a numerical scheme for a singular control problem investment consumption under proportional transaction costs abstract this paper concerns the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional transaction costs we consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption the problem is mainly governed by a time dependent hamilton jacobi bellman equation with gradient constraints we propose a numerical method which is composed of monte carlo simulation to take advantage of the high dimensional properties and finite difference method to approximate the gradients of the value function numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in dai et al 2009 and chen and dai 2013

small groups of finite morley rank with involutions by analogy with thompson s classification of nonsolvable finite n groups we classify groups of finite morley rank with solvable local subgroups of even and mixed type we also consider several aspects reminiscent of small groups of finite morley rank of odd type

a diffusion model for optimal portfolio selection in the presence of brokerage fees we consider a financial market model with two assets one has deterministic rate of growth while the rate of growth of the second asset is governed by a brownian motion with drift we can shift money from one asset to another however there are losses of money brokerage fees involved in shifting money from the risky to the nonrisky asset we want to maximize the expected rate of growth of funds it is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort

the stochastic maximum principle in singular optimal control with recursive utilities abstract in this paper we consider stochastic recursive optimal control problem in which the control variable has two components with the first absolutely continuous and the second singular the control domain of the first component needs not to be convex by using a spike variation on the absolutely continuous part of the control and a convex perturbation on the singular one respectively we obtain a stochastic maximum principle of the optimal control also we give the relationship of the backward variational equation the adjoint equation and forward variational equation

rebalancing with linear and quadratic costs we consider a financial market consisting of one safe and one risky asset which offer constant investment opportunities taking into account both proportional transaction costs and linear price impact we derive optimal rebalancing policies for representative investors with constant relative risk aversion and a long horizon

introduction to financial optimization mathematical programming special issue optimization models are effective for solving significant problems in finance including long term financial planning and other portfolio problems prominent examples include asset liability management for pension plans and insurance companies integrated risk management for intermediaries and long term planning for individuals several applications will be briefly mentioned three distinct approaches are available for solving multi stage financial optimization models 1 dynamic stochastic control 2 stochastic programming and 3 optimizing a stochastic simulation model we briefly review the pros and cons of these approaches discuss further applications of financial optimization and conclude with topics for future research

optimal portfolio and consumption decisions for a small investor on a finite horizon a general consumption investment problem is considered for an agent whose actions cannot affect the market prices and who strives to maximize total expected discounted utility of both consumption

stochastic optimal control of ultradiffusion processes with application to dynamic portfolio management we consider theoretical and approximation aspects of the stochastic optimal control of ultradiffusion processes in the context of a prototype model for the selling price of a european call option within a continuous time framework the dynamic management of a portfolio of assets is effected through continuous or point control activation costs and phase delay the performance index is derived from the unique weak variational solution to the ultraparabolic hamilton jacobi equation the value function is the optimal realization of the performance index relative to all feasible portfolios an approximation procedure based upon a temporal box scheme finite element method is analyzed numerical examples are presented in order to demonstrate the viability of the approach

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

determining mixed linear nonlinear coupled differential equations from multivariate discrete time series sequences abstract a new method is described for extracting mixed linear nonlinear coupled differential equations from multivariate discrete time series data it is assumed in the present work that the solution of the coupled ordinary differential equations can be represented as a multivariate volterra functional expansion a tractable hierarchy of moment equations is generated by operating on a suitably truncated volterra functional expansion the hierarchy facilitates the calculation of the coefficients of the coupled differential equations in order to demonstrate the method s ability to accurately estimate the coefficients of the governing differential equations it is applied to data derived from the numerical solution of the lorenz equations with additive noise the method is then used to construct a dynamic global mid and high magnetic latitude ionospheric model where nonlinear phenomena such as period doubling and quenching occur it is shown that the estimated inhomogeneous coupled second order differential equation model for the ionospheric fof2 peak plasma density can accurately forecast the future behaviour of a set of ionosonde stations which encompass the earth finally the method is used to forecast the future behaviour of a portfolio of japanese common stock prices the hierarchy method can be used to characterise the observed behaviour of a wide class of coupled linear and mixed linear nonlinear phenomena

optimal investment with high watermark performance fee we consider the problem of optimal investment and consumption when the investment opportunity is represented by a hedge fund charging proportional fees on profit the value of the fund evolves as a geometric brownian motion and the performance of the investment and consumption strategy is measured using discounted power utility from consumption on infinite horizon the resulting stochastic control problem is solved using dynamic programming arguments we show by analytical methods that the associated hamilton jacobi bellman equation has a smooth solution and then obtain the existence and representation of the optimal control in feedback form using verification arguments

explicit solution of a general consumption investment problem this paper solves a general consumption and investment decision problem in closed form an investor seeks to maximize total expected discounted utility of consumption there are n distinct risky investments modelled by dependent geometric brownian motion processes and one riskless deterministic investment the analysis allows for a general utility function and general rates of return the model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy so this paper generalizes many of the results of lehoczky sethi and shreve lehoczky j s sethi s shreva 1983 optimal consumption and investment policies allowing consumption constraints and bankruptcy math oper res 8 613 636 the value function is determined explicitly as are the optimal consumption and investment policies the analysis is extended to consider more general risky investments under certain conditions the value functions derived for geometric brownian motion are shown to provide upper and lower bounds on the value functions in the more general context

stochastic maximum principle for optimal control problems of forward backward systems involving impulse controls we consider a stochastic optimal control problem of a forward backward system in which the control variable consists of two components the continuous control and the impulse control the domain of the control is assumed to be convex necessary optimality conditions of the pontryagin maximum principle type are obtained for this stochastic optimal control problem we also give additional conditions under which the necessary optimality conditions turn out to be sufficient

groups of finite morley rank with a generically sharply multiply transitive action abstract we prove that if g is a group of finite morley rank that acts definably and generically sharply n transitively on a connected abelian group v of morley rank n with no involutions then there is an algebraically closed field f of characteristic 2 such that v has the structure of a vector space of dimension n over f and g acts on v as the group gl n f in its natural action on f n

optimal trading strategy for european options with transaction costs abstract the european option with transaction costs is studied the cost of making a transaction is taken to be proportional by a factor to the value in dollars of stock traded when there are no transaction costs i e when 0 the well known black scholes strategy tells how to hedge the option since no non trivial perfect hedging strategy exists when 0 see ann appl probab 5 2 1995 327 we instead try to maximize the expected utility attainable we seek to understand the effect transaction costs have on the maximum attainable expected utility over all strategies when is small but non zero it turns out that transaction costs diminish the expected utility by an amount which has the order of magnitude 2 3 we will compute that correction explicitly modulo an error which is small compared to 2 3 we will exhibit an explicit strategy whose expected utility differs from the maximum attainable expected utility by an error small in comparison to 2 3

a stochastic programming model using an endogenously determined worst case risk measure for dynamic asset allocation we present a new approach to asset allocation with transaction costs a multiperiod stochastic linear programming model is developed where the risk is based on the worst case payoff that is endogenously determined by the model that balances expected return and risk utilizing portfolio protection and dynamic hedging an investment portfolio similar to an option like payoff structure on the initial investment portfolio is characterized the relative changes in the expected terminal wealth worst case payoff and risk aversion are studied theoretically and illustrated using a numerical example this model dominates a static mean variance model when the optimal portfolios are evaluated by the sharpe ratio

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

multi asset investment consumption model with transaction costs abstract in this paper we consider the multi asset optimal investment consumption model a riskless asset and d risky assets when the initial time is t 0 for a proportional transaction costs and discount factors we proof that the value function of the model is a unique viscosity solution of a hamilton jacobi bellman hjb equations

groupes de rang de morley fini de type pair avec un sous groupe faiblement inclus resume we show that a minimal counterexample to the cherlin zil ber conjecture of even type and with a weakly embedded subgroup is isomorphic to psl 2 k where k is an algebraically closed field of characteristic 2

optimal consumption and portfolio with both fixed and proportional transaction costs we consider a market model with one risk free and one risky asset in which the dynamics of the risky asset are governed by a geometric brownian motion in this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets we suppose that these transfers involve a fixed transaction cost k 0 independent of the size of the transaction plus a cost proportional to the size of the transaction the objective is to maximize the cumulative expected utility of consumption over a planning horizon we formulate this problem as a combined stochastic control impulse control problem which in turn leads to a nonlinear quasi variational hamilton jacobi bellman inequality qvhjbi we prove that the value function is the unique viscosity solution of this qvhjbi finally numerical results are presented

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

stochastic perron s method for the probability of lifetime ruin problem under transaction costs we apply the stochastic perron s method to a singular control problem where an individual targets at a given consumption rate invests in a risky financial market in which trading is subject to proportional transaction costs and seeks to minimize her probability of lifetime ruin without relying on the dynamic programming principle we characterize the value function as the unique viscosity solution of an associated hamilton jacobi bellman variational inequality we also provide a complete proof of the comparison principle which is the main assumption of the stochastic perron s method

introduction to convex optimization in financial markets convexity arises naturally in financial risk management in risk preferences concerning random cash flows convexity corresponds to the fundamental diversification principle convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints moreover modern securities markets are based on trading protocols that result in convex trading costs the first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms the second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems

homogenization and asymptotics for small transaction costs we consider the classical merton problem of lifetime consumption portfolio optimization with small proportional transaction costs the first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one dimensional case unlike the existing literature we consider a general utility function and general dynamics for the underlying assets our arguments are based on ideas from homogenization theory and use convergence tools from the theory of viscosity solutions the multidimensional case is studied in our companion paper d possamai h m soner and n touzi homogenization and asymptotics for small transaction costs the multidimensional case arxiv 1212 6275v2 math ap preprint 2012 using the same approach

derivation of a new mertons optimal problem presented by fractional stochastic stock price and its applications in this article a new model of mertons optimal problem is derived this derivation is based on stock price presented by fractional order stochastic differential equation an extension of hamiltonjacobibellman is used to transfer our proposed model to a fractional partial differential equation as an application of our proposed model two optimal problems are discussed and solved analytically

asset liquidity and the valuation of derivative securities we consider the valuation of european style derivative securities under limited asset liquidity through the dynamic management of a portfolio of assets effected through continuous transaction the valuation arises from the optimal realization of a performance index relative to the set of all feasible portfolio trajectories an approximation procedure based upon the method of lines finite element method is developed and analyzed numerical examples are presented in order to demonstrate the viability of the approach

a spectral method for an optimal investment problem with transaction costs under potential utility abstract this paper concerns the numerical solution of the finite horizon optimal investment problem with transaction costs under potential utility the problem is initially posed in terms of an evolutive hjb equation with gradient constraints in day and yi 2009 the problem is reformulated as a non linear parabolic double obstacle problem posed in one spatial variable and defined in an unbounded domain where several explicit properties and formulae are obtained the restatement of the problem in polar coordinates allows to pose the problem in one spatial variable in a finite domain avoiding some of the technical difficulties of the numerical solution of the previous statement of the problem if high precision is required the spectral numerical method proposed becomes more efficient than simpler methods as finite differences for example

tame minimal simple groups of finite morley rank abstract we consider tame minimal simple groups of finite morley rank and of odd type we show that the prufer 2 rank of such a group is bounded by 2 we also find all potential nonalgebraic configurations there are essentially four of them and we delineate them with some precision

on solvable groups of finite morley rank we investigate solvable groups of finite morley rank we find conditions on g for g to split in g in particular if g is abelian and z g 1 we prove that g g t for some t and the ring z t anng is intepretable in g we exploit the methods used in proving these results to find more information about solvable groups

cosets genericity and the weyl group abstract in a connected group of finite morley rank in which generically elements belong to connected nilpotent subgroups proper normalizing cosets of definable subgroups are not generous we explain why this is true and what consequences this has for the abstract theory of weyl groups in groups of finite morley rank

convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem we consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs numerical approximations for the value function using the markov chain approximation method of kushner and dupuis are studied the main result of the paper shows that the value function of the markov decision problem mdp corresponding to the approximating controlled markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits all our convergence arguments are probabilistic the main assumption that we make is that the value function be finite and continuous in particular uniqueness of the solutions of the associated hjb equations is neither needed nor available in the generality under which the problem is considered specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function degeneracies in the dynamics mixed boundary dirichlet neumann conditions and presence of both singular and absolutely continuous controls in the dynamics finally schemes for computing the value function and optimal control policies for the mdp are presented and illustrated with a numerical study

optimality necessary conditions in singular stochastic control problems with nonsmooth data the present paper studies the stochastic maximum principle in singular optimal control where the state is governed by a stochastic differential equation with nonsmooth coefficients allowing both

generalized euler lagrange equations for fractional variational problems with free boundary conditions in this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems fvps with completely free boundary conditions have an extremum the fractional derivatives are defined in the caputo sense first we present the necessary conditions for the problem with only one dependent variable and then we generalize them to problems with multiple dependent variables we also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables it is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives the new results reduce to the known necessary conditions some examples are presented to demonstrate the applicability of the new formulations

singular optimal controls of stochastic recursive systems and hamilton jacobi bellman inequality abstract in this paper we study the optimal singular controls for stochastic recursive systems in which the control has two components the regular control and the singular control under certain assumptions we establish the dynamic programming principle for this kind of optimal singular controls problem and prove that the value function is a unique viscosity solution of the corresponding hamilton jacobi bellman inequality in a given class of bounded and continuous functions at last an example is given for illustration

survival and growth with a liability optimal portfolio strategies in continuous time we study the optimal behavior of an investor who is forced to withdraw funds continuously at a fixed rate per unit time e g to pay for a liability to consume or to pay dividends the investor is allowed to invest in any or all of a given number of risky stocks whose prices follow geometric brownian motion as well as in a riskless asset which has a constant rate of return the fact that the withdrawal is continuously enforced regardless of the wealth level ensures that there is a region where there is a positive probability of ruin in the complementary region ruin can be avoided with certainty call the former region the danger zone and the latter region the safe region we first consider the problem of maximizing the probability that the safe region is reached before bankruptcy which we call the survival problem while we show among other results that an optimal policy does not exist for this problem we are able to construct explicit e optimal policies for any e 0 in the safe region where ultimate survival is assured we turn our attention to growth among other results we find the optimal growth policy for the investor i e the policy which reaches another higher valued goal as quickly as possible other variants of both the survival problem as well as the growth problem are also discussed our results for the latter are intimately related to the theory of constant proportions portfolio insurance

shadow prices and well posedness in the problem of optimal investment and consumption with transaction costs we revisit the optimal investment and consumption model of davis and norman math oper res 15 1990 pp 676 713 and shreve and soner ann appl probab 4 1994 pp 609 692 following a shadow price approach similar to that of kallsen and muhle karbe ann appl probab 20 2010 pp 1341 1358 making use of the completeness of the model without transaction costs we reformulate and reduce the hamilton jacobi bellman equation for this singular stochastic control problem to a nonstandard free boundary problem for a first order ode with an integral constraint having shown that the free boundary problem has a smooth solution we use it to construct the solution of the original optimal investment consumption problem in a self contained manner and without any recourse to the dynamic programming principle furthermore we provide an explicit characterization of model parameters for which the value function is finite

finite horizon optimal investment and consumption with transaction costs this paper concerns continuous time optimal investment and the consumption decision of a constant relative risk aversion crra investor who faces proportional transaction costs and a finite time horizon in the no consumption case it has been studied by liu and loewenstein review of financial studies 15 2002 pp 805 835 and dai and yi j differential equations 246 2009 pp 1445 1469 mathematically it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints the problem gives rise to two free boundaries which stand for the optimal buying and selling strategies respectively we present an analytical approach to analyze the behaviors of free boundaries the regularity of the value function is studied as well our approach is essentially based on the connection between singular control and optimal stopping which is first revealed in the present problem

groups of mixed type the cherlin zil ber conjecture states that an infinite simple group of finite morley rank is an algebraic group over an algebraically closed field the proofs of the following three conjectures would provide a proof of the cherlin zil ber conjecture conjecture 1 there exist no nonsolvable connected groups of finite morley rank all of whose proper definable connected subgroups are nilpotent such groups are called bad groups 2 conjecture 2 there exists no structure k q a of finite morley rank where k is an algebraically closed field and a is an infinite proper definable subgroup of the multiplicative group of k such structures are called bad fields

optimal trend following trading rules we develop an optimal trend following trading rule in a bull bear switching market where the drift of the stock price switches between two parameters corresponding to an uptrend bull market and a downtrend bear market according to an unobservable markov chain we consider a flnite horizon investment problem and aim to maximize the expected return of the terminal wealth we start by restricting to allowing at and long positions only and describe the trading decisions using a sequence of stopping times indicating the time of entering and exiting long positions assuming trading all available funds we show that the optimal trading strategy is a trend following system characterized by the conditional probability in the uptrend crossing two threshold curves the thresholds can be obtained by solving the associated hjb equations in addition we examine trading strategies with short selling in terms of an approximation simulations and empirical experiments are conducted and reported

consistency issues for numerical methods for variance control with applications to optimization in finance the paper is concerned with numerical algorithms for the optimal control of diffusion type processes when the noise variance also depends on the control this problem is of increasing importance in applications particularly in financial mathematics we discuss the construction of numerical algorithms guaranteed to converge to the true minimum as the discretization level decreases and with acceptable numerical properties the algorithms are based on the popular markov chain approximation method the basic criterion the algorithms must satisfy is a weak local consistency condition which is essential for convergence to the true optimal cost function this condition is often hard to satisfy by simple algorithms with essentially only local transitions when the variance is also controlled numerical noise can be introduced by the more convenient approximations this question of numerical noise also called numerical viscosity is dealt with in detail and methods for eliminating or greatly reducing it are discussed

on the uniqueness of unbounded viscosity solutions arising in an optimal terminal wealth problem with transaction costs we study the uniqueness of viscosity solutions of a hamilton jacobi bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility

on groups of finite morley rank with weakly embedded subgroups the present paper is a contribution to the classification of tame stable groups of finite morley rank a group of finite morley rank is called tame if it involves no bad groups and no bad fields in the sense of 1 see also 1 it is conjectured that an infinite simple tame group of finite morley rank is isomorphic to an algebraic group over an algebraically closed field it is known that a minimal counterexample is either of even type meaning that the sylow 2 subgroups are of bounded exponent or of odd type meaning that the sylow 2 subgroups are finite extensions of divisible abelian groups we deal here with tame simple groups of even type having a weakly embedded subgroup a notion related to the notion of strongly embedded subgroup in the finite case but having no exact analog in that case in 10 the classication of simple tame groups with a strongly embedded subgroup was derived in an inductive framework as follows

numerical solution of an optimal investment problem with proportional transaction costs this paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite horizon optimal investment problem with proportional transaction costs the problem is initially posed in terms of an evolutive hjb equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable the proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain a characteristics method for time discretization to deal with the large gradients of the solution a newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle free boundary feature moreover piecewise linear lagrange finite elements for spatial discretization are considered numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in dai and yi 2009 6

groupes de type mixte resume we show that a minimal counterexample to the cherlin zil ber conjecture tame or not cannot be of mixed type

finite horizon optimal investment with transaction costs a parabolic double obstacle problem abstract this paper concerns optimal investment problem of a crra investor who faces proportional transaction costs and finite time horizon from the angle of stochastic control it is a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies this enables us to make use of the well developed theory of obstacle problem to attack the problem the c 2 1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized

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

the stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls this paper is concerned with a pontryagin s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls this kind of control problems is motivated by some interesting phenomena arising from economics and finance we establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis to explain the theoretical results we apply them to a production and consumption choice problem

minimal connected simple groups of finite morley rank with strongly embedded subgroups let g be a simple k group of finite morley rank of odd type which is not algebraic then g has prufer 2 rank at most two this improves on an earlier result of the second and third authors for the tame case in the critical case g is minimal connected simple of odd type with a proper definable strongly embedded subgroup the bulk of the analysis relies on the first author s theory of 0 unipotence and the related 0 sylow subgroup theory as well as the so called bender method adapted to this context

bounded variation singular stochastic control and dynkin game we consider a bounded variation singular stochastic control problem with value v in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation bsde associated with this is a dynkin game with value u we establish the well known relation frac partial partial x v u for this general situation a saddle point for the dynkin game is given by the pair of first action times of an optimal control the methods are from stochastic analysis and include a priori estimates pathwise construction and comparison theorems for forward stochastic differential equations fsde and bsde

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

a numerical method for solving singular stochastic control problems singular stochastic control has found diverse applications in operations management economics and finance however in all but the simplest of cases singular stochastic control problems cannot be solved analytically in this paper we propose a method for numerically solving a class of singular stochastic control problems we combine finite element methods that numerically solve partial differential equations with a policy update procedure based on the principle of smooth pasting to iteratively solve hamilton jacobi bellman equations associated with the stochastic control problem a key feature of our method is that the presence of singular controls simplifies the procedure we illustrate the method on two examples of singular stochastic control problems one drawn from economics and the other from queueing systems

groups in supersimple and pseudofinite theories we consider groups g interpretable in a supersimple nite rank theory t such that t eq eliminates 9 1 it is shown that g has a denable soluble radical if g has rank 2 then if g is pseudonite it is soluble by nite and partial results are obtained under weaker hypotheses such as functional unimodularity of the theory a classication is obtained when t is pseudonite and g has a denable and denably

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given