusion of the optimal hedging strategy allows one to reduce the financial risk associated with option trading and for the very same reason reduces considerably the variance of our hmc scheme as compared to previous methods the explicit accounting of the hedging cost naturally converts the objective probability into the risk neutral one this allows a consistent use of purely historical time series to price derivatives and obtain their residual risk the method can be used to price a large class of exotic options including those with path dependent and early exercise features

numerical analysis of american option pricing in a jump diffusion model we discuss pricing formulae for american options in merton s jump diffusion model with the help of variational inequalities we derive some regularity properties of price functions using the finite difference method a discretization scheme is presented and a convergence theorem for the first order derivatives is proved numerical methods and results are also discussed

existence and uniqueness of the universal w algebra a precise definition of the universal classical w algebra for the wn series is given and its existence and uniqueness are proved the main observation is that there is a natural reduction from wn to wn 1 that allows us to define the universal w algebra as an inverse limit this universal w algebra is in a sense the smallest w algebra from which all wn can be obtained by reduction these results extend to other w algebras obtained by reducing the gel fand dickey brackets as well as to w superalgebras obtained from the supersymmetric gel fand dickey brackets

minimal martingale measure pricing and hedging in a pure jump model under restricted information abstract financial asset price movements are considered in a model with partial information the risky assets price is modeled as a marked point process the underlying event arrival process which is again a marked point process is assumed to be unobserved by the market agents taking into account the presence of catastrophic events the possibility of common jump times between the risky assets price process and the arrival process is allowed the problem of characterizing the equivalent martingale measures is investigated and in particular the minimal martingale measure is studied the hedging and the pricing problem of a european contingent claim are discussed the arbitrage free price is identified with the conditional expectation with respect to the observations under the minimal martingale measure

utility maximization with discretionary stopping utility maximization problems of mixed optimal stopping control type are considered which can be solved by reduction to a family of related pure optimal stopping problems sufficient conditions for the existence of optimal strategies are provided in the context of continuous time ito process models for complete markets the mathematical tools used are those of optimal stopping theory continuous time martingales convex analysis and duality theory several examples are solved explicitly including one which demonstrates that optimal strategies need not always exist

mean variance hedging when there are jumps in this paper we consider the problem of mean variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by brownian motion and doubly stochastic poisson processes this problem is formulated as a stochastic control problem and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations the results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modeled by the arrivals of poisson processes with stochastic intensities applications to the problem of hedging default risk are also discussed

equilibrium pricing in incomplete markets under translation invariant preferences we propose a general discrete time framework for deriving equilibrium prices of financial securities it allows for heterogeneous agents unspanned random endowments and convex trading constraints we give a dual characterization of equilibria and provide general results on their existence and uniqueness in the special case where all agents have preferences of the same type and in equilibrium all random endowments are replicable by trading in the financial market we show that a one fund theorem holds and give an explicit expression for the equilibrium pricing kernel

on the structure of general mean variance hedging strategies we provide a new characterization of mean variance hedging strategies in a general semimartingale market the key point is the introduction of a new probability measure p which turns the dynamic asset allocation problem into a myopic one the minimal martingale measure relative to p coincides with the variance optimal martingale measure relative to the original probability measure p

model risk and discretisation of locally risk minimising strategies we consider two models for the price process a time continuous jump diffusion and a time discretisation of it then we study the robustness of the related locally risk minimising strategy to this model choice where we focus mainly on hedging asian and spread options using the discretisation scheme and the convergence results on backward stochastic differential equations as studied in khedher and vanmaele 2016 we show that the discrete time locally risk minimising strategies converge to the corresponding continuous time strategies in an l 2 sense we present different numerical examples to illustrate our results

subordination on ultrametric spaces using the formalism of continuous time random walks ctrw the authors discuss particle diffusion on ultrametric spaces ums for broad waiting time distributions psi t approximately t alpha 1 the temporal alpha and the energetic gamma parameters combine multiplicatively subordinate i e s t the mean number of ums sites visited follows a s t approximately talpha gamma law this mirrors previous findings for ctrw on fractals

approximation pricing and the variance optimal martingale measure let x be a semimartingale and let be the space of all predictable x integrable processes v such that so dx is in the space s 2 of semi martingales we consider the problem of approximating a given random variable h l 2 p by the sum of a constant c and a stochastic integral s 0 t v s dx s with respect to the l 2 p norm this problem comes from financial mathematics where the optimal constant v 0 can be interpreted as an approximation price for the contingent claim h an elementary computation yields v 0 as the expectation of h under the variance optimal signed martingale measure p and this leads us to study p in more detail in the case of finite discrete time we explicitly construct p by backward recursion and we show that p is typically not a probability but only a signed measure in a continuous time framework the situation becomes rather different we prove that p is nonnegative if x has continuous paths and satisfies a very mild no arbitrage condition as an application we show how to obtain the optimal integrand in feedback form with the help of a backward stochastic differential equation

p adic models of ultrametric diffusion constrained by hierarchical energy landscapes we demonstrate that the p adic analysis is a natural basis for the construction of a wide variety of models of ultrametric diffusion constrained by hierarchical energy landscapes a general analytical description in terms of the p adic analysis is given for a class of models two exactly solvable examples i e the ultrametric diffusion constrained by the linear energy landscape and the ultrametric diffusion with a reaction sink are considered we show that such models can be applied to both the relaxation in complex systems and the rate processes coupled to rearrangement of the complex surrounding

random hierarchical matrices spectral properties and relation to polymers on disordered trees we study the statistical and dynamic properties of the systems characterized by an ultrametric space of states and translationary non invariant symmetric transition matrices of the parisi type subjected to locally constant randomization using the explicit expression for eigenvalues of such matrices we compute the spectral density for the gaussian distribution of matrix elements we also compute the averaged survival probability sp having sense of the probability to find a system in the initial state by time t using the similarity between the averaged sp for locally constant randomized parisi matrices and the partition function of directed polymers on disordered trees we show that for times t tcr where tcr is some critical time a lacunary structure of the ultrametric space occurs with the probability 1 const t this means that the escape from some bounded areas of the ultrametric space of states is locked and the kinetics is confined in these areas for infinitely long time the explosion of the interest to the statistical properties of ensembles of random matrices around 1950s has been motivated in first turn by a series of physical problems in nuclear physics later the physical applications of the random matrix theory rmt were extended to mesoscopics and statistical physics one can mention recent works 1 2 which deal with physical applications of the eigenvalue statistics of ensembles of random matrices the typical problems of rmt concern the evaluation of the averaged spectral density as well as distributions of level spacings under the supposition that all matrix elements are independent randomly distributed entries which take values in some specific ensemble 3 despite the standard rmt can be applied to a wide range of physical phenomena rmt in its basic form does not cover a particular class of hierarchical complex systems important for glasses real networks and proteins the description of such complex systems deals with the concept of hierarchical organization of energy landscapes 4 5 a complex system is assumed to have a large number of metastable states corresponding to local minima in the potential energy landscape with respect to the transition rates the minima are suggested to be clustered in hierarchically nested basins of minima i e larger basins consist of smaller basins each of these consists of even smaller ones etc the basins of local energy minima are separated by a hierarchically arranged set of barriers large basins are separated by high barriers and smaller basins within each larger one are separated by lower barriers since the transitions between the basins are determined by the passages over the highest barriers separating them the transitions between any two local minima obey the strong triangle inequality for ultrametric distances hence the hierarchy of basins possesses the ultrametric geometry which has the natural visualization by the uniform p adic i e p 1 branching cayley tree

p adic description of characteristic relaxation in complex systems this work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p adic analysis we show that three types of relaxation fitted into the kohlrausch williams watts law the power decay law and the logarithmic decay law are similar random processes inherently these processes are ultrametric and are described by the p adic master equation the physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape we also discuss relations between the relaxation kinetics and the energy landscapes

markowitz s mean variance portfolio selection with regime switching from discrete time models to their continuous time limits we study a discrete time version of markowitz s mean variance portfolio selection problem where the market parameters depend on the market mode regime that jumps among a finite number of states the random regime switching is delineated by a finite state markov chain based on which a discrete time markov modulated portfolio selection model is presented such models either arise from multiperiod portfolio selections or result from numerical solution of continuous time problems the natural connections between discrete time models and their continuous time counterpart are revealed since the markov chain frequently has a large state space to reduce the complexity an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a two time scale problem we prove that the process of interest yields a switching diffusion limit using weak convergence methods next based on the optimal control of the limit process obtained from our recent work we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality

martingale and duality methods for utility maximization in a incomplete market the problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market this paper studies the same problem in an incomplete market containing a bond and a finite number of stocks whose prices are driven by a multidimensional brownian motion process w the coefficients of the bond and stock processes are adapted to the filtration history of w and incompleteness arises when the number of stocks is strictly smaller than the dimension of w it is shown that there is a way to complete the market by introducing additional fictitious stocks so that the optimal portfolio for the thus completed market coincides with the optimal portfolio for the original incomplete market the notion of a least favorable completion is introduced and is shown to be closely related to the existence question for an optimal portfolio in the incomplete market this notion is expounded upon using martingale techniques several equivalent characterizations are provided

a new approach to deformation equations of noncommutative kp hierarchies partly inspired by sato s theory of the kadomtsev petviash vili kp hierarchy we start with a quite general hierarchy of linear ordinary differential equatio ns in a space of matrices and derive from it a matrix riccati hierarchy the latter is then shown to exhibit an underlying weakly nonassociative wna algebra structure from which we can conclude refering to previous work that any solution of the riccati system also solves the potential kp hierarchy in the corresponding matrix algebra we then turn to the case where the components of the matrices are multiplied using a generalized star product associated with the deformation parameters there are additional symmetries flow equations which enlarge the respective kp hierarchy they have a compact formulation in terms of the wna structure we also present a formulation of the kp hierarchy equations themselves as deformation flow equations

a physical interpretation of age dependent master equations abstract physical systems described by fast and slow relaxing variables are considered the underlying microscopic equations are assumed to be markovian considering that the macroscopic state of the system is characterized by the set of slow relaxing variables we show that the age state probability density obeys a system of age dependent master equations with time and age dependent transition rates if the time scales attached to the slow and fast relaxing variables are well differentiated there is a strong correlation between the age of a fluctuation state and the corresponding number of transition events an alternative approach based on haken s slaving principle is developed for time homogeneous microscopic equations the adiabatic elimination of fast relaxing variables leads to a system of age dependent master equations with time independent and age dependent transition rates these equations are equivalent to a generalized master equation with an inhomogeneous term

portfolio selection in the enlarged markovian regime switching market we study a portfolio selection problem in a continuous time markovian regime switching model the market in this model is in general incomplete we adopt a method to complete the market based on an enlargement of the market using a set of geometric markovian jump securities we solve the portfolio selection problem in the enlarged market for a power utility and a logarithmic utility closed form solutions for the optimal portfolio strategies and the value functions are obtained in both cases we also establish the relationship between the optimal portfolio problems in the enlarged market and the original market

on quadratic cost criteria for option hedging consider an option with maturity time t corresponding to a contingent claim h in an incomplete market a fair hedging price for h should take into account an optimal dynamical hedging plan against h let ct be the cumulative cost and t be the set of events of the history up to time t you can choose the plan at time t such that you minimize i e ct 1 ct 2 t ii e ct ct 2 t or iii e ct c0 2 sufficient conditions on the underlying stochastic process in discrete time are provided such that the fair hedging price does not depend on the choice of i ii or iii which fact should increase its acceptability

the algebra of q pseudodifferential symbols and the q wkp n algebra in this paper we continue with the program to explore the topography of the space of w type algebras in the present case the starting point is the work of khesin lyubashenko and roger on the algebra of q deformed pseudodifferential symbols and their associated integrable hierarchies the analysis goes on by studying the associated hamiltonian structures for which compact expressions are found the fundamental poisson brackets yield q deformations of wkp and related w type algebras which in specific cases coincide with the ones constructed by frenkel and reshetikhin the construction underlies a continuous correspondence between the hamiltonian structures of the toda lattice and the kp hierarchies

optimization problems in the theory of continuous trading a unified approach based on stochastic analysis to the problems of option pricing consumption investment and equilibrium in a financial market with asset prices modelled by continuous semi martingales is presented for the first of these problems the valuation of both european and american contingent claims is discussed the former can be exercised only at a specified time t the maturity date whereas the latter can be exercised at any time in 0 t notions and results from the theory of optimal stopping are employed in the treatment of american options a general consurption investment problem is also considered for an agent whose actions cannot affect the market prices and whose intention is to maximize total expected utility of both consumption and terminal wealth under very general conditions on the utility functions of the agent it is shown how to approach the above problem by considering separately the two more elementary ones of maximizing utility from consumption only and of maxim

on l 2 projections on a space of stochastic integrals let x be an ir d valued continuous semimartingale t a fixed time horizon and the space of all ir d valued predictable x integrable processes such that the stochastic integral g d xis a square integrable semimartingale a re cent paper of delbaen monat schachermayer schweizer stricker 1996 gives necessary and sufficient conditions on x for gt to be closed in l 2 p in this paper we describe the structure of the l 2 projection mapping an ft measurable random variable h l 2 p o ngt and provide the resulting integrand h in feedback form this is related to variance optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on x our proofs use the variance optimal martin gale measure p for x and weighted norm inequalities relating p to the original measure p

hedging with a correlated asset solution of a nonlinear pricing pde hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk even if the residual risk is considered diversifiable the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument if the residual risk is not considered diversifiable then this risk can be priced using an actuarial standard deviation principle in infinitesimal time in both cases these models result in the same nonlinear partial differential equation pde a fully implicit monotone discretization method is developed for solution of this pricing pde this method is shown to converge to the viscosity solution certain grid conditions are required to guarantee monotonicity an algorithm is derived which given an initial grid inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied at each timestep the nonlinear discretized algebraic equations are solved using an iterative algorithm which is shown to be globally convergent monte carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option

quadratic hedging and mean variance portfolio selection with random parameters in an incomplete market this paper concerns the problems ofquadratic hedging andpricing andmean variance portfolio selection in anincomplete market setting with continuous trading multiple assets and brownian information in particular we assume throughout that the parameters describing the market model may be random processes we approach these problems from the perspective oflinear quadratic lq optimal control and backward stochastic differential equations bsdes that is we focus on the so calledstochastic riccati equation sre associated with the problem excepting certain special cases solvability of the sre remains an open question our primary theoretical contribution is a proof of existence and uniqueness of solutions of the sre associated with the quadratic hedging and mean variance problems in addition we derive closed form expressions for the optimal portfolios and efficient frontier in terms of the solution of the sre a generalization of themutual fund theorem is also obtained

option pricing with a general marked point process this paper examines the impact of a random number of price changes on the options valuation the model introduces the structure of the general marked point processes mpp this kind of model allows us to take account of more general distributions of interarrival times than usual jump diffusion models in particular stock price variations can be correlated with the times of transactions thus the investors can decide to trade according to the history of market values and the sizes of price variations can also depend on the past times of transactions by using the special decomposition of predictable processes with respect to a marked point process the determination of all risk neutral probabilities is detailed derivative prices are calculated in this context with different basic examples

classical n 2 w superalgebras from superpseudodifferential operators we study the supersymmetric gelfand dickey algebras associated with the superpseudodifferential operators of positive as well as negative leading order we show that upon the usual constraint these algebras contain the n 2 super virasoro algebra as a subalgebra when the leading order is odd the decompositions of the coefficient functions into n 1 primary fields are then obtained by covariantizing the superpseudodifferential operators we discuss the problem of identifying n 2 supermultiplets and work out a couple of supermultiplets by explicit computations

a one parameter family of hamiltonian structures for the kp hierarchy and a continuous deformation of the nonlinear wkp algebra the kp hierarchy is hamiltonian relative to a one parameter family of poisson structures obtained from a generalized adler map in the space of formal pseudodifferential symbols with noninteger powers the resulting w algebra is a one parameter deformation of wkp admitting a central extension for generic values of the parameter reducing naturally to w n for special values of the parameter and contracting to the centrally extended w1 w and further truncations in the classical limit all algebras in the one parameter family are equivalent and isomorphic towkp the reduction induced by setting the spin one field to zero yields a one parameter deformation of hat w infty which contracts to a new nonlinear algebra of the w type

option pricing with regime switching by trinomial tree method we present a fast and simple tree model to price simple and exotic options in markov regime switching model mrsm with multi regime we modify the trinomial tree model of boyle 1986 12 by controlling the risk neutral probability measure in different regime states to ensure that the tree model can accommodate the data of all different regimes at the same time preserving its combining tree structure in mrsm the market might not be complete therefore we provide some ideas and discussions on managing the regime switching risk in support of our results

minimal variance hedging of options with student t underlying i explicitly work out closed form solutions for the optimal hedging strategies in the sense of bouchaud and sornette in the case of european call options where the underlying is modeled by unbiased iid additive returns with the student t distributions the results may serve as illustrative examples for option pricing in the presence of fat tails

mean variance hedging and stochastic control beyond the brownian setting we show for continuous semimartingales in a general filtration how the mean variance hedging problem can be treated as a linear quadratic stochastic control problem the adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process and we give necessary and sufficient conditions for the solvability of these generalized stochastic riccati equations motivated from mathematical finance this paper takes a first step toward linear quadratic stochastic control in more general than brownian settings