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We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best [Formula: see text] of all states receive a fixed prize. Within the mean field limit version of the game, we compute an explicit equilibrium, a threshold strategy that consists of choosing the maximal fluctuation intensity when the state is below a given threshold and the minimal intensity otherwise. We show that for large n, the symmetric n-tuple of the threshold strategy provides an approximate Nash equilibrium of the n-player game. We also derive the rate at which the approximate equilibrium reward and the best-response reward converge to each other, as the number of players n tends to infinity. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two-player case. Funding: Support from the Deutsche Forschungsgemeinschaft [Grant AN 1024/5-1] is acknowledged. The research of N. Kazi-Tani is supported by the Agence Nationale de la Recherche [Grant ANR-21-CE46-0002-03]. The research of C. Zhou is supported by the National Natural Science Foundation of China [Grant 11871364], the Singapore Ministry of Education [Grants A-8000453-00-00, A-0004273-00-00 and A-0004277-00-00] and the MERLION 2020 award [Grant A-0004589-00-00].

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

International audience; We consider the problem of controlling the diffusion coefficient of a diffusion with constant negative drift rate such that the probability of hitting a given lower barrier up to some finite time horizon is minimized. We assume that the diffusion rate can be chosen in a progressively measurable way with values in the interval [0, 1]. We prove that the value function is regular, concave in the space variable, and that it solves the associated HJB equation. To do so, we show that the heat equation on a right triangle, with a boundary condition that is discontinuous in the corner, possesses a smooth solution.

We consider a stochastic control problem with time-inhomogeneous linear dynamics and a long-term average quadratic cost functional. We provide sufficient conditions for the problem to be well-posed. We describe an explicit optimal control in terms of a bounded and non-negative solution of a Riccati equation on $$[0, \infty )$$ [ 0 , ∞ ) , without an initial and terminal condition. We show that, in contrast to the time-homogeneous case, in the inhomogeneous case the optimally controlled state dynamics are not necessarily ergodic.

We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward–backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a monotonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.

We provide a weak law of large numbers for arrays of nonnegative and pairwise negatively associated (in each row) random variables under a rather weak domination condition.

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of coin tossing Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of $1/4$ with respect to every $p$-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than $1/4$. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points., Comment: To appear in J. Math. Anal. Appl

We consider the problem of optimally stopping a continuous-time process with a stopping time satisfying a given expectation cost constraint. We show, by introducing a new state variable, that one can transform the problem into an unconstrained control problem and hence obtain a dynamic programming principle. We characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order. We prove a classical verification theorem and illustrate its applicability with several examples.

We study a stopping problem arising from a sequential testing of two simple hypotheses H0 and H1 on the drift rate of a Brownian motion. We impose an expectation constraint on the stopping rules allowed and show that an optimal stopping rule satisfying the constraint can be found among the rules of the following type: stop if the posterior probability for H1 attains a given lower or upper barrier; or stop if the posterior probability comes back to a fixed intermediate point after a sufficiently large excursion. Stopping at the intermediate point means that the testing is abandoned without accepting H0 or H1. In contrast to the unconstrained case, optimal stopping rules, in general, cannot be found among interval exit times. Thus, optimal stopping rules in the constrained case qualitatively differ from optimal rules in the unconstrained case.

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process in natural scale can be approximated with such Markov chains arbitrarily well. The functional limit theorem applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. Our results allow to practically approximate such processes with irregular behavior; we illustrate this with Markov processes exhibiting sticky features, e.g., sticky Brownian motion and a Brownian motion slowed down on the Cantor set., To appear in Ann. Inst. Henri Poincar\'e Probab. Stat

We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form $d A_t =\mu (t, A_t) d t + \sigma(t, A_t) d W_t$. We provide sufficient conditions guaranteeing that for a given probability measure $\nu$ on $\mathbb{R}$ there exists a bounded stopping time $\tau$ and a real $a$ such that the solution $(A_t)$ of the SDE with initial value $a$ satisfies $A_\tau \sim \nu$. We hereby distinguish the cases where $(A_t)$ is a solution of the SDE in a weak or strong sense. Our construction of embedding stopping times is based on a solution of a fully coupled forward-backward SDE. We use the so-called method of decoupling fields for verifying that the FBSDE has a unique solution. Finally, we sketch an algorithm for putting our theoretical construction into practice and illustrate it with a numerical experiment., Comment: 39 pages, 2 pictures, To appear in Annales de l'Institut Henri Poincare (B) Probability and Statistics

We consider the problem of how to optimally close a large asset position in a market with a linear temporary price impact. We take the perspective of an agent who obtains a signal about the future price evolvement. By means of classical stochastic control we derive explicit formulas for the closing strategy that minimizes the expected execution costs. We compare agents observing the signal with agents who do not see it. We compute explicitly the expected additional gain due to the signal, and perform a comparative statics analysis.

We consider the dynamic control problem of attaining a target position at a finite time T , while minimizing a linear-quadratic cost functional depending on the position and speed. We assume that the coefficients of the linearquadratic cost functional are stochastic processes adapted to a Brownian filtration. We provide a probabilistic solution in terms of two coupled backward stochastic differential equations possessing a singularity at the terminal time T . We verify optimality of the candidate control by using a penalization argument. Special cases for which the problem has explicit solutions are discussed. Finally we illustrate our results in financial applications, where we derive optimal trading strategies for closing financial asset positions in markets with stochastic price impact and non-zero returns.

We consider a diffusion control problem, where the controller totally determines the state's diffusion coefficient but has no influence on the state's drift rate. By using the Pontryagin maximum principle we characterize an optimal control in terms of the adjoint forward-backward stochastic differential equation (FBSDE), turning out to be fully coupled. We use the method of decoupling fields for proving that the adjoint FBSDE possesses a solution.

We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between \(\sigma _1\) and \(\sigma _2\), where \(0 < \sigma _1 < \sigma _2\). The optimal control consists in choosing the minimal volatility \(\sigma _1\) when the process is above the threshold, and the maximal volatility if it is below. We give a rigorous proof using classical verification and provide integral formulas for the maximal expected occupation time above the threshold.

Soit $X_{1},X_{2},\ldots$ une suite de variables aleatoires independantes avec esperance $E(X_{i})=0$, et $Y^{N}_{k+1}=Y^{N}_{k}+a_{N}(Y^{N}_{k})X_{k+1}$ une marche aleatoire renormalisee avec une fonction $a_{N}:\mathbb{R}\to\mathbb{R}_{+}$. On montre, sous certaines conditions legeres sur la loi de $X_{i}$, que l’on peut choisir le facteur $a_{N}$ d’une facon que $(Y^{N}_{\lfloor Nt\rfloor})_{t\in\mathbb{R}_{+}}$ converge en loi, quand $N$ tend vers l’infini, vers une diffusion $(M_{t})_{t\in\mathbb{R}_{+}}$ etant la solution d’une equation differentielle stochastique avec des coefficients irreguliers. A cet effet, nous plongeons la marche aleatoire renormalisee dans la diffusion $M$ par une suite de temps d’arret ayant un pas de temps avec esperance $1/N$.

We consider the problem of optimally stopping a one-dimensional regular continuous strong 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. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a backward stochastic differential equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [Stochastic Process. Appl., 116 (2006), pp. 2014--2056] and [Ann. Probab., 35 (2007), pp. 1071--1117]. We perform a verification and discuss special cases for which the control problem has explicit solutions.

International audience; We set up a game theoretical model to analyze the optimal attacking intensity of sports teams during a game. We suppose that two teams can dynamically choose among more or less offensive actions and that the scoring probability of each team depends on both teams' actions. We assume a zero sum setting and characterize a Nash equilibrium in terms of the unique solution of an Isaacs equation. We present results from numerical experiments showing that towards the end of game it is optimal for the loosing team to start playing more offensively and for the winning team to play more defensively. This is even the case if the effects of the action changes cancel out so that the scoring intensities remain the same.

Consider an agent with a forward position of an illiquid asset (e.g. a commodity) that has to be closed before delivery. Suppose that the liquidity of the asset increases as the delivery date approaches. Assume further that the agent has two possibilities for hedging the risk inherent in the forward position: first, he can enter customized forward contracts; second, he can acquire standardized and liquidly traded forward contracts. We assume that purchasing customized forwards perfectly eliminates the risk, but entails high liquidity costs charged by the counterparty. The standardized forwards can be acquired at considerably lower costs, but do not perfectly match the agent's risk and hence entail basis risk. By means of stochastic control we show how to obtain an optimal trade-off between liquidity costs and basis risk. To this end we reduce the hedging problem to a family of stopping problems. In two case studies we consider simple liquidity dynamics for which optimal hedging strategies can be calculated explicitly.

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.

When managing risk, frequently only imperfect hedging instruments are at hand. We show how to optimally cross-hedge risk when the spread between the hedging instrument and the risk is stationary. For linear risk positions we derive explicit formulas for the hedge error, and for nonlinear positions we show how to obtain numerically efficient estimates. Finally, we demonstrate that even in cases with no clear-cut decision concerning the stationarity of the spread, it is better to allow for mean reversion of the spread rather than to neglect it.

This article deals with the Skorokhod embedding problem in bounded time for the Brownian motion with drift Xt = κt + Wt: Given a probability measure μ we aim at finding a stopping time τ such that the law of Xτ is μ, and τ is almost surely smaller than some given fixed time horizon T > 0. We provide necessary and sufficient conditions on the distribution μ for the existence of such bounded stopping times.

The problem of finding the mean variance optimal portfolio in a multiperiod model can not be solved directly by means of dynamic programming. In order to find a solution we therefore first introduce independent market clones having the same distributional properties as the original market, and we replace the portfolio mean and variance by their empirical counterparts. We then use dynamic programming to derive portfolios maximizing a weighted sum of the empirical mean and variance. By letting the number of market clones converge to infinity we are able to solve the original mean variance problem.

This paper is concerned with the study of quadratic hedging of contingent claims with basis risk. We extend existing results by allowing the correlation between the hedging instrument and the underlying of the contingent claim to be random itself. We assume that the correlation process ρ evolves according to a stochastic differential equation with values between the boundaries −1 and 1. We keep the correlation dynamics general and derive an integrability condition on the correlation process that allows to describe and compute the quadratic hedge by means of a simple hedging formula that can be directly implemented. Furthermore, we show that the conditions on ρ are fulfilled by a large class of dynamics. The theory is exemplified by various explicitly given correlation dynamics.

Let μ be a Poisson random measure, let $\mathbb{F}$ be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let $\mathbb{G}$ be the initial enlargement of $\mathbb{F}$ with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the $\mathbb{G}$ -compensator relative to the $\mathbb{F}$ -compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the filtration is enlarged from $\mathbb{F}$ to $\mathbb{G}$ . In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable $\mathbb{F}$ -martingale is a $\mathbb{G}$ -semimartingale that belongs to the normed space $\mathcal{S}^{1}$ relative to $\mathbb{G}$ .

International audience; This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs needed for that representation allows for quadratic growth generators and jumps at random times. Since the existence and uniqueness theory for this class of BSDEs has not yet been developed to the required generality, the first part of the paper is devoted to fill that gap. By using a simple constructive algorithm, and known results on continuous quadratic BSDEs, we provide sufficient conditions for the existence and uniqueness of quadratic BSDEs with discontinuities at random times.

We consider insurance derivatives depending on an external physical risk process, for example a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indierence prices of the derivatives, and interpret them in terms of diversification pressure. Moreover we check the optimal investment strategies for standard admissibility criteria. Finally we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.

We solve Skorokhod's embedding problem for Brownian motion with linear drift (Wt + κ t)t≥0 by means of techniques of stochastic control theory. The search for a stopping time T such that the law of WT + κ T coincides with a prescribed law μ possessing the first moment is based on solutions of backward stochastic differential equations of quadratic type. This new approach generalizes an approach by Bass [3] of the classical version of Skorokhod's embedding problem using martingale representation techniques.

We show that the maximal expected utility satisfies a monotone continuity property with respect to increasing information. Let be a sequence of increasing filtrations converging to , and let u n (x) and u ∞(x) be the maximal expected utilities when investing in a financial market according to strategies adapted to and , respectively. We give sufficient conditions for the convergence u n (x) → u ∞(x) as n → ∞. We provide examples in which convergence does not hold. Then we consider the respective utility-based prices, π n and π∞, of contingent claims under (G t n ) and (G t ∞). We analyse to what extent π n → π∞ as n → ∞.

We consider flnancial markets with two kinds of small traders: regular traders who perceive the (continuous) asset price process S through its natural flltration, and insiders who possess some information advantage which makes the flltrations through which they experience the evolution of the market richer. We discuss the link between (NFLVR), the semimartingale property of S viewed from the agent’s perspective, and bounded expected utility. We show that whenever an agent’s expected utility is flnite, S is a semimartingale with a Doob-Meyer decomposition featuring a martingale part and an information drift. The expected utility gain of an insider with respect to a regular trader is calculated in a completely general setting. In particular, for the logarithmic utility function, utility gain is a function of the relative information drift alone, regardless of whether the market admits arbitrage.

We reveal pitfalls in the hedging of insurance contracts with a minimum return guarantee on the underlying investment, e.g. an external mutual fund. We analyze basis risk entailed by hedging the guarantee with a dynamic portfolio of proxy assets for the funds. We also take account of liquidity risk which arises since the insurer may need to advance funds for performing the hedge. Based on a least-squares Monte Carlo simulation, we study the economic implications of basis and liquidity risks. We demonstrate that both risks may be surprisingly high and show how the design of the contract and the hedging strategy may help to alleviate them.

We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion $X$: given a distribution $\rho$, we construct a stopping time $\tau$ such that the stopped process $X_{\tau}$ has the distribution $\rho$. Our solution method makes use of martingale representations (in a similar way to Bass (In Seminar on Probability XVII. Lecture Notes in Math. 784 (1983) 221-224 Springer) who solves the SEP for Brownian motion) and draws on law uniqueness of weak solutions of SDEs. Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Then, we fully characterize the distributions that can be embedded with integrable stopping times. Finally, we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding., Comment: Published at http://dx.doi.org/10.3150/14-BEJ598 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [7] and [8]. We perform a verification and discuss special cases for which the control problem has explicit solutions.

Frequently, dynamic hedging strategies minimizing risk exposure are not given in closed form, but need to be approximated numerically. This makes it difficult to estimate residual hedging risk, also called basis risk, when only imperfect hedging instruments are at hand. We propose an easy to implement and computationally efficient least-squares Monte Carlo algorithm to estimate residual hedging risk. The algorithm approximates the variance minimal hedging strategy within general diffusion models. Moreover, the algorithm produces both high-biased and low-biased estimators for the residual hedging error variance, thus providing an intrinsic criterion for the quality of the approximation. In a number of examples we show that the algorithm delivers accurate hedging error characteristics within seconds.

We reveal pitfalls in the hedging of insurance contracts with a minimum return guarantee on the underlying investment, e.g.\ an external mutual fund. We analyze basis risk entailed by hedging the guarantee with a dynamic portfolio of proxy assets for the funds. We also take account of liquidity risk which arises since the insurer may need to advance funds for performing the hedge. Based on a least-squares Monte Carlo simulation, we study the economic implications of basis and liquidity risk. We demonstrate that both risks may be surprisingly high and show how the design of the contract and the hedging strategy may help to alleviate them.

We consider the stochastic control problem of how to optimally close a large asset position in an illiquid market with price impact. We assume that the risk attributed to an open position depends on the price evolvement since the beginning of the trading period. Within a continuous-time model with a linear temporary price impact we show how to obtain an optimal trade off between liquidity costs and price risk. The optimal trading rates, turning out to be price sensitive, can be characterized in terms of a PDE describing by how much they differ from a linear trading rate. The PDE, in general, does not possess a closed-form solution. We provide a uniqueness result for solutions in the viscosity sense, allowing in the following to identify the value function and optimal trading rates. Finally we show that optimal strategies from discrete model approximations converge to the continuous-time optimal trading rates.

When managing energy or weather related risk often only imperfect hedging instruments are available. In the first part we illustrate problems arising with imperfect hedging by studying a toy model. We consider an airline’s problem with covering income risk due to fluctuating kerosine prices by investing into futures written on heating oil with closely correlated price dynamics. In the second part we outline recent results on exponential utility based cross hedging concepts. They highlight in a generalization of the Black- Scholes delta hedge formula to incomplete markets. Its derivation is based on a purely stochastic approach of utility maximization. It interprets stochastic control problems in the BSDE language, and profits from the power of the stochastic calculus of variations.

We consider the problem of how to close a large asset position in an illiquid market in such a way that very high liquidation costs are unlikely. To this end we introduce a discrete-time model that provides a simple device for designing and controlling the distribution of the revenues/costs from unwinding the position. By appealing to dynamic programming we derive semi-explicit formulas for the optimal execution strategies. We then present a numerical algorithm for approximating optimal execution rates as functions of the price. We provide error bounds and prove convergence. Finally, examples for the liquidation of forward positions in illiquid energy markets illustrate the efficiency of the algorithm.

We consider backward stochastic differential equations (BSDE) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. In case the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. In case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in case of non-uniqueness classical solutions that fail to be measure solutions can coexists with different measure solutions., 32 pages

This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical 'delta hedge' in complete markets.

Let (Gt) be an enlargement of the filtration (Ft). Jeulin and Jacod discussed a sufficient criterion for the inheritance of the semimartingale property when passing to the larger filtration. We provide alternative proofs of their results in a more general setting by using decoupling measures and Girsanov’s changes of measure. We derive necessary and sufficient conditions for the embedding of vector spaces of (Ft)-semimartingales into spaces of (Gt)-semimartingales to be continuous in terms of generalized entropies of the information increment.

We review a general mathematical link between utility and information theory appearing in a simple financial market model with two kinds of small investors: insiders, whose extra information is stored in an enlargement of the less informed agents’filtration. The insider’s expected logarithmic utility increment is described in terms of the information drift, i.e. the drift one has to eliminate in order to perceive the price dynamics as a martingale from his perspective. We describe the information drift in a very general setting by natural quantities expressing the conditional laws of the better informed view of the world. This on the other hand allows to identify the additional utility by entropy related quantities known from information theory.

The utility maximisation problem is considered for investors with anticipative additional information. We distinguish between models with conditional measures and models with enlarged filtrations. The dual functions of the maximal expected utility are determined with the help of f-divergences. We assume that our measures are absolutely continuous with respect to a local martingale measure (LMM), but not necessarily equivalent. Thus we do not exclude arbitrage.

The background for the general mathematical link between utility and information theory investigated in this paper is a simple financial market model with two kinds of small traders: less informed traders and insiders, whose extra information is represented by an enlargement of the other agents' filtration. The expected logarithmic utility increment, that is, the difference of the insider's and the less informed trader's expected logarithmic utility is described in terms of the information drift, that is, the drift one has to eliminate in order to perceive the price dynamics as a martingale from the insider's perspective. On the one hand, we describe the information drift in a very general setting by natural quantities expressing the probabilistic better informed view of the world. This, on the other hand, allows us to identify the additional utility by entropy related quantities known from information theory. In particular, in a complete market in which the insider has some fixed additional information during the entire trading interval, its utility increment can be represented by the Shannon information of his extra knowledge. For general markets, and in some particular examples, we provide estimates of maximal utility by information inequalities., Published at http://dx.doi.org/10.1214/009117905000000648 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Let M be a purely discontinuous martingale relative to a filtration ( F t ) . Given an arbitrary extension ( G t ) of the filtration ( F t ) , we will provide sufficient conditions for M to be a semimartingale relative to ( G t ) . Moreover we describe methods of how to find the Doob–Meyer decomposition with respect to the enlarged filtration. To this end we prove a new and more explicit version of the predictable representation property of Poisson random measures. Finally some concrete examples will show how the method developed may be applied.

International audience; We consider the problem of how to optimally close a large assetposition in a market with a linear temporary price impact. We take the perspectiveof an agent who obtains a signal about the future price evolvement.By means of classical stochastic control we derive explicit formulas for the closingstrategy that minimizes the expected execution costs. We compare agentsobserving the signal with agents who do not see it. We compute explicitly theexpected additional gain due to the signal, and perform a comparative staticsanalysis.

We consider Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDE to be differentiable in the variational sense (Malliavin differentiable)., Comment: This the revised version