Stochastic Analysis of Dynamical Systems, Stochastic Control and Games

For a fixed given deterministic final horizon T > 0, let us consider the following multidimensional SDE: dXt = b(t,Xt)dt+ σ(t,Xt)dWt, t ∈ [0, T ] (1) where b : [0, T ] × Rd → Rd, σ : [0, T ] × Rd → Rd ⊗ Rd are bounded coefficients that are measurable in time and Hölder continuous in space (this last condition will be possibly relaxed for the drift term b) and W is a Brownian motion on some filtered probability space (Ω,F , (Ft)t≥0). Also a(t, x) = σσ∗(t, x) is assumed to be uniformly elliptic. In particular those assumptions guarantee that (1) admits a unique weak solution. We now introduce, for a given ε > 0, a perturbated version of (1) with dynamics: dX (ε) t = bε(t,X (ε) t )dt+ σε(t,X (ε) t )dWt, t ∈ [0, T ] (2) STOCHASTIC ANALYSIS OF DYNAMICAL SYSTEMS, STOCHASTIC CONTROL AND GAMES5 where bε : [0, T ] × Rd → Rd, σε : [0, T ] × Rd → Rd ⊗ Rd satisfy at least the same assumptions as b, σ and are in some sense meant to be close to b, σ where ε is small. Our goal is to investigate how the closeness of (bε, σε) and (b, σ) is reflected on the respective densities of the associated processes. Speaker: Stephane Menozzi, University of Evry Title: Weak Error for Stable Driven SDEs with Hölder coefficients Abstract: We study the weak error of the Euler Scheme for a non-degenerate SDE driven by a symmetric stable process with bounded Hölder coefficients. We precisely quantify the distance between the densities of the SDE and its Euler scheme, establishing a convergence rate depending non trivially on both the Hölder exponent of the coefficients and the stability index of the driving process. Our analysis relies on pointwise estimates of the fractional derivatives of the underlying heat kernel in this setting. These controls have independent interest and seem, to our best knowledge, new. Joint Work with L. Huang (Michigan State University) Speaker: Jan Palczewski, University of Leeds Title: Impulse control maximising average cost per unit time: a nonuniformly ergodic case Abstract: I will talk about maximisation of an average-cost-per-unit-time ergodic functional over impulse strategies controlling a Feller-Markov process. The uncontrolled process is assumed to be ergodic but, unlike the extant literature, the convergence to invariant measure does not have to be uniformly geometric in total variation norm; in particular, I allow for non-uniform geometric or polynomial convergence. Cost of an impulse may be unbounded, e.g., proportional to the distance the process is shifted. I will show that the optimal value does not depend on the initial point and provide optimal or ε-optimal strategies. This talk is based on joint research with Lukasz Stettner. Speaker: Etienne Pardoux, University Aix-Marseille Title: Random evolution of a population in a changing environment Abstract: We consider a model of the evolution of a population, whose fitness, in the absence of mutations, degradates continuously, due to a constant modification of the ecological conditions (e.g. global warming). We superimpose mutations, which arise according to a Poisson random measure, and get fixed according to a probability which depends upon how much the proposed mutation will improve the fitness. We neglect the time taken for fixation of mutations, and assume that the population is constantly monomorphic. This leads us to consider an SDE driven by a Poisson Point Process, either in dimension 1 or in dimension 2. We study the large time behaviour of the 6 LEEDS, 24TH–26TH OCTOBER 2016 Markov solution of that equation (i.e. its transience/recurrence property). We also study the asymptotic of small/frequent mutations. This is joint work with Elma Nassar and Michael Kopp. Speaker: Goran Peskir, University of Manchester Title: Constrained Dynamic Optimality and Binomial Terminal Wealth Abstract: Assuming that the stock price follows a geometric Brownian motion and the bond price compounds exponentially, we recently derived a dynamically optimal control for the investor aiming to minimise the variance of his terminal wealth over all admissible controls such that the expectation of the terminal wealth is bounded below by a given constant. We showed that the dynamically optimal wealth process solves a meander type equation which makes the wealth process hit the given constant exactly at the terminal time. This was done under no pathwise constraint on the wealth process which could take low/negative values of unlimited size. In this talk we consider the analogous variance minimising problem upon imposing the guarantee that the (discounted) wealth process always stays above a given constant regardless of whether the investment is unfavourable. We show that the dynamically optimal wealth process can be characterised as the unique (strong) solution to a stochastic differential equation with time-dependent coefficients. By analysing this stochastic differential equation (extending Feller’s test to time-inhomogeneous diffusions that is of independent interest) we prove that the dynamically optimal terminal wealth can only take two values. This binomial nature of the dynamically optimal strategy stands in sharp contrast with other known portfolio selection strategies encountered in the literature. A direct comparison shows that the dynamically optimal (time-consistent) strategy outperforms the statically optimal (timeinconsistent) strategy in the minimising variance problem. Joint work with Jesper Pedersen (Copenhagen) Speaker: Alexey Piunovskiy, University of Liverpool Title: Constrained Optimal Control of Piecewise Deterministic Markov Processes (PDMP) Abstract: The PDMP X(t) in the state space X ⊂ Rd develops according to a fixed (uncontrolled) vector flow between the jumps, which occur according to the transition rate q(dy|x, a). Here and below, a ∈ A is the action (control). Other jumps take place when the process X(·) reaches the boundary of X. The after-jump state in the latter case realises according to the stochastic kernel Q(dy|x, a). Assume that C j (x, a) are the ”gradual” cost rates of the types j = 0, 1, . . . , p, to be integrated w.r.t. time, and Ci j(x, a) are the ”impulsive” costs at the jump epochs from the state x ∈ X. We investigate the constrained optimal control problem: minimize the total expected discounted cost of type j = 0 subject to inequalities on the similar functionals of types j = 1, 2, . . . , p. This problem is reformulated as STOCHASTIC ANALYSIS OF DYNAMICAL SYSTEMS, STOCHASTIC CONTROL AND GAMES7 a Linear Program in the space of so called occupation measures. Under the standard continuity-compactness conditions, the LP approach provides the solution. In the unconstrained case, when p = 0, the Dynamic Programming is another powerful method. The presented results were published in [SIAM J. Control Optim. 2016, V.54, pp.1444-1474]. One can find other models of PDMP and meaningful examples in [M.Davis. Markov Models and Optimization, Chapman and Hall, 1993] and in [O.Costa, F.Dufour. Continuous Average Control of Piecewise Deterministic Markov Processes. Springer, 2010]. Speaker: Agnes Sulem, INRIA Title: Game options in an imperfect financial market with default Abstract: We study pricing and superhedging strategies for game options in an imperfect financial market with default. We extend the results obtained by Kifer in the case of a perfect market model to the case of imperfections in the market taken into account via the nonlinearity of the wealth dynamics. In this framework, the pricing system is expressed as a nonlinear Eg-expectation/evaluation induced by a nonlinear Backward Stochastic Differential Equation (BSDE) with jump with driver g. We prove that the superhedging price of a game option coincides with the value function of a corresponding generalized Dynkin game expressed in terms of the Eg-evaluation. The proofs of these results are based on links between generalized Dynkin games and doubly reflected BSDEs. We then address the case of ambiguity on the model, for example an ambiguity on the default probability and characterize the superhedging price of a game option as the value function of a mixed generalized Dynkin game. Joint work with Roxana Dumitrescu (Humboldt-Universität zu Berlin) and Marie-Claire Quenez (Université Paris-Diderot). Speaker: Stephane Villeneuve, Toulouse School of Economics Title: Optimal stopping games with unknown drift Abstract: The paper studies the classical optimal stopping (or Dynkin) game for Markov process when the drift of the underlying is unobservable. It mainly focuses on the description of the optimal strategies for both players and gives sufficient conditions for the existence of a Nash equilibrium. Furthermore, the regularity of the value of the game -the so-called smooth-fit principleis analyzed. Speaker: Tomasz Zastawniak, University of York Title: Game options with gradual exercise and cancellation under transaction costs Abstract: We consider game (Israeli) options in a multi-asset market model 8 LEEDS, 24TH–26TH OCTOBER 2016 with proportional transaction costs in the case when the buyer is allowed to exercise the option and the seller has the right to cancel the option, both of them gradually, i.e. at a so-called mixed (or randomised) stopping time, rather than instantly at an ordinary stopping time. Allowing gradual exercise and cancellation leads to increased flexibility in hedging, and hence results in tighter bounds on the option price as compared to instant exercise and cancellation. Algorithmic constructions can be developed to compute the bid and ask prices and the associated hedging strategies and optimal mixed stopping times for both exercise and cancellation. Probabilistic dual representations for bid and ask prices can also be established in this setting. (joint with Alet Roux) Speaker: Mihail Zervos, London School of Economics Title: Dynamic contracting under moral hazard Abstract: We consider a contracting problem that a firm faces in the presence of managerial moral hazard and stochastic cashflows. We first develop a general contracting setting. We then restrict attention to contracts that admit appropriate state space representations. In the latter context, we establish the link between the optimal contract and the solution to a suitable stochastic control problem, which we explicitly solve. Speaker: Tusheng Zhang, University of Manchester Title: Lattice Approximations of Reflected Stochastic Partial Differential Equations Driven by Space-Time White Noise Abstract: We introduce a discretization/approximation scheme for reflected stochastic partial differential equations driven by space-time white noise through systems of reflecting stochastic differential equations. To establish the convergence of the scheme, we study the existence and uniqueness of solutions of Skorohod-type deterministic systems on time-dependent domains. We also need to establish the convergence of an approximation scheme for deterministic parabolic obstacle problems. 2. Contributed talks (alphabetical order) Speaker: Goncalo dos Reis, University of Edinburgh Title: Equilibrium pricing under relative performance concerns Abstract: We investigate the effects of the social interactions of a finite set of agents on an equilibrium pricing mechanism. A derivative written We investigate the effects of the social interactions of a finite set of agents on an equilibrium pricing mechanism. A derivative written on non-tradable underlyings is introduced to the market and priced in an equilibrium framework by agents who assess risk using convex dynamic risk measures expressed by Backward Stochastic Differential Equations (BSDE). Each agent is not only exposed to financial and non-financial risk factors, but she also faces performance concerns with respect to the other agents. Within our proposed model we prove the existence and uniqueness of an equilibrium STOCHASTIC ANALYSIS OF DYNAMICAL SYSTEMS, STOCHASTIC CONTROL AND GAMES9 whose analysis involves systems of fully coupled multi-dimensional quadratic BSDEs. We extend the theory of the representative agent by showing that a non-standard aggregation of risk measures is possible via weighted-dilated infimal convolution. We analyze the impact of the problem’s parameters on the pricing mechanism, in particular how the agents’ performance concern rates affect prices and risk perceptions. In extreme situations, we find that the concern rates destroy the equilibrium while the risk measures themselves remain stable. This is a joint work with Jana Bielagk (HU-Berlin) and Arnaud Lionnet (INRIA Paris). arXiv:1511.04218 Speaker: Miryana Grigorova, Centre for Risk and Insurance-Hannover Title: Reflected BSDEs and optimal stopping with g-expectations: beyond the right-continuous case Abstract: Backward stochastic differential equations (BSDEs) have found number of applications in finance, among which pricing and hedging of European options, recursive utilities, dynamic risk measures. Reflected backward stochastic differential equations (RBSDEs) can be seen as a variant of BSDEs in which the (first component of the) solution is constrained to remain greater than or equal to a given process called the obstacle. Compared to the case of (non-reflected) BSDEs, there is an additional nondecreasing predictable process which keeps the (first component of the) solution above the obstacle. RBSDEs have been introduced by El Karoui et al. (1997) in the case of a continuous obstacle and have proved useful, for instance, in the study of American options. There have been several extensions of this work to the case of a discontinuous obstacle in all of which an assumption of right-continuity on the obstacle is made. In the first part of the talk we consider a further extension of the theory of RBSDEs to the case where the obstacle is not necessarily right-continuous. Compared to the right-continuous case, the additional nondecreasing process, which “pushes” the (first component of the) solution to stay above the obstacle, is no longer right-continuous. We establish existence and uniqueness of the solution in appropriate Banach spaces. To prove our results we use tools from the general theory of processes such as Mertens decomposition of strong optional (but not necessarily right-continuous) supermartingales (generalizing DoobMeyer decomposition), some tools from optimal stopping theory, as well as a generalization of Itô’s formula due to Gal’chouk and Lenglart. In the second part of the talk we make some links between the RBSDEs presented in the first part and an optimal stopping problem in which the risk of a financial position ξ (where ξ is not assumed to be right-continuous) is assessed by a g-conditional expectation. Recall that g-conditional expectations are non-linear operators defined through (non-reflected) BSDEs. We characterize the “value function” of our optimal stopping problem in terms of the solution to the RBSDE of the first part. Under an additional assumption 10 LEEDS, 24TH–26TH OCTOBER 2016 on ξ, we show the existence of an optimal stopping time. If time permits, we will also present the case of Doubly Reflected BSDEs whose barriers are not right-continuous and some links with Dynkin games. The talk is based on a joint work with P. Imkeller, E. Offen, Y. Ouknine, and M.C. Quenez. Speaker: Joao Guerra, CEMAPRE and ISEG, University of Lisbon Title: Option pricing in jump-diffusion models with transaction costs and stochastic control Abstract: We present an approach for pricing a European call option in presence of proportional transaction costs, when the stock price follows a jump-diffusion model. The value of the option is found using the concept of indifference pricing. This requires the solution of two stochastic singular control problems in finite time, satisfying the same Hamilton-Jacobi-Bellman equation with different terminal conditions. We solve numerically the continuous time optimization problem, using the Markov chain approximation method. Numerical results are presented for the case of a stock dynamics following an exponential Merton jump-diffusion model. Speaker: Elena Issoglio, University of Leeds Title: FBSDEs with distributional coefficients Abstract: In this talk I will present some recent results about systems of forward-backward stochastic differential equations (FBSDEs) where some of the coefficients are Schwartz distributions, in particular they are elements of a fractional Sobolev space of negative order. A notion of virtual solution is introduced in order to make sense of the singular integrals that appear in the FBSDE. In this setting we show the validity of the so-called non-linear Feynman-Kac formula. (This talk is based on a joint work with Shuai Jing (ArXiv:1605.01558).) Speaker: Maike Klein, University of Jena Title: Three points suffice Abstract: We consider the problem of optimally stopping a one-dimensional continuous-time Markov process with a stopping time satisfying an expectaWe consider the problem of optimally stopping a one-dimensional continuous-time Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. In other words, stopping at three points is enough. For the proof we extend the balayage method of Chacon and Walsh for solving the Skorokhod embedding problem to general Markov processes. This talk is based on a joint work with Stefan Ankirchner, Nabil Kazi-Tani and Thomas Kruse. STOCHASTIC ANALYSIS OF DYNAMICAL SYSTEMS, STOCHASTIC CONTROL AND GAMES 11 Speaker: Martin Lopez Garcia, University of Leeds Title: Analysing Markov chains on networks: the SIR epidemic process as an example Abstract: When analysing a continuous-time Markov chain (CTMC) on a network, where the nodes in the network can be in M different states, and these states change randomly over time, the space of states of this Markov chain contains MN states. This makes any analytical or exact approach unfeasible for even moderate values of M or N . The aim in this talk is to present the SIR epidemic model as an example of this type of processes, and to show how to analyse this process when the population under study is formed by a small highly heterogeneous group of N individuals. This approach makes special focus on algorithmic issues, and requires a creative order for the states within the state space S = {S, I,R}N of the CTMC. We will illustrate this approach by studying the spread of the nosocomial pathogen Methicillin-resistant Staphylococcus Aureus among the patients within an intensive care unit (ICU). The interest here is in analysing the effectiveness of different control strategies which intrinsically incorporate heterogeneities among the patients within the ICU. Speaker: Mario Maurelli, WIAS & Technische Universität Berlin Title: Regularization by noise for scalar conservation laws Abstract: We say that a regularization by noise phenomenon occurs for a possibly ill-posed differential equation if this equation becomes well-posed (in a pathwise sense) under addition of noise. Most of the results in this direction are limited to SDEs and associated linear SPDEs. In this talk, we show a regularization by noise result for a nonlinear SPDE, namely a stochastic scalar conservation law on Rd with a space-irregular flux: ∂tv + b · ∇[v] +∇v ◦ Ẇ = 0, where b = b(x) is a given deterministic, possibly irregular vector field, W is a d-dimensional Brownian motion (◦ denotes Stratonovich integration) and v = v(t, x, ω) is the scalar solution. More precisely we prove that, under suitable Sobolev assumptions on b and integrability assumptions on its divergence, the equation admits a unique entropy solution. The result is false without noise. The proof of uniqueness is based on a careful combination of arguments used in the linear case: first we show the renormalization property for the kinetic formulation of the equation, then we use second order PDE estimates and a duality argument to conclude. If time permits, we will discuss also some open questions. Joint work with Benjamin Gess 12 LEEDS, 24TH–26TH OCTOBER 2016 Speaker: Neofytos Rodosthenous Title: Optimal stopping problems with random time-horizon under spectrally negative Lévy models Abstract: We study the optimal exercising of an American call option with a random time-horizon under exponential spectrally negative Lévy models. The random time-horizon is modelled as the so-called Omega default clock in insurance, which is the first time when the occupation time of the Lévy process below a level y exceeds an independent exponential time with mean q. We show that the shape of the value function and exercise strategy vary qualitatively with different values of y and q. In particular, we show that apart from the optimality of traditional up-crossing strategies, we may have two disconnected waiting regions for certain values of y and q, resulting in the optimality of two-sided exercise strategies. We give a complete characterisation of all optimal exercise thresholds. Speaker: Alet Roux, University of York Title: Derivative pricing with minimization of exponential regret under proportional transaction costs Abstract: We consider an investor with a position in a derivative (represented by a payment stream in discrete time), who is allowed to inject additional funds over the lifetime of the derivative to ”top up” their position in the underlying assets. The total regret (disutility, or negative utility) of these injections can be used as a measurement of investment performance, which allows the formulation of an optimal investment problem, leading to indifference pricing. Our setting contains classical utility indifference pricing of European options as a special case. In a discrete time model with finite state space, we present a dynamic programming procedure for computing the indifference price. The dynamic programming procedure arises from a dual formulation of the indifference pricing problem, which involves the sum of the relative entropy of martingale measures with respect to the real-world probability. We also present some numerical examples. This is joint work-in-progress with Zhikang Xu. Speaker: Jacco Thijssen, University of York Title: Quick or Persistent? On the Feedback Effects between First and Second Mover Advantages in a Stochastic Investment Game Abstract: We analyse a dynamic model of investment under uncertainty in a duopoly where firms have an option to switch from one market to another. We construct a subgame perfect equilibrium in Markovian mixed strategies and show that both preemption and attrition can occur along typical equilibrium paths. This changes the nature of stopping problems to be solved, compared to existing strategic real option models. Equilibrium outcomes STOCHASTIC ANALYSIS OF DYNAMICAL SYSTEMS, STOCHASTIC CONTROL AND GAMES 13 differ qualitatively from those of the model’s deterministic version. Competition endogenously determines the firms’ exposure to the two markets’ risk factors, one of which is indeed eliminated by the mixed strategies in equilibrium. (Joint with Jan-Henrik Steg, Center for Mathematical Economics, Bielefeld University) Speaker: Tiziano Vargiolu, University of Padova Title: Verification theorem for stochastic impulse non-zero sum games and applications Abstract: In this talk, we provide a general framework for impulse stochastic differential nonzero sum games. Within this setting, we investigate the notion of Nash equilibrium through the corresponding quasi-variational inequalities (QVI). Related former works address the case of non-zero sum optimal stopping games (Bensoussan-Friedman 1977) and zero-sum stochastic differential games with impulse control (Cosso 2013), but to our knowledge, the class of non-zero sum stochastic impulse games have not yet been addressed in the literature in its generality. We fill this gap by providing the right system of QVI with the right smoothness required to have both classical solutions and working examples. Finally, we present some applications.(joint with R. Aid, M. Basei, G. Callegaro, L. Campi) 3. Posters (alphabetical order) Speaker: Alessandro Balata, University of Leeds Title: Regress Later Monte Carlo for Controlled Diffusions Abstract: In this poster I will present a method to avoid the curse of dimensionality introduced in regression Monte Carlo methods by the endogeneity of controlled diffusions (i.e. we avoid the discretisation of the state space). We build on the method introduced in Kharroubi et al. (2013) and Langrene at al. (2015) without however using any control randomization; rather, we use regression to decouple the value of the controlled diffusion in one time step to the following one allowing the backward decision on the control to be dependent on the present state only and the future information to be used solely for training. The advantages provided by the greater flexibility of the algorithm comprise: computation of only one regression per time step; an enhanced policy iteration algorithm for the case in which the control only acts on degenerate components of the diffusion; small number of sample points required to achieve precision comparable to other methods. I will motivate this approach showing two examples: one related with the management of the state of charge of a battery used for grid application, the other related with the control of the output of a power plant in order to match its committed capacity and minimise its costs. 14 LEEDS, 24TH–26TH OCTOBER 2016 Speaker: Hinesh Chotai, Imperial College Title: FBSDEs and applications to carbon emissions markets Abstract: Forward-backward stochastic differential equations (FBSDEs) are a class of dynamical system with a wide range of applications in areas such as stochastic control and mathematical finance. At the same time, carbon markets are currently being implemented worldwide to mitigate the effects of climate change. Motivated by the work of Carmona, R., Delarue, F., Espinosa, G.E. and Touzi, N., ‘Singular forward-backward stochastic differential equations and emissions derivatives’, and related works, this poster exhibits a class of models for cap-and-trade schemes for which the solution to the pricing problem arises from the solution of an appropriate coupled FBSDE. The main application is in modelling a (single-period) emissions trading system such as the system in force in the European Union (EU). In contrast to the classes of FBSDEs considered in most of the literature, these FBSDEs are significant in two ways: the terminal condition of the backward equation is given by a discontinuous function of the terminal value of the forward equation, and the forward dynamics may not be strongly elliptic, not even in the neighbourhood of the singularities of the terminal condition. After presenting an overview of the main existence result for this class of FBSDE, the economic context of the model will be explained and some numerical results presented. Speaker: Zeyu He, University of Leeds Title: Time-consistent Mean-variance optimisation in discrete time Abstract:We study the time-consistent equilibrium control for a meanvariance (MV) problem in discrete time. We have characterized the discrete time condition for open-loop control and compared it with the closed-loop control. The result turns out to be that the open-loop control is always more risky than the closed-loop at any time. We argue this is because of the different short-term investment strategies for open-loop and closed-loop equilibriums. Also, since the equilibrium control is derived backwards from the maturity time, we study the asymptotical behaviour as time goes backwards to infinite. The recent study on equilibrium control for both open-loop and closed-loop both indicate that, as time goes infinite, the control would decrease to 0, and increase rapidly as the time goes closed to the mature. We argue that this observation is unrealistic from an economic point of view. This result suggests the investor to make a completely different investment strategy even with the same business situation. To deal with this situation, we suggest to re-scale the original MV functional by adjusting the risk-aversion and study such infinitely games with present-biased (β-δ) preference. Speaker: Elena Karachanskaya, Far Eastern State Transport University, Khabarovsk STOCHASTIC ANALYSIS OF DYNAMICAL SYSTEMS, STOCHASTIC CONTROL AND GAMES 15 Title: The Itô Wentzell formula with jumps (generalized Ito Wentzell formula): proofs and applications Abstract: In this report I will tell about a few approaches for obtaining the generalized Ito Wentzell formula (GIWF). We have result for a multidimensional case and for an Ito’s equation with Wiener process and Poisson’s jumps together. The first version of the GIWF was represented by Valery Doobko in 2002 (for equation Ito with centered Poisson measure), the one for non centered PM was obtained in 2011 (E. K.) and we got another versions for this formula later. All approaches are different and rigorously proven. Next, I will show an application of the GIWF for constructing a program control with Prob. 1 for stochastic systems and use of the GIWF in studies of diffusion in coherent random fields. Speaker: David Zoltan Szabo Title: Pricing put options for electrical power systems balancing reserve Abstract:TBA Speaker: Hynek Walner, Czech Academy of Sciences Title: optimal stochastic control of tumor growth Abstract: We study a model of non-linear tumor growth, which is governed by controlled SDE corresponding to Gompertz growth. We attain both analytical results using Potryagin’s principle for simplified form and numerical results for various control functions using Forward-Backward scheme for stochastic control problems. Speaker: Zimeng Wang, University of Nottingham Title: Auxiliary HJB Equation in Stochastic Controls with Delay Abstract: This work uses the method of dynamic programming to investigate a class of stochastic optimal control problems, where the state equations are described by stochastic differential equations with (time) delay and the dependence on the pasts is only through a single point X(s− δ) of the state process. We introduce an auxiliary function associated with the value function of the delayed control problem and then apply dynamic programming to derive a finite-dimensional Hamilton-Jacobi-Bellman (HJB) equation satisfied by the auxiliary function. We give the corresponding stochastic verification theorem and obtain the connection to the corresponding stochastic maximum principle. Furthermore, we apply the result to a delayed linearquadratic (LQ) stochastic optimal control problem with numerical results.