approximation of cvar minimization for hedging under exponential levy models in this paper we study the hedging problem based on the cvar in incomplete markets as the superhedging is quite expensive in terms of initial capital we construct a self financing strategy that minimizes the cvar of hedging risk under a budget constraint on the initial capital in incomplete markets no explicit solution can be provided to approximate the problem we apply the neyman pearson lemma approach with a specific equivalent martingale measure afterwards we explicit the solution for call options hedging under the exponential levy class of price models this approach leads to an efficient and easy to implement method using the fast fourier transform we illustrate numerical results for the merton model

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

brief paper the asymptotic behavior of stochastically perturbed di sir epidemic models with saturated incidences in this paper we consider a class of di sir epidemic models with saturated incidences and parameter perturbation we investigate the asymptotic behavior according to the perturbation and the reproductive number r 0 when the perturbation is large the infective in every group decays exponentially to zero while the susceptible converges weakly to stationary distribution regardless of the magnitude of r 0 when the perturbation is small we get the same exponential stability and weak convergence if r 0 1 and we use a new class of stochastic lyapunov functions to obtain the ergodicity and positive recurrence if r 0 1

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

k fuzzy ideals in semirings abstract we introduce the k fuzzy ideal of a semiring also we construct an extension of a fuzzy ideal in a k semiring and study the quotient structure of a k semiring by a fuzzy ideal

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

the characterizations of h hemiregular hemirings and h intra hemiregular hemirings in this paper the concepts of fuzzy h bi ideals and fuzzy h quasi ideals of a hemiring are introduced and related properties are investigated the notion of h intra hemiregularity of a hemiring which is a generalization of the notion of intra regularity of a ring is provided some characterizations of h hemiregular hemirings h intra hemiregular hemirings and hemirings that are both h hemiregular and h intra hemiregular are derived in terms of fuzzy left fuzzy right h ideals fuzzy h bi ideals and fuzzy h quasi ideals

pricing and hedging european style options in levy based stochastic volatility models considering the leverage effect abstract this paper develops a semi analytical exact solution for the discrete time variance optimal hedging strategy based on two dimensional fourier cosine series expansions this approach is novel and is more efficient than those previously suggested in the literature numerical studies for different models also suggest that the algorithm has better accuracy and stability for the pricing and hedging problems a levy based stochastic volatility model considering the leverage effect is also proposed we have employed data on apple stock prices and the corresponding option contracts to test this model to evaluate its ability to perform pricing and hedging in summary our model design is relatively reliable in the context of the known empirical patterns such as the existence of jumps the volatility smile and the negative correlation between asset returns and volatilities

exponential stock models driven by tempered stable processes abstract we investigate exponential stock models driven by tempered stable processes which constitute a rich family of purely discontinuous levy processes with a view of option pricing we provide a systematic analysis of the existence of equivalent martingale measures under which the model remains analytically tractable this includes the existence of esscher martingale measures and martingale measures having minimal distance to the physical probability measure moreover we provide pricing formulae for european call options and perform a case study

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

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

on an investment consumption model with transaction costs this paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log normal diffusions we suppose that transactions between the assets incur a cost proportional to the size of the transaction the problem is to maximize the total utility of consumption dynamic programming leads to a variational inequality for the value function existence and uniqueness of a viscosity solution are proved the variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods numerical results are displayed for n 1 and n 2

stability in distribution of neutral stochastic functional differential equations with markovian switching abstract stability in distribution of stochastic differential equations with markovian switching and stochastic differential delay equations with markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems there are several papers which study this stability for stochastic differential equations with markovian switching and stochastic differential delay equations with markovian switching technically in our paper we are concerned with the general neutral stochastic functional differential equations with markovian switching and we derive the sufficient conditions for stability in distribution at the end of our paper one example is established to illustrate the theory of our work

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

admissible strategies in semimartingale portfolio selection the choice of admissible trading strategies in mathematical modeling of financial markets is a delicate issue going back to harrison and kreps j econom theory 20 1979 pp 381 408 in the context of optimal portfolio selection with expected utility preferences this question has been the focus of considerable attention over the last 20 years we propose a novel notion of admissibility that has many pleasant features admissibility is characterized purely under the objective measure p each admissible strategy can be approximated by simple strategies using a finite number of trading dates the wealth of any admissible strategy is a supermartingale under all pricing measures local boundedness of the price process is not required neither strict monotonicity strict concavity nor differentiability of the utility function is necessary the definition encompasses both the classical mean variance preferences and the monotone expected utility for utility functions finite on mathbb r our class represents a minimal set containing simple strategies which also contains the optimizer under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function

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

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

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

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

generalized fuzzy h bi ideals and h quasi ideals of hemirings a new kind of generalized fuzzy h ideals of a hemiring namely the set membership variant set membership variantlogical orq fuzzy h bi ideal resp h quasi ideal is studied and the relationships between these generalized fuzzy h ideals are described some characterization theorems of prime and semiprime set membership variant set membership variantlogical orq fuzzy h bi ideals resp h quasi ideals of a hemiring are also given in particular we show that the h hemiregular hemirings and h intra hemiregular hemirings can be described by using some of their generalized fuzzy h ideals finally the implication based fuzzy h bi ideals resp h quasi ideals of a hemiring are considered

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

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

stochastic exponentials and logarithms on stochastic intervals a survey abstract stochastic exponentials are defined for semimartingales on stochastic intervals and stochastic logarithms are defined for semimartingales up to the first time the semimartingale hits zero continuously in the case of nonnegative local supermartingales these two stochastic transformations are inverse to each other the reciprocal of a stochastic exponential on a stochastic interval is again a stochastic exponential on a stochastic interval

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

risk aversion asymptotics for power utility maximization we consider the economic problem of optimal consumption and investment with power utility we study the optimal strategy as the relative risk aversion tends to infinity or to one the convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models the limits are related to exponential and logarithmic utility to derive these results we combine approaches from optimal control convex analysis and backward stochastic differential equations bsdes

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

stability of degenerate diffusions with state dependent switching this paper deals with the ergodic properties of hybrid systems modelled by diffusion processes with state dependent switching we investigate the sufficient conditions expressed in terms of the parameters of the underlying process which would ensure the existence of a unique invariant probability and stability in distribution of the flow it turns out that the conditions would depend on certain averaging mechanisms over the states of the discrete component of the hybrid system

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

characterizations of hemirings by νq k fuzzy ideals in this paper we define q k fuzzy h sub hemirings q k fuzzy h bi ideals q k fuzzy h quasi ideals we characterize h hemiregular and h intra hemiregular hemirings by the properties of their q k fuzzy h ideals q k fuzzy h bi idealsand q k fuzzy h quasi ideals some of these characterizations are generalizations of the characterizations given in 36 x ma j zhan generalized fuzzyh bi ideals and h quasi ideals of hemirings inform sci 179 2009 1249 1268

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

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

discretisation of fbsdes driven by cadlag martingales abstract we study the discretisation of forward backward stochastic differential equations fbsdes driven by cadlag martingales we prove that under certain conditions imposed on the parameters of the fbsde the time discrete scheme we consider converges to the time continuous equation in the l 2 sense moreover we show that the l 2 norm of the error is of the order of the time step

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

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

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

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

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

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

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

stability in distribution for a class of singular diffusions a verifiable criterion is derived for the stability in distribution of singular diffusions that is for the weak convergence of the transition probability p t x dy as t rightarrow infty to a unique invariant probability for this we establish the following i tightness of p t x dy t geq 0 and ii asymptotic flatness of the stochastic flow when specialized to highly nonradial nonsingular diffusions the results here are often applicable where has minskii s well known criterion fails when applied to traps a sufficient condition for stochastic stability of nonlinear diffusions is derived which supplements has minskii s result for linear diffusions we also answer a question raised by l stettner originally posed to him by h j kushner is the diffusion stable in distribution if the drift is bx where b is a stable matrix and sigma cdot is lipschitzian sigma underline 0 neq 0 if not what additional conditions must be imposed

stability of a random diffusion with linear drift we consider a linear system with markovian switching which is perturbed by gaussian type noise if the linear system is mean square stable then we show that under certain conditions the perturbed system is also stable we also show that under certain conditions the linear system with markovian switching can be stabilized by such noisy perturbation

stability for a random evolution equation with gaussian perturbation in this paper we consider a random evolution equation which is perturbed by gaussian type noise and show how to construct its coupled solutions on the basis of the coupling results we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation in addition we also prove the feller continuity and the existence and uniqueness of invariant measures of the solutions

fuzzy k ideals of semirings the aim of this paper is to introduce the notions of fuzzy k ideals prime fuzzy k ideals and semiregularity of semirings and to characterize prime fuzzy k ideals semiregular semirings noetherian semirings and artinian semirings by fuzzy k ideals

on fuzzy h ideals in hemirings the fuzzy setting of a left h ideal in a hemiring is constructed and basic properties are investigated using a collection of left h ideals of a hemiring s fuzzy left h ideals of s are established the notion of a finite valued fuzzy left h ideal is introduced and its characterization is given fuzzy relations on a hemiring s are discussed

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

pricing options in jump diffusion models an extrapolation approach we propose a new computational method for the valuation of options in jump diffusion models the option value function for european and barrier options satisfies a partial integrodifferential equation pide this pide is commonly integrated in time by implicit explicit imex time discretization schemes where the differential diffusion term is treated implicitly while the integral jump term is treated explicitly in particular the popular imex euler scheme is first order accurate in time second order accuracy in time can be achieved by using the imex midpoint scheme in contrast to the above approaches we propose a new high order time discretization scheme for the pide based on the extrapolation approach to the solution of odes that also treats the diffusion term implicitly and the jump term explicitly the scheme is simple to implement can be added to any pide solver based on the imex euler scheme and is remarkably fast and accurate we demonstrate our approach on the examples of merton s and kou s jump diffusion models the diffusion extended variance gamma model as well as the two dimensional duffie pan singleton model with correlated and contemporaneous jumps in the stock price and its volatility by way of example pricing a one year double barrier option in kou s jump diffusion model our scheme attains accuracy of 10 5 in 72 time steps in 0 05 seconds in contrast it takes the first order imex euler scheme more than 1 3 million time steps in 873 seconds and the second order imex midpoint scheme 768 time steps in 0 49 seconds to attain the same accuracy our scheme is also well suited for bermudan options combining simplicity of implementation and remarkable gains in computational efficiency we expect this method to be very attractive to financial engineering modelers

characterizations of h intra and h quasi hemiregular hemirings the concepts of c email protected q d fuzzy h bi h quasi ideals of hemirings are introduced some new characterization theorems of these kinds of fuzzy h ideals are also given in particular some characterizations of the h intra hemiregular and h quasi hemiregular hemirings are investigated by these kinds of fuzzy h ideals

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

characterizations of hemirings by their h ideals in this paper we characterize hemirings in which all h ideals or all fuzzy h ideals are idempotent it is proved among other results that every h ideal of a hemiring r is idempotent if and only if the lattice of fuzzy h ideals of r is distributive under the sum and h intrinsic product of fuzzy h ideals or equivalently if and only if each fuzzy h ideal of r is intersection of those prime fuzzy h ideals of r which contain it we also define two types of prime fuzzy h ideals of r and prove that a non constant h ideal of r is prime in the second sense if and only if each of its proper level set is a prime h ideal of r

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

optimal control on the l infty norm of a diffusion process stochastic control problems are considered where the cost to be minimized is either a running maximum of the state variable or more generally a running maximum of a function of the state variable and the control in both cases it is proved that the value function which must be defined on an augmented state space to take care of the non markovian feature of the running maximum is the unique viscosity solution of the associated bellman equation which turns out to be in the second case a variational inequality with an oblique derivative boundary condition most of this work consists of proving the convergence of lp approximations and this is done by purely partial differential equation pde methods

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

numerical methods for an optimal investment consumption model this paper examines some numerical techniques for an investment consumption problem considered by fleming and zariphopoulou the value function v x satisfies the differential equation of dynamic programming for x 0 special monotonicity and concavity features of the problem allow us to prove convergence not only of discrete approximations to v x but of the corresponding discrete approximations to optimal investment and consumption policies

stability in distribution of stochastic functional differential equations abstract in this paper we investigate the stability in distribution for a class of stochastic functional differential equations sfdes which include stochastic differential delay equations sddes although stability in distribution has been studied by several authors recently there is so far no stability in distribution criterion on sfdes where the terms involved the delay components are highly nonlinear not bounded by linear functions in this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear sfdes two examples will be given to illustrate our new results we also explain the reason why the existing stability in distribution criteria are not applicable by these two examples

optional decomposition of supermartingales and hedging contingent claims in incomplete security markets letm x be the family of all equivalent local martingale measuresq for some locally boundedd dimensional processx andv be a positive process the main result of the paper theorem 2 1 states that the processv is a supermartingale whateverq m x if and only if this process admits the following decomposition v t whereh is an integrand forx andc is an adapted increasing process we call such a representationoptional because in contrast to the doob meyer decomposition it generally exists only with an adapted optional processc we apply this decomposition to the problem of hedging european and american style contingent claims in the setting ofincomplete security markets

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

the ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence abstract in this paper we include stochastic perturbations into sir and seir epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number r 0 the long time behavior of the two stochastic systems is studied mainly we utilize stochastic lyapunov functions to show under some conditions the solution has the ergodic property as r 0 1 while exponential stability as r 0 1 at last we make simulations to conform our analytical results

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

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

fuzzy h ideals of hemirings a characterization of an h hemiregular hemiring in terms of a fuzzy h ideal is provided some properties of prime fuzzy h ideals of h hemiregular hemirings are investigated it is proved that a fuzzy subset z of a hemiring s is a prime fuzzy left right h ideal of s if and only if z is two valued z 0 1 and the set of all x in s such that z x 1 is a prime left right h ideal of s finally the similar properties for maximal fuzzy left right h ideals of hemirings are considered

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