stepsize control for mean square numerical methods for stochastic differential equations with small noise a strategy for controlling the stepsize in the numerical integration of stochastic differential equations sdes is presented it is based on estimating the pth mean of local errors the strategy leads to stepsize sequences that are identical for all computed paths for the family of euler schemes for sdes with small noise we derive computable estimates for the dominating term of the pth mean of local errors and show that the strategy becomes efficient for reasonable stepsizes numerical experience is reported for test examples including scalar sdes and a stochastic circuit model

stochastic integrators with stationary independent increments lipschitzian stochastic differential equations driven by a process with stationary independent increments permit a priori growth and stability estimates up to any sure time and the solutions depend differentiably on parameters provided the levy measure of the driving term has suitable moments

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

the composite euler method for stiff stochastic differential equations in this paper we present the composite euler method for the strong solution of stochastic differential equations driven by d dimensional wiener processes this method is a combination of the semi implicit euler method and the implicit euler method at each step either the semi implicit euler method or the implicit euler method is used in order to obtain better stability properties we give criteria for selecting the semi implicit euler method or the implicit euler method for the linear test equation the convergence properties of the composite euler method depend on the criteria for selecting the methods numerical results suggest that the convergence properties of the composite euler method applied to nonlinear sdes is the same as those applied to linear equations the stability properties of the composite euler method are shown to be far superior to those of the euler methods and numerical results show that the composite euler method is a very promising method

pathwise stochastic taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic pdes in this paper we study a new type of taylor expansion for ito type random fields up to the second order we show that an ito type random field with reasonably regular integrands can be expanded up to the second order to the linear combination of increments of temporal and spatial variables as well as the driven brownian motion around even a random t x point also the remainder can be estimated in a pathwise manner we then show that such a taylor expansion is valid for the solutions to a fairly large class of stochastic differential equations with parameters or even fully nonlinear stochastic partial differential equations whenever they exist using such analysis we then propose a new definition of stochastic viscosity solution for fully nonlinear stochastic pdes in the spirit of its deterministic counterpart we prove that this new definition is actually equivalent to the one proposed in our previous works 2 and 3 at least for a class of quasilinear spdes

asymptotic properties of monte carlo estimators of diffusion processes abstract this paper studies the limit distributions of monte carlo estimators of diffusion processes we examine two types of estimators based on the euler scheme one applied to the original processes the other to a doss transformation of the processes we show that the transformation increases the speed of convergence of the euler scheme we also study estimators of conditional expectations of diffusions after characterizing expected approximation errors we construct second order bias corrected estimators we also derive new convergence results for the mihlstein scheme illustrations of the results are provided in the context of simulation based estimation of diffusion processes

adaptive schemes for the numerical solution of sdes a comparison the efficient numerical solution of stochastic differential equations is important for applications in many fields adaptive schemes well developed in the deterministic setting may be one possible way to reduce computational cost we review the two main step size control algorithms that have been proposed in recent years for stochastic differential systems and compare their efficiency in a simulation study

a new adaptive runge kutta method for stochastic differential equations in this paper we will present a new adaptive time stepping algorithm for strong approximation of stochastic ordinary differential equations we will employ two different error estimation criteria for drift and diffusion terms of the equation both of them based on forward and backward moves along the same time step we will use step size selection mechanisms suitable for each of the two main regimes in the solution behavior which correspond to domination of the drift based local error estimator or diffusion based one numerical experiments will show the effectiveness of this approach in the pathwise approximation of several standard test problems

backward doubly stochastic differential equations and systems of quasilinear spdes we introduce a new class of backward stochastic differential equations which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations thus extending the feynman kac formula for linear spde s

on volterra equations driven by semimartingales on considere des equations stochastiques de la forme x t h t o t f t s x dzs ou z est une semimartingale de dimension finie h est un processus cadlag et f est une fonctionnelle

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

superhedging and dynamic risk measures under volatility uncertainty we consider dynamic sublinear expectations i e time consistent coherent risk mea sures whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty we derive a cadlag nonlinear martingale which is also the value process of a superhedg ing problem the superhedging strategy is obtained from a representation similar to the optional decomposition furthermore we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward sde the uniqueness of dynamic extensions of static sublinear expectations is also studied

an adaptive timestepping algorithm for stochastic differential equations we introduce a variable timestepping procedure using local error control for the pathwise strong numerical integration of a system of stochastic differential equations forced by a single wiener process the milstein method is used to advance the numerical solution and the stepsizes are determined via two local error estimates that roughly correspond to leading order deterministic and stochastic local error components one advantage of using two error controls is an increased flexibility that allows for the treatment of both drift and diffusion dominated regimes in a consistent manner numerical results are presented and the generalization of this approach to wider classes of problems and methods is discussed

a transition to sharp timing in stochastic leaky integrate and fire neurons driven by frozen noisy input the firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current mainen sejnowski 1995 this suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking cecchi et al 2000 tiesinga 2002 fellous rudolph destexhe sejnowski 2003 in order to investigate this interplay we focus on the stochastic leaky integrate and fire neuron and identify the holder exponent h of the integrated input as the key mathematical property dictating the regime of firing of a single unit neuron we have recently provided numerical evidence taillefumier magnasco 2013 for the existence of a phase transition when becomes less than the statistical holder exponent associated with internal gaussian white noise h 1 2 here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent h 0 in doing so we account for the existence of a transition between two regimes of firing when h 1 2 and we show that spiking times have a continuous density when the holder exponent satisfies h 1 2 the transition at h 1 2 formally separates rate codes for which the neural firing probability varies smoothly from temporal codes for which the neuron fires at sharply defined times regardless of the intensity of internal noise

martingale optimal transport duality we obtain a dual representation of the kantorovich functional defined for functions on the skorokhod space using quotient sets our representation takes the form of a choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets these sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given another important ingredient of our analysis is a regularized version of jakubowski s s topology on the skorokhod space

a finiteness theorem in the galois cohomology of algebraic number fields in this note we show that if k is an algebraic number field with algebraic closure k and m is a finitely generated free z module with continuous gal k k action then the continuous galois cohomology group hl k m is a finitely generated z module under certain conditions on m see theorem 1 below also we present a simpler construction of a mapping due to s bloch which relates torsion algebraic cycles and etale cohomology the galois cohomology of an algebraic number field is well known to be important in the study of the arithmetic of the field and of algebraic varieties defined over the field unfortunately this cohomology is not always well behaved for example the galois cohomology group hl k z l lim hl gal k k y

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

quantum field theory of treasury bonds the heath jarrow morton hjm formulation of treasury bonds in terms of forward rates is recast as a problem in path integration the hjm model is generalized to the case where all the forward rates are allowed to fluctuate independently the resulting theory is shown to be a two dimensional gaussian quantum field theory the no arbitrage condition is obtained and a functional integral derivation is given for the price of a futures and an options contract

step size control in the numerical solution of stochastic differential equations abstract we introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations the algorithm is based on a new pair of embedded explicit runge kutta methods of strong order 1 5 1 0 where the method of strong order 1 5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error we show that convergence of our method is preserved though the discretization times are not stopping times any more and further we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

numerical solutions of stochastic differential equations implementation and stability issues stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise there has been much work done recently on developing numerical methods for solving sdes this paper will focus on stability issues and variable stepsize implementation techniques for numerically solving sdes effectively

stochastic string models with continuous semimartingales abstract this paper reformulates the stochastic string model of santa clara and sornette using stochastic calculus with continuous semimartingales we present some new results such as a the dynamics of the short term interest rate b the pde that must be satisfied by the bond price and c an analytic expression for the price of a european bond call option additionally we clarify some important features of the stochastic string model and show its relevance to price derivatives and the equivalence with an infinite dimensional hjm model to price european options

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

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

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

comment on numerical methods for stochastic differential equations wilkie phys rev e 70 017701 2004 used a heuristic approach to derive runge kutta based numerical methods for stochastic differential equations based on methods used for solving ordinary differential equations the aim was to follow solution paths with high order we point out that this approach is invalid in the general case and does not lead to high order methods we warn readers against the inappropriate use of deterministic calculus in a stochastic setting

the sensitivity analysis of propagator for path independent quantum finance model abstract quantum finance successfully implements the imperfectly correlated fluctuation of forward interest rates at different maturities by replacing the wiener process with a two dimensional quantum field interest rate derivatives can be priced at a more realistic value under this new framework the quantum finance model requires three main ingredients for pricing the initial forward interest rates the volatility of forward interest rates and the correlation of forward interest rates at different maturities however the hedging strategy only focused on fluctuation of forward interest rates this hedging method is based on the assumption that the propagator the covariance of forward interest rates has an ergodic property since inserting the propagator is the main characteristic that distinguishes quantum finance from the libor market model lmm and the heath jarrow and morton hjm model understanding the impact of propagator dynamics on the price of interest rate derivatives is crucial this research is the first step in developing a hedge strategy with respect to the evolution of the propagator we analyze the dynamics of the propagator from libor futures data and the integrated propagator from zero coupon bond rate data then we study the sensitivity of the implied volatility of caplets and swaptions according to the three dominant dynamics of the propagator and the change of the zero coupon bond option price according to the two dominant dynamics of the integrated propagator

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

hedging libor derivatives in a field theory model of interest rates abstract we investigate libor based derivatives using a parsimonious field theory interest rate model capable of instilling imperfect correlation between different maturities delta and gamma hedge parameters are derived for libor caps against fluctuations in underlying forward rates an empirical illustration of our methodology is conducted to demonstrate the influence of correlation on the hedging of interest rate risk

sharp inequalities for the distribution of a stochastic integral in which the integrator is a bounded submartingale we obtain a sharp probability bound on the maximal function of a strong subordinate of a bounded submartingale an analogous inequality also holds for stochastic integrals in which the integrator is a bounded submartingale and the integrand is a bounded predictable process

mean square stability properties of an adaptive time stepping sde solver we consider stability properties of a class of adaptive time stepping schemes based upon the milstein method for stochastic differential equations with a single scalar forcing in particular we focus upon mean square stability for a class of linear test problems with multiplicative noise we demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls one for the drift term and one for the diffusion

stochastic optimal control of state constrained systems in this article we consider the problem of stochastic optimal control in continuous time and state action space of systems with state constraints these systems typically appear in the area of robotics where hard obstacles constrain the state space of the robot a common approach is to solve the problem locally using a linear quadratic gaussian lqg method we take a different approach and apply path integral control as introduced by kappen kappen h j 2005a path integrals and symmetry breaking for optimal control theory journal of statistical mechanics theory and experiment 2005 p11011 kappen h j 2005b linear theory for control of nonlinear stochastic systems physical review letters 95 200201 we use hybrid monte carlo sampling to infer the control we introduce an adaptive time discretisation scheme for the simulation of the controlled dynamics we demonstrate our approach on two examples a simple particle in a halfspace and a more complex two joint manipulator and we show tha

a fast algorithm for the first passage times of gauss markov processes with holder continuous boundaries even for simple diffusion processes treating first passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly here we present a novel numerical method that is faster and has more tightly controlled accuracy our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non negative homogeneously holder continuous boundaries by gauss markov processes these include the ornstein uhlenbeck process underlying the ubiquitous leaky integrate and fire model of neuronal excitation our method evaluates discrete points in a sample path exactly and refines this representation recursively only in regions where a passage is rigorously estimated to be probable e g when close to the boundary

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

one modification of the martingale transform and its applications to paraproducts and stochastic integrals abstract in this paper we introduce a variant of burkholder s martingale transform associated with two martingales with respect to different filtrations even though the classical martingale techniques cannot be applied we show that the discussed transformation still satisfies some expected l p estimates then we apply the obtained inequalities to general dilation twisted paraproducts particular instances of which have already appeared in the literature as another application we construct stochastic integrals 0 t h s d x s y s associated with certain continuous time martingales x t t 0 and y t t 0 the process x t y t t 0 is shown to be a good integrator although it is not necessarily a semimartingale or even adapted to any convenient filtration

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

probabilistic interpretation for solutions of fully nonlinear stochastic pdes in this article we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation 2bdsde we prove existence and uniqueness of the solution under a lipschitz type assumption on the generator and we investigate the links between the 2bdsdes and a class of parabolic fully nonlinear stochastic pdes precisely we show that the markovian solution of 2bdsdes provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear spdes

a variable stepsize implementation for stochastic differential equations stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise much work has been done recently on developing higher order runge kutta methods for solving sdes numerically fixed stepsize implementations of numerical methods have limitations when for example the sde being solved is stiff as this forces the stepsize to be very small this paper presents a completely general variable stepsize implementation of an embedded runge kutta pair for solving sdes numerically in this implementation there is no restriction on the value used for the stepsize and it is demonstrated that the integration remains on the correct brownian path

higher order time integration of coulomb collisions in a plasma using langevin equations the extension of langevin equation monte carlo algorithms for coulomb collisions from the conventional euler maruyama time integration to the next higher order of accuracy the milstein scheme has been developed implemented and tested this extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the fixed frame spherical coordinate velocity variables results from the numerical implementation show the expected improvement o dt vs o dt 1 2 in the strong convergence rate both for the speed v and angular components of the scattering an important result is that this improved convergence is achieved for the angular component of the scattering if and only if the area integral terms in the milstein scheme are included the resulting milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency these include both algorithms with improved convergence in the averages weak convergence and multi time level schemes the latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional monte carlo schemes and their performance is improved considerably when the milstein algorithm is used for the underlying time advance versus the euler maruyama algorithm a new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy this method while being useful in its own right because of its relative simplicity is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration

parameter estimation for partially observed hypoelliptic diffusions hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis we study parameter estimation for such processes in situations where we observe some components of the solution at discrete times since exact likelihoods for the transition densities are typically not known approximations are used that are expected to work well in the limit of small intersample times t and large total observation times n t hypoellipticity together with partial observation leads to ill conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated we combine these in a deterministic scan gibbs sampler alternating between missing data in the unobserved solution components and parameters numerical experiments illustrate asymptotic consistency of the method when applied to simulated data the paper concludes with an application of the gibbs sampler to molecular dynamics data

a note on strong approximation of sdes with smooth coefficients that have at most linearly growing derivatives abstract in the literature on strong approximation of sdes polynomial error rate results for numerical schemes are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition in the present article we show that the latter condition is far from being sufficient for polynomial error rates we construct an sde with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can at best be approximated with a logarithmic rate whatever numerical method based on observations of the driving brownian motion at finitely many fixed times is used most interestingly it furthermore turns out that using a method that properly adjusts the number of evaluations of the driving brownian motion to its actual path the latter sde can in fact be approximated with polynomial rate 1 in terms of the average number of evaluations that are used to the best of our knowledge this is only the second example in the literature of an sde for which there exist adaptive methods that perform superior to non adaptive ones with respect to the convergence rate

adaptive stepsize based on control theory for stochastic differential equations the numerical solution of stochastic differential equations sdes has been focussed recently on the development of numerical methods with good stability and order properties these numerical implementations have been made with fixed stepsize but there are many situations when a fixed stepsize is not appropriate in the numerical solution of ordinary differential equations much work has been carried out on developing robust implementation techniques using variable stepsize it has been necessary in the deterministic case to consider the best choice for an initial stepsize as well as developing effective strategies for stepsize control the same of course must be carried out in the stochastic case in this paper proportional integral pi control is applied to a variable stepsize implementation of an embedded pair of stochastic runge kutta methods used to obtain numerical solutions of nonstiff sdes for stiff sdes the embedded pair of the balanced milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change the extension of these stepsize controllers from a digital filter theory point of view via pi with derivative pid control will also be implemented the implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

quantum field theory of forward rates with stochastic volatility in a recent formulation of a quantum field theory of forward rates the volatility of the forward rates was taken to be deterministic the field theory of the forward rates is generalized to the case of stochastic volatility two cases are analyzed first when volatility is taken to be a function of the forward rates and second when volatility is taken to be an independent quantum field since volatility is a positive valued quantum field the full theory turns out to be an interacting nonlinear quantum field theory in two dimensions the state space and hamiltonian for the interacting theory are obtained and shown to have a nontrivial structure due to the manifold moving with a constant velocity the no arbitrage condition is reformulated in terms of the hamiltonian of the system and then exactly solved for the nonlinear interacting case

volterra equations driven by semimartingales existence and uniqueness of solutions is established for stochastic volterra integral equations driven by right continuous semimartingales this resolves in the affirmative a conjecture of m berger and v mizel

the stochastic string model as a unifying theory of the term structure of interest rates we present the stochastic string model of santa clara and sornette 2001 as reformulated by bueno guerrero et al 2015 as a unifying theory of the continuous time modeling of the term structure of interest rates we provide several new results such as a an orthogonality condition for the volatilities in the heath jarrow and morton 1992 hjm model b the interpretation of multi factor hjm models as approximations to a full infinite dimensional model c a result of consistency based on hilbert spaces and d a theorem for option valuation

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

stability of random attractors under perturbation and approximation abstract the comparison of the long time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals however one can still compare their asymptotic behaviour using the global attractor and this is now standard in the deterministic autonomous case for random dynamical systems there is an additional problem since the convergence of numerical methods for such systems is usually given only on average in this paper the deterministic approach is extended to cover stochastic differential equations giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi continuous with respect to a given family of perturbations or approximations

optimal approximation of stochastic differential equations by adaptive step size control we study the pathwise strong approximation of scalar stochastic differential equations with respect to the global error in the l2 norm for equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving brownian motion as a consequence higher order methods do not exist if the global error is analyzed we introduce an adaptive step size control for the euler scheme which performs asymptotically optimally in particular the new method is more efficient than an equidistant discretization this superiority is confirmed in simulation experiments for equations with additive noise as well as for general scalar equations

differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities let x and y be two continuous time martingales if quadratic variation of x minus that of y is a nondecreasing and nonnegative function of time we say that y is differentially subordinate to x and prove that y p p 1 x p for 1 p where p p v q and q is the conjugate of p this inequality contains burkholder s l p inequality for stochastic integrals which implies that the above inequality is sharp we also extend his concept of strong differential subordination and several other of his inequalities and sharpen an inequality of banuelos

a direct proof of the bichteler dellacherie theorem and connections to arbitrage we give an elementary proof of the celebrated bichteler dellacherie theorem which states that the class of stochastic processes s allowing for a useful integration theory consists precisely of those processes which can be written in the form s m a where m is a local martingale and a is a finite variation process in other words s is a good integrator if and only if it is a semi martingale we obtain this decomposition rather directly from an elementary discrete time doob meyer decomposition by passing to convex combinations we ob tain a direct construction of the continuous time decomposition which then yields the desired decomposition as a by product of our proof we obtain a characterization of semi martingales in terms of a variant of no free lunch thus extending a result from ds94

the central limit theorem for stochastic integrals with respect to levy processes let m be a symmetric independently scattered random measure on lbrack 0 1 rbrack with control measure m which is uniformly in the domain of normal attraction of a stable measure of index p in 0 2 rbrack let f be a non anticipating process with respect to x t m lbrack 0 t rbrack if m is continuous and a previsible process in general satisfying int 1 0 e f p dm infty then the stochastic integral int t 0 f dm can be defined as a process in d lbrack 0 1 rbrack and is in the domain of normal attraction of a stable process of order p in d lbrack 0 1 rbrack in the sense of of weak convergence of probability measures if m is gaussian and continuous in probability then the central limit theorem holds in c lbrack 0 1 rbrack in particular ito and diffusion processes satisfy the clt our main tool is an upper bound for the weak l p norm of sup 0 leq t leq 1 int t 0 f dm in terms of the l p p times m norm of f

reflected brownian motion in a wedge semimartingale property in this paper the object of study is reflected brownian motion in a two dimensional wedge with constant direction of reflection on each side of the wedge the basic question considered here is when is this process a semimartingale it is first shown that a related process defined by specifying the corner of the wedge to be an absorbing state rather than an instantaneous one is a semimartingale conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by vardhan and williams brownian motion in a wedge with oblique reflection comm pure appl math to appear under these conditions it is shown that starting away from the corner the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge this equivalence is also shown to hold starting from the corner except in one unresolved case for which the wedge angle exceeds and the directions of reflection are exactly opposed

successive approximations for solutions of stochastic integral equations of volterra type in an abstract wiener space i h b where w s is a b valued wiener process the first integral is in the sense of ito see kuo 5 and the second is in the sense of bochner the determinist case is discussed by chen 1 and a finite dimensional stochastic case is discussed by kravec 3 4 conditions which guarantee the convergence in maximal quadratic mean of picard s successive approximations to a solution of 1 1 are given our assumptions made on rp ki are like that of chen 11

pathwise stochastic control problems and stochastic hjb equations in this paper we study a class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise or information such a phenomenon can be illustrated by considering a particular investor who wants to take advantage of certain extra information but in a completely legal manner we show that such a control problem may not even have a minimizing sequence but nevertheless the bellman dynamical programming principle still holds we then show that the corresponding hamilton jacobi bellman equation is a stochastic partial differential equation as was predicted by lion and souganidis c r acad sci paris se r i math 327 1998 pp 735 741 our main device is a doss sussmann type transformation introduced in our previous work stochastic process appl 93 2001 pp 181 204 and stochastic process appl 93 2001 pp 205 228 with the help of such a transformation we reduce the pathwise control problem to a more standard relaxed control problem from which we are able to verify that the value function of the pathwise stochastic control problem is the unique stochastic viscosity solution to this stochastic partial differential equation in the sense of stochastic process appl 93 2001 pp 181 204 and stochastic process appl 93 2001 pp 205 228

valuation of caps and swaptions under a stochastic string model abstract we develop a gaussian stochastic string model that provides exact closed form expressions for the prices and hedging portfolios of caps and swaptions under certain conditions our pricing expressions reduce to black 1976 formulas we also propose a stochastic string libor market model that generalizes the models of brace et al 1997 and longstaff et al 2001a we provide a possible explanation for some problems involved in the relative valuation of these derivatives we also attain the observational equivalence of kerkhof and pelsser 2002 and prove that in our setup the models proposed by brace et al 1997 and longstaff et al 2001a are more parsimonious than stated in the original papers

on infinitely divisible semimartingales stricker s theorem states that a gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment gaussian process and a gaussian process of finite variation see stricker z wahrsch verw geb 64 3 303 312 1983 we consider extensions of this result to non gaussian infinitely divisible processes first we show that the class of infinitely divisible semimartingales is so large that the natural analog of stricker s theorem fails to hold then as the main result we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation consequently the natural analog of stricker s theorem holds for all strictly representable processes as defined in this paper since gaussian processes are strictly representable due to hida s multiplicity theorem the classical stricker s theorem follows from our result another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property when a certain associated infinitely divisible process is of finite variation this gives the key to characterize the semimartingale property for many processes of interest along these lines using basse o connor and rosinski stoch process appl 123 6 1871 1890 2013a we characterize semimartingales within a large class of stationary increment infinitely divisible processes this class includes many infinitely divisible processes of interest including linear fractional processes mixed moving averages and supou processes as particular cases the proof of the main theorem relies on series representations of jumps of cadlag infinitely divisible processes given in basse o connor and rosinski ann probab 41 6 4317 4341 2013b combined with techniques of stochastic analysis