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The aim of this paper is to analyse a WIS-stochastic differential equation driven by fractional Brownian motion with H>0.5. For this, we summarise the theory of fractional white noise and prove a fundamental L^2-estimate for WIS-integrals. We apply this to prove the existence and uniqueness of a solution in L^2(P) of a WIS-stochastic differential equation driven fractional Brownian motion with H>0.5 under Lipschitz conditions on its coefficients.

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean-Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon-Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process. For some particular McKean-Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.

We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $\alpha \in (0,2)$. We consider the equation in the sense of distribution, and we find an explicit expression for the $\mathcal{S}'$-valued solution $Y(t,x)$, where $\mathcal{S}'$ is the space of tempered distributions. Following the terminology of Y. Hu \cite{Hu}, we say that the solution is \emph{mild} if $Y(t,x) \in L^2(\mathbb{P})$ for all $t,x$, where $\mathbb{P}$ is the probability law of the underlying time-space Brownian motion. It is well-known that in the classical case with $\alpha = 1$, the solution is mild if and only if the space dimension $d=1$. We prove that if $\alpha \in (1,2)$ the solution is mild if $d=1$ or $d=2$. If $\alpha < 1$ we prove that the solution is not mild for any $d$.

In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida-Malliavin calculus, which are useful tools for studying such equations. As an illustration I apply white noise calculus to find an explicit solution of an SPDE describing population growth in an environment subject to time-space white noise.

3rd edition; International audience; In this third edition, we have expanded and updated the second edition and includedmore recent developments within stochastic control and its applications.Specifically, we have replaced Section1.5on application tofinance by a morecomprehensive presentation offinancial markets modeled by jump diffusions (thenew Chap.2). We have added a new chapter on backward stochastic differentialequations, convex risk measures, and recursive utilities (Chap.4). Moreover, wehave expanded the optimal stopping chapter (was Chap. 2, now Chap.3) and thestochastic control chapter (was Chap. 3, now Chap.5) and added a new chapter onstochastic differential games (Chap.6). In addition, we have corrected errors andupdated and improved the presentation throughout the book.

The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of the order ow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.

In this paper we present a new verification theorem for optimal stopping problems for Hunt processes. The approach is based on the Fukushima-Dynkin formula, and its advantage is that it allows us to verify that a given function is the value function without using the viscosity solution argument. Our verification theorem works in any dimension. We illustrate our results with some examples of optimal stopping of reflected diffusions and absorbed diffusions.

The purpose of these lectures is threefold: We first give a short survey of the Hida white noise calculus, and in this context we introduce the Hida-Malliavin derivative as a stochastic gradient with values in the Hida stochastic distribution space $(\mathcal{S}% )^*$. We show that this Hida-Malliavin derivative defined on $L^2(\mathcal{F}_T,P)$ is a natural extension of the classical Malliavin derivative defined on the subspace $\mathbb{D}_{1,2}$ of $L^2(P)$. The Hida-Malliavin calculus allows us to prove new results under weaker assumptions than could be obtained by the classical theory. In particular, we prove the following: (i) A general integration by parts formula and duality theorem for Skorohod integrals, (ii) a generalised fundamental theorem of stochastic calculus, and (iii) a general Clark-Ocone theorem, valid for all $F \in L^2(\mathcal{F}_T,P)$. As applications of the above theory we prove the following: A general representation theorem for backward stochastic differential equations with jumps, in terms of Hida-Malliavin derivatives; a general stochastic maximum principle for optimal control; backward stochastic Volterra integral equations; optimal control of stochastic Volterra integral equations and other stochastic systems.

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. - In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. - For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

The continuous-time version of Kyle's (1985) model of asset pricing with asymmetric information is studied, and generalized by allowing time-varying noise trading. From rather simple assumptions we are able to derive the optimal trade for an insider; the trading intensity satisfies a deterministic integral equation, given perfect inside information, which we give a closed form solution to. We use a new technique called forward integration in order to find the optimal trading strategy. This is an extension of the stochastic integral which takes account of the informational asymmetry inherent in this problem. The market makers' price response is found by the use of filtering theory. The novelty is our approach, which could be extended in scope.

The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of the order flow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.

We study optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in case of partial information control. One important novelty of our problem is represented by the introduction of general mean-field operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove the existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We apply our results to find the explicit optimal control for an optimal harvesting problem.

We present a new approach to the optimal portfolio problem for an insider with logarithmic utility. Our method is based on white noise theory, stochastic forward integrals, Hida-Malliavin calculus and the Donsker delta function.

International audience

International audience

We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida-Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.

We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process $X(t)$ and a \emph{predictive mean-field} backward SDE (BSDE) in the unknowns $Y(t), Z(t), K(t,\cdot)$. The driver of the BSDE at time $t$ may depend not just upon the unknown processes $Y(t), Z(t), K(t,\cdot)$, but also on the predicted future value $Y(t+\delta)$, defined by the conditional expectation $A(t):= E[Y(t+\delta) | \mathcal{F}_t]$. \\ We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems:\\ (i) Optimal portfolio in a financial market with an \emph{insider influenced asset price process.} \\ (ii) Optimal consumption rate from a cash flow modeled as a geometric It\^ o-L\' evy SDE, with respect to \emph{predictive recursive utility}.

We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow $(\varepsilon_t)_{t\geq0}\subseteq(\mathcal{F}_t)_{t\geq0}$. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the problem of maximizing the expected utility of terminal wealth has a solution up to a stopping time, if and only if the marginal utility of the terminal wealth is the density of a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control under partial information. We then show that the financial market is globally viable if and only if there exists a partial information local martingale deflator (PILMD), which can be explicitly constructed. In the case of bounded coefficients, the latter turns out to be the density process of a global PIEMM. We illustrate our results by means of an explicit example.; On considère un marché financier constitué d'un actif risqué dont le prix est modélisé par un processus de diffusion avec saut à coefficients bornés. On suppose que les investisseurs n'ont accès qu'à une information partielle $(\varepsilon_t)_{t\geq0}\subseteq(\mathcal{F}_t)_{t\geq0}$. Pour toute fonction d'utilité on démontre que ce marché est localement viable, dans le sens où le problème de maximisation de l'espérance de l'utilité de la richesse terminale a une solution jusqu'à un temps d'arrêt si et seulement si l'utilité marginale de la richesse terminale est la densité d'une mesure martingale équivalente sous information partielle. On prouve cette équivalence de manière constructive au moyen de principes de maximum stochastique pour le contrôle en information partielle. On démontre ensuite que le marché financier est globalement viable si et seulement s'il existe un déflateur martingale local sous information partielle, que l'on peut construire explicitement. Dans le cas de coefficients bornés, ce dernier correspond au processus de densité d'une mesure martingale équivalente sous information partielle. Nous illustrons ces résultats sur un exemple.

International audience

International audience

This paper studies singular mean field control problems and singular mean field stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Applications are given to optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and to mean-field singular investment games. In particular, a simple singular mean-field investment game is studied where the Nash equilibrium exists but is not unique.

This paper is mainly a survey of recent research developments regarding methods for risk minimization in financial markets modeled by It\^o-L\'evy processes, but it also contains some new results on the underlying stochastic maximum principle. The concept of a convex risk measure is introduced, and two representations of such measures are given, namely: (i) the dual representation and (ii) the representation by means of backward stochastic differential equations (BSDEs) with jumps. Depending on the representation, the corresponding risk minimal portfolio problem is studied, either in the context of stochastic differential games or optimal control of forward-backward SDEs. The related concept of recursive utility is also introduced, and corresponding recursive utility maximization problems are studied. In either case the maximum principle for optimal stochastic control plays a crucial role, and in the paper we prove a version of this principle which is stronger than what was previously known. The theory is illustrated by examples, showing explicitly the risk minimizing portfolio in some cases.

By permission. Copyright (c) Applied Probability Trust (2014). We consider explicit formulae for equilibrium prices in a continuous-time vertical contracting model. A manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits. Demand is an Itô–Lévy process, and to increase realism, information is delayed. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein– Uhlenbeck process, both with time-variable coefficients. Moreover, explicit solution formulae are given, so these results are operational. An interesting finding is that information that is more precise may be a considerable disadvantage for the retailer. time-variable coeficients. Moreover, these results are operational because we are able to offer explicit solution formulas. An interesting finding is that information that is more precise may be a considerable disadvantage for the retailer.

We study optimal stochastic control problems of general coupled systems of forward-backward stochastic differential equations with jumps. By means of the It\^o-Ventzell formula the system is transformed to a controlled backward stochastic partial differential equation (BSPDE) with jumps. Using a comparison principle for such BSPDEs we obtain a general stochastic Hamilton-Jacobi- Bellman (HJB) equation for such control problems. In the classical Markovian case with optimal control of jump diffusions, the equation reduces to the classical HJB equation. The results are applied to study risk minimization in financial markets.

25 pages; A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: \begin{myenumerate} \item The optimal terminal wealth $X^*(T) : = X_{\varphi^*}(T)$ of the classical problem to maximize the expected $U$-utility of the terminal wealth $X_{\varphi}(T)$ generated by admissible portfolios $\varphi(t); 0 \leq t \leq T$ in a market with the risky asset price process modeled as a semimartingale \item The optimal scenario $\frac{dQ^*}{dP}$ of the dual problem to minimize the expected $V$-value of $\frac{dQ}{dP}$ over a family of equivalent local martingale measures $Q$. Here $V$ is the convex dual function of the concave function $U$. \end{myenumerate} In this paper we consider markets modeled by Itô-Lévy processes, and in the first part we give a new proof of the above result in this setting, based on the maximum principle in stochastic control theory. An advantage with our approach is that it also gives an explicit relation between the optimal portfolio $\varphi^*$ and the optimal measure $Q^*$, in terms of backward stochastic differential equations. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a relation between them. In particular, we show explicitly how to get from the solution of one of the problems to the solution of the other. We illustrate the results with explicit examples.

We study optimal stochastic control problems of general coupled systems of forward- backward stochastic di erential equations with jumps. By means of the Itô-Ventzell formula the system is transformed to a controlled backward stochastic partial di eren- tial equation (BSPDE) with jumps. Using a comparison principle for such BSPDEs we obtain a general stochastic Hamilton-Jacobi- Bellman (HJB) equation for such control problems. In the classical Markovian case with optimal control of jump di usions, the equation reduces to the classical HJB equation. The results are applied to study risk minimization in nancial markets.

We consider a problem of optimal control of an infinite horizon system governed by forward-backward stochastic differential equations with delay. Sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived. We illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay.

We prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information. The results are applied to partial information optimal consumption and portfolio problems in infinite horizon.

We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results by an application to the optimal consumption rate from an economic quantity.

In this paper, we prove a maximum principle for general stochastic differential Stackelberg games, and apply the theory to continuous time newsvendor problems. In the newsvendor problem, a manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits under a random demand rate. Our demand rate is an Itô-Lévy process, and to increase realism information is delayed, e.g., due to production time. A special feature of our time-continuous model is that it allows for a price-dependent demand, thereby opening for strategies where pricing is used to manipulate the demand. Revised edition 7 February 2012.

The single auction equilibrium of Kyle's (1985) is studied, in which noise traders may be partially informed, or alternatively they can be manipulated. Unlike Kyle's assumption that the quantity traded by the noise traders is independent of the asset value, we assume that the noise traders are able to correlate their trade with the true price. This has several implications for the equilibrium, one being that the insider's expected profits decrease as the noise traders' ability to correlate positively improve. In the limit, the noise traders do not lose on average, and the insider makes zero expected profits. When the correlation is negative, we interpret this as manipulation. In this case the insider makes the highest expected profits, and the informativeness of prices is at its minimum.

The single auction equilibrium of Kyle's (1985) is studied, in which noise traders may be partially informed, or alternatively they can be manipulated. Unlike Kyle's assumption that the quantity traded by the noise traders is independent of the asset value, we assume that the noise traders are able to correlate their trade with the true price. This has several implications for the equilibrium, one being that the insider's expected pro ts decrease as the noise traders' ability to correlate positively improve. In the limit, the noise traders do not loose on average, and the insider makes zero expected pro ts. When the correlation is negative, we interpret this as manipulation. In this case the insider makes the highest expected pro ts, and the informativeness of prices is at its maximum.

The continuous-time version of Kyle's [6] model, known as the Back's [2] model, of asset pricing with asymmetric information, is studied. A larger class of price processes and a larger classes of noise traders' processes are studied. The price process, as in Kyle's [6] model, is allowed to depend on the path of the market order. The process of the noise traders' is considered to be an inhomogeneous Lévy process. The solutions are found with the use of the Hamilton-Jacobi-Bellman equations. With the informed agent being risk-neutral, the price pressure is constant over time, and there is no equilibirium in the presence of jumps. If the informed agent is risk-averse, there is no equilibirium in the presence of either jumps or drift in the process of the noise traders'.

The continuous-time version of Kyle's (1985) model of asset pricing with asymmetric information is studied, and generalized in various directions, i.e., by allowing time-varying liquidity trading, and by having weaker a priori assumptions on the model. This extension is made possible by the use of filtering theory. We derive the optimal trade for an insider and the corresponding price of the risky asset; the insider's trading intensity satisfies a deterministic integral equation, given perfect inside information.

In this paper we introduce Skorohod-semimartingales as an expanded concept of classical semimartingales in the setting of Lévy processes. We show under mild conditions that Skorohod-semimartingales similarly to semimartingales admit a unique decomposition.

In this paper we suggest a general stochastic maximum principle for optimal control of anticipating stochastic differential equations driven by a Lévy type of noise. We use techniques of Malliavin calculus and forward integration. We apply our results to study a general optimal portfolio problem of an insider. In particular, we find conditions on the insider information filtration which are sufficient to give the insider an infinite wealth. We also apply the results to find the optimal consumption rate for an insider.

In the first part of the paper, we obtain existence and characterizations of an optimal control for a linear quadratic control problem of linear stochastic Volterra equations. In the second part, using the Malliavin calculus approach, we deduce a general maximum principle for optimal control of general stochastic Volterra equations. The result is applied to solve some stochastic control problem for some stochastic delay equations.

In this paper, we study backward stochastic differential equations with respect to general filtrations. The results are used to find the optimal consumption rate for an insider from a cash flow modeled as a generalized geometric Itô-Lévy process.

In this paper we study the Cauchy problem for the wave equation with space-time Lévy noise initial data in the Kondratiev space of stochastic distributions. We prove that this problem has a strong and unique C2-solution, which takes an explicit form. Our approach is based on the use of the Hermite transform.