optimal control for partially observed diffusions stochastic control problems are considered in which a state process x t and an observation process y t are governed by ito sense stochastic differential equations driven by independent brownian motions the control u t enters linearly in the dynamics of x t a separated control problem is introduced in which the state at any time t is a measure lambda t representing an unnormalized conditional distribution for x t given y s u s for s leqq t the method depends on introducing a pathwise version of lambda t which depends continuously on observation and control trajectories y u existence of an optimal control is obtained in a suitable class larger than the usual class of controls admissible in the strict sense that u t is measurable on the sigma algebra mathcal f t y generated by observations y s s leqq t the dynamics of lambda t are studied using a method of forward and backward partial differential equations under a suitable nondegeneracy condition the m

control of the stochastic burgers model of turbulence we consider a control problem for a stochastic burgers equation this problem is motivated by a model from the control of turbulence see choi et al j fluid mech 253 1993 pp 509 543 we study a sequence of approximated hamilton jacobi equations by using dynamic programming

quasi random methods for estimating integrals using relatively small samples much of the recent work dealing with quasi random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite dimensional integral is replaced by a finite sum of integrand values in contrast with this perspective to concentrate on asymptotic convergence rates this paper emphasizes quasi random methods that are effective for all sample sizes throughout the paper the problem of estimating finite dimensional integrals is used to illustrate the major ideas although much of what is done applies equally to the problem of solving certain fredholm integral equations some new techniques based on error reducing transformations of the integrand are described that have been shown to be useful both in estimating high dimensional integrals and in solving integral equations these techniques illustrate the utility of carrying over to the quasi monte carlo method certain devices that have proven to be very valuable in statistical pseudoran

a computational approach to approximate and plausible reasoning with applications to expert systems the intended purpose of this paper is twofold proposing a common basis for the modeling of uncertainty and imprecision and discussing various kinds of approximate and plausible reasoning schemes in this framework together with probability different kinds of uncertainty measures credibility and plausibility functions in the sense of shafer possibility measures in the sense of zadeh and the dual measures of necessity sugeno s g fuzzy measures are introduced in a unified way the modeling of imprecision in terms of possibility distribution is then presented and related questions such as the measure of the uncertainty of fuzzy events the probability and possibility qualification of statements the concept of a degree of truth and the truth qualification of propositions are discussed at length deductive inference from premises weighted by different kinds of measures by uncertainty or by truth values in the framework of various multivalued logics is fully investigated then deductive inferences from imprecise or fuzzy premises are dealt with patterns of reasoning where both uncertainty and imprecision are present are also addressed the last section is devoted to the combination of uncertain or imprecise pieces of information given by different sources on the whole this paper is a tentative survey of quantitative approaches in the modeling of uncertainty and imprecision including recent theoretical proposals as well as more empirical techniques such as the ones developed in expert systems such as mycin or prospector the management of uncertainty and imprecision in reasoning patterns being a key issue in artificial intelligence

optimal control of a stochastic delay heat equation with boundary noise and boundary control in this paper a controlled stochastic delay heat equation with neumann boundary noise and boundary control is considered the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of the backward stochastic differential equations which is applied to the optimal control problem

some results on non linear optimal control problems and hamilton jacobi equations in infinite dimensions abstract the optimal control of a distributed parameter system is connected to the solution of the corresponding hamilton jacobi equation this is a first order equation in infinite dimensions with discontinuous coefficients we study the hamilton jacobi equation of a system governed either by a semilinear or by a monotone dynamics replacing the unbounded terms of this equation by their yosida approximations we prove convergence of the approximate solutions to the value function of the original problem by using the uniqueness result for viscosity solutions examples and applications are included

on viscosity solutions of path dependent pdes in this paper we propose a notion of viscosity solutions for path dependent semi linear parabolic pdes this can also be viewed as viscosity solutions of non markovian backward sdes and thus extends the well known nonlinear feynman kac formula to non markovian case we shall prove the existence uniqueness stability and comparison principle for the viscosity solutions the key ingredient of our approach is a functional ito calculus recently introduced by dupire functional ito calculus 2009 preprint

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

practical error estimation in adaptive multidimensional quadrature routines abstract in this paper we discuss various ways of improving the reliability of adaptive multidimensional quadrature routines we discuss the reasons that estimates of the error over a given region of integration may fail in representing an upper bound for the actual error on basis of this discussion we suggest different actions to be taken in order to improve the reliability of the quadrature routine in the numerical experiments we study how these new error estimates work in practice

adaptive integration for multi factor portfolio credit loss models we propose algorithms of adaptive integration for calculation of the tail probability in multi factor credit portfolio loss models we first modify the classical genz malik rule a deterministic multiple integration rule suitable for portfolio credit models with number of factors less than 8 later on we arrive at the adaptive monte carlo integration which essentially replaces the deterministic integration rule by antithetic random numbers the latter can not only handle higher dimensional models but is also able to provide reliable probabilistic error bounds both algorithms are asymptotic convergent and consistently outperform the plain monte carlo method

priority assignment a conditioned sets approach abstract the author suggests how conditioned sets of fuzzy variables may be used to provide priority weights a case is presented where the conditioned sets approach is contrasted with the prioritization information generated by the eigenvalue priority assignment model

partial regularity of viscosity solutions for a class of kolmogorov equations arising from mathematical finance we study value functions which are viscosity solutions of certain kolmogorov equations using pde techniques we prove that they are c1 regular on special finite dimensional subspaces the problem has origins in hedging derivatives of risky assets in mathematical finance

dynamic programming for optimal control of stochastic mckean vlasov dynamics we study the optimal control of general stochastic mckean vlasov equation such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles players in mean field interaction under common noise our first main result is to state a dynamic programming principle for the value function in the wasserstein space of probability measures which is proved from a flow property of the conditional law of the controlled state process next by relying on the notion of differentiability with respect to probability measures due to p l lions 32 and ito s formula along a flow of conditional measures we derive the dynamic programming hamilton jacobi bellman equation and prove the viscosity property together with a uniqueness result for the value function finally we solve explicitly the linear quadratic stochastic mckean vlasov control problem and give an application to an interbank systemic risk model with common noise

an algebraic synthesis of the foundations of logic and probability abstract this paper develops a unified algebraic theory of logic and probability based upon an initial boolean algebra logic of propositions together with its extensionally associated boolean algebra of models probabilistic events that satisfy the axioms of the boolean algebra of propositions a proposition may be true in some models and false in others but each model must unambiguously assign true or false to each proposition thus the examples models in which different propositions are true may be all some or none according as those propositions are necessary possible or impossible a probability measure p on the collection of models events induces a probability measure on the propositions the crucial if then relation is defined so that p if p then q equals p q p the conditional probability of q given p this conditional probability is less than the probability of the material conditional q or not p unless p p 1 or p q p 1 at these two extremes the two propositional constructions coalesce thus an additional relation q p is needed in logic and probability namely q given p this is accomplished in the algebraic logic generated by the various sum ideals entailments of various propositions that may serve as the explicit condition of a conditional proposition q p the corresponding relation b a in the model realm defines the notion of a conditional event in this context of algebraic logic and model theory a new algebraic structure called the conditional closure l l of a boolean logic l is defined consisting of the ordered pairs q p of all propositions of l excluding those for which the condition p is equivalent to 0 that is impossible the conditional closure is not altogether a boolean algebra not altogether distributive for instance but it has many subalgebras that are boolean including l itself this paper contains partially assumed formulas for operating on the conditional propositions q p using and or not and of course if then this allows complex conditional expressions to be reduced to simple conditionals of boolean propositions which have a conditional probability an example calculation from the dice table illustrates how to simplify a compound conditional proposition and compute its probability the structure of l l is explored including the finite conditional closures of finite boolean logics formulas are given for the probabilistic difference between familiar conditional propositions that are equivalent in the two valued logic where all propositions are either necessary or impossible but not equivalent in the conditional closure of a logic with possible propositions the probability that a proposition p is a theorem true in all models is contrasted from the probability of the models in which the proposition is true the latter being p p while the former is p p 1 this leads to a natural connection with fuzzy set theory since p 1 can also be expressed as p 1 where 1 is the sum ideal filter of necessary propositions given the boolean axioms this paper ends with a representation of each conditional proposition q p as the p measurable two valued truth function of q restricted to those models that satisfy p

a maximum entropy approach to nonmonotonic reasoning an approach to nonmonotonic reasoning that combines the principle of infinitesimal probabilities with that of maximum entropy thus extending the inferential power of the probabilistic interpretation of defaults is proposed a precise formalization of the consequences entailed by a conditional knowledge base is provided the computational machinery necessary for drawing these consequences is developed and the behavior of the maximum entropy approach is compared to related work in default reasoning the resulting formalism offers a compromise between two extremes the cautious approach based on the conditional interpretations of defaults and the bold approach based on minimizing abnormalities

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

partially observed control of markov processes iv abstract a class of degenerate elliptic nonlinear second order equations is considered these equations arising from a related class of probabilistic variational problems are defined on the space of probability measures on rd the existence of c1 1 viscosity solutions is established as well as the existence uniqueness and feedback characterization of the minima of the variational problems

partially observed control of markov processes iii let v denote the value function of a partially observed control problem if v is one differentiable in a certain direction b then optimal controls are characterized by a feedback involving the directional derivative b v it is also shown that v satisfies the corresponding bellman equation an infinite dimensional pde on the space of measures in the viscosity sense of crandall and lions

optimal control of semilinear stochastic evolution equations the aim of this paper is to initiate a semigroup theory based approach to characterization of optimal markov controls for controlled semilinear stochastic evolution equations it may be recalled that markov controls are those that depend only on the current state at each time for finite dimensional controlled stochastic differential equations with a nondegenerate diffusion matrix this task is traditionally achieved through the hamilton jacobi bellman equation of dynamic programming associated with the problem and an accompanying verification theorem the latter states that an optimal markov control can be explicitly obtained by the pointwise minimization of a hamiltonian derivable from the solution of the hjb equation moreover any optimal markov control is obtainable in this manner

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

regularity results for the minimum time function of a class of semilinear evolution equations of parabolic type under suitable controllability and smoothness assumptions the minimum time function t x of a semilinear control system is proved to be locally lipschitz continuous and semiconcave on the controllable set these properties are then applied to derive optimality conditions relating optimal trajectories to the superdifferential of t

generalized solutions of hjb equations applied to stochastic control on hilbert space abstract in this paper we prove the existence of borel measurable optimal control laws for infinite dimensional nonstationary stochastic control problems on hilbert space this is based on some recent results on existence of generalized solutions of hjb equations on hilbert space l2 h where is a suitable reference measure on h for illustration some examples are presented

optimal control problems for stochastic reaction diffusion systems with non lipschitz coefficients by using the dynamic programming approach we study a control problem for a class of stochastic reaction diffusion systems with coefficients having polynomial growth in the cost functional a non lipschitz term appears and this allows us to treat the quadratic case which is of interest in the applications the corresponding hamilton jacobi bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates the main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system

regularity for obstacle problems in infinite dimensional hilbert spaces in this paper we study fully nonlinear obstacle type problems in hilbert spaces we introduce the notion of q elliptic equation and prove existence uniqueness and regularity of viscosity solutions of q elliptic obstacle problems in particular we show that solutions of concave problems with semiconvex obstacles are in the space

infinite dimensional hamilton jacobi equations with large zeroth order coefficient abstract we establish the existence and uniqueness of the c 1 1 solution of an infinite dimensional hamilton jacobi equation defined on the space of probability measures on r d when the zeroth order coefficient is sufficiently large

advances in multidimensional integration this paper gives a personal bird s eye view of some aspects of multivariate numerical integration it will sketch what happened during the past 50 years and point the reader s attention to what did not happen

regular solutions of second order stationary hamilton jacobi equations abstract we study a second order stationary hamilton jacobi equation in infinite dimension this equation is nonlinear and convex with respect to the first order term we use properties of the transition semigroup associated to the linear equation to write the hamilton jacobi equation in integral form and we prove existence uniqueness and regularity of a solution by the theory of maximal monotone operators we also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems finally the solution is the value function of the associated optimal stochastic control problem some examples are given

on the existence of optimal controls for partially observed diffusions the problem of optimally controlling a partially observed diffusion process is shown to have a solution in two cases when the set of admissible controls is compact or when the set of admissible controls is the set of randomized controls

probability updating using second order probabilities and conditional event algebra abstract the second order probability technique is a procedure whereby an incompletely specified finite argument probability function is modeled as a random vector over the associated natural simplex of all possible probability functions of a fixed common number of arguments often the distribution of the random vector is assumed to be a uniform one as in the case of goodman s treatment of the probabilistic syllogism or penguin triangle problem 1998 but when compatibility with a non uniform prior distribution is sought the uniform random vector assumption must be replaced by a different one in a related but distinct direction conditional event algebra is a new mathematical tool which allows the direct incorporation of conditional relations expressed through conditional probabilities with other such conditional relations or unconditional events i r goodman h t nguyen journal of uncertainty fuzziness and knowledge based systems 3 3 1995 247 339 i r goodman r p mahler h t nguyen mathematics of data fusion kluwer academic publishers dordrecht holland 1997 the judy benjamin jb problem as originally posed by van fraasen british journal of philosophy of science 32 1981 375 379 has been recently addressed by grove and halpern g h proceedings of the 13th conference uncertainty in ai 1997 pp 208 214 and has elements in it conducive to analysis via both second order probability and conditional event algebra the problem involves the issue of how to update a particular event of interest having a known prior probability part of a known uniform prior probability function upon a known conditional probability statement when the latter is given through a probability function otherwise unknown and distinct from the prior probability function in the problem as stated the event of interest is disjoint from the antecedent event in the conditional probability statement g h make use of a trust principle in order to be able to apply in effect the second order probability technique with an associated uniform distribution g h s result differs sharply from that of van fraasen who employs instead a minimal cross entropy approach to the problem in that g h show no change in probability values between the prior and posterior assessments this paper considers in a more general context the application of the second order probability technique using the dirichlet family generalization of the uniform distribution over a simplex in updating an event upon a conditional probability statement when no such disjointness as in the jb problem holds and compatibility is sought with a non uniform prior when this is specialized to the disjoint event case even with a non uniform prior the independence result of g h extends to this context the jb problem is also considered via a conditional event algebra viewpoint which avoids the trust principle in conjunction with the second order probability technique and general agreement with that of g h s independence result is established for the case of disjoint event of interest from the antecedent and consequence of the conditional form these relations are also extended to a more general setting

nonmonotonic reasoning preferential models and cumulative logics abstract many systems that exhibit nonmonotonic behavior have been described and studied already in the literature the general notion of nonmonotonic reasoning though has almost always been described only negatively by the property it does not enjoy i e monotonicity we study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties we concentrate on a number of families of nonmonotonic consequence relations defined in the style of gentzen 13 both proof theoretic and semantic points of view are developed in parallel the former point of view was pioneered by gabbay 10 while the latter has been advocated by shoham 38 five such families are defined and characterized by representation theorems relating the two points of view one of the families of interest that of preferential relations turns out to have been studied by adams 2 the preferential models proposed here are a much stronger tool than adams probabilistic semantics the basic language used in this paper is that of propositional logic the extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper

optimal control problems for stochastic delay evolution equations in banach spaces we consider the optimal control for a banach space valued stochastic delay evolution equation the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of backward stochastic differential equations an application to optimal control of stochastic delay partial differential equations is also given

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

viscosity solutions of hamilton jacobi equations in infinite dimensions v unbounded linear terms and b continuous solutions we continue the systematic study of first order hamilton jacobi equations in infinite dimensions begun in parts i iv of this series 121 our approach relies on as before an appropriate interpretation of the notion of viscosity solutions as introduced in 11 lo in the interim the strong development of viscosity solution theory in many directions has continued unabated in addition to the many references given in earlier papers in this series which we will not repeat here see 12 we would like to point out that a complete theory of second order fully nonlinear and possibly degenerate elliptic equations in finite dimensions has been made possible by the use of viscosity solutions s 9 18 20 22 19 32 331 for example and there is now a substantial body of work concerning second order equations in infinite dimensions 25 271 however in this work we come back to the particular situation studied in 12 part iv namely we come back to hamilton jacobi equations in

hamilton jacobi bellman equations for the optimal control of the duncan mortensen zakai equation abstract we study a class of hamilton jacobi bellman hjb equations associated to stochastic optimal control of the duncan mortensen zakai equation the equations are investigated in weighted l 2 spaces we introduce an appropriate notion of weak viscosity solution of such equations and prove that the value function is the unique solution of the hjb equation we apply the results to stochastic optimal control problems with partial observation and correlated noise

explicit solutions for continuous time mean variance portfolio selection with nonlinear wealth equations abstract this paper concerns the continuous time mean variance portfolio selection problem with a special nonlinear wealth equation this nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in ji 2010 does not work we invoke the hjb equation of this problem and give an explicit viscosity solution of the hjb equation furthermore via this explicit viscosity solution we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem finally we show that our nonlinear wealth equation can cover three important cases

the existence and uniqueness of the solution for nonlinear kolmogorov equations abstract by means of backward stochastic differential equations the existence and uniqueness of the mild solution are obtained for the nonlinear kolmogorov equations associated with stochastic delay evolution equations applications to optimal control are also given

stationary hamilton jacobi equations in hilbert spaces and applications to a stochastic optimal control problem we study an infinite horizon stochastic control problem associated with a class of stochastic reaction diffusion systems with coefficients having polynomial growth the hamiltonian is assumed to be only locally lipschitz continuous so that the quadratic case can be covered we prove that the value function v corresponding to the control problem is given by the solution of the stationary hamilton jacobi equation associated with the state system to this purpose we write the hamilton jacobi equation in integral form and by using the smoothing properties of the transition semigroup relative to the state system and the theory of m dissipative operators we show that it admits a unique solution moreover the value function v is obtained as the limit of minima for some approximating control problems which admit unique optimal controls and states

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

comparison of viscosity solutions of semi linear path dependent pdes this paper provides a probabilistic proof of the comparison result for viscosity solutions of path dependent semilinear pdes we consider the notion of viscosity solutions introduced in 8 which considers as test functions all those smooth processes which are tangent in mean when restricted to the markovian case this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in 1 2 our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison this is most remarkable in the context of the linear path dependent heat equation as a key ingredient for our methodology we introduce a notion of punctual differentiation similar to the corresponding concept in the standard viscosity solutions 3 and we prove that semimartingales are almost everywhere punctually differentiable this smoothness result can be viewed as the counterpart of the aleksandroff smoothness result for convex functions a similar comparison result was established earlier in 8 the result of this paper is more general and more importantly the arguments that we develop do not rely on any representation of the solution

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

optimal relaxed controls for infinite dimensional stochastic systems of zakai type in this paper we present some new results on partially observed control problems for infinite dimensional stochastic systems in hilbert space using a fundamental result of da prato and zabczyk on an infinite dimensional kolmogorov operator we prove the existence of optimal relaxed controls for an infinite dimensional zakai equation following a semigroup approach and the theory of measurable selections this result is also extended to differential inclusion we also present some necessary conditions of optimality

extension of the concept of propositional deduction from classical logic to probability an overview of probability selection approaches abstract establishing a rigorous framework for propositional deduction which also agrees with what is termed commonsense reasoning poses a difficult challenge this paper is the first of a two part effort in considering the issue and proposing in future part 2 a particular approach via the use of second order probabilities i e distributions of probability measures as opposed to probability selection approaches especially that of maximum entropy and adams high probability schemes in the second order probability approach usually all probability distributions are assumed a priori to be either equally likely or more generally to be distributed via the dirichlet family up to the constraints involved in the premise set of the potential deduction considered

nonlinear semigroups associated with optimal stopping of controlled diffusions under partial observation abstract we treat optimal stopping problems of controlled diffusions under partial observation two kinds of admissibility strict sense and wide sense are considered using the girsanov transformation the unnormalized conditional probability turns out the state process subject to the zakai equation and we can derive a nonlinear semigroup which presents the bellman principle in our problems in the diffusion case an optimal stopping time is given appealing to a snell envelope