Optimal control of predictive mean-field equations and applications to finance

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}

GANs and Closures: Micro-Macro Consistency in Multiscale Modeling

Sampling the phase space of molecular systems -- and, more generally, of complex systems effectively modeled by stochastic differential equations -- is a crucial modeling step in many fields, from protein folding to materials discovery. These problems are often multiscale in nature: they can be described in terms of low-dimensional effective free energy surfaces parametrized by a small number of "slow" reaction coordinates; the remaining "fast" degrees of freedom populate an equilibrium measure on the reaction coordinate values. Sampling procedures for such problems are used to estimate effective free energy differences as well as ensemble averages with respect to the conditional equilibrium distributions; these latter averages lead to closures for effective reduced dynamic models. Over the years, enhanced sampling techniques coupled with molecular simulation have been developed. An intriguing analogy arises with the field of Machine Learning (ML), where Generative Adversarial Networks can produce high dimensional samples from low dimensional probability distributions. This sample generation returns plausible high dimensional space realizations of a model state, from information about its low-dimensional representation. In this work, we present an approach that couples physics-based simulations and biasing methods for sampling conditional distributions with ML-based conditional generative adversarial networks for the same task. The "coarse descriptors" on which we condition the fine scale realizations can either be known a priori, or learned through nonlinear dimensionality reduction. We suggest that this may bring out the best features of both approaches: we demonstrate that a framework that couples cGANs with physics-based enhanced sampling techniques can improve multiscale SDE dynamical systems sampling, and even shows promise for systems of increasing complexity.Comment: 21 pages, 10 figures, 3 table

Kyle-Back Models with risk aversion and non-Gaussian Beliefs

We show that the problem of existence of equilibrium in Kyle's continuous time insider trading model ([31]) can be tackled by considering a system of quasilinear parabolic equation and a Fokker-Planck equation coupled via a transport type constraint. By obtaining a stochastic representation for the solution of such a system, we show the well-posedness of solutions and study the properties of the equilibrium obtained for small enough risk aversion parameter. In our model, the insider has exponential type utility and the belief of the market on the distribution of the price at final time can be non-Gaussian.Comment: Some typos in Introduction and Subsection 5.1 were correcte

Optimal investment with inside information and parameter uncertainty

An optimal investment problem is solved for an insider who has access to noisy information related to a future stock price, but who does not know the stock price drift. The drift is filtered from a combination of price observations and the privileged information, fusing a partial information scenario with enlargement of filtration techniques. We apply a variant of the Kalman-Bucy filter to infer a signal, given a combination of an observation process and some additional information. This converts the combined partial and inside information model to a full information model, and the associated investment problem for HARA utility is explicitly solved via duality methods. We consider the cases in which the agent has information on the terminal value of the Brownian motion driving the stock, and on the terminal stock price itself. Comparisons are drawn with the classical partial information case without insider knowledge. The parameter uncertainty results in stock price inside information being more valuable than Brownian information, and perfect knowledge of the future stock price leads to infinite additional utility. This is in contrast to the conventional case in which the stock drift is assumed known, in which perfect information of any kind leads to unbounded additional utility, since stock price information is then indistinguishable from Brownian information

Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems

This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse, the optimal strategies of the agents are solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers solves a nonstandard "quadratic" backward stochastic differential equation. The main result of the paper is the existence of a Markovian solution to this forward-backward system on an arbitrary time interval, which is obtained via a fixed-point argument on the space of absolutely continuous distribution functions. Moreover, the equilibrium obtained in this paper is able to explain several stylized facts which are not captured by the current asymmetric information models.Comment: Published at http://dx.doi.org/10.1214/15-AAP1138 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Kyle models with terminal trading constraints

Department of Mathematical SciencesWe study Kyle models with terminal trading constraints that are variations of Kyle (1985) and Back (1992) where the insider has no trading constraint. We find that the constraint produces new features to our model. First, it turns out that we need a new state process in the structure of equilibria. Second, we show that our insider places a block order at terminal time, \Delta \theta_T =\tilde{a} ??? \theta_{T-}, to satisfy her constraint. We prove the existence of equilibria in both discrete time and continuous time settings. For the continuous time model, we establish the explicit equilibrium by deriving an autonomous system of first-order nonlinear ordinary differential equations (ODEs). Moreover, we obtain results associated with empirical findings, for example, autocorrelated aggregate holdings, decreasing price impact function, and U-shaped trading patterns.clos

On pricing rules and optimal strategies in general Kyle-Back models

The folk result in Kyle-Back models states that the value function of the insider remains unchanged when her admissible strategies are restricted to absolutely continuous ones. In this paper we show that, for a large class of pricing rules used in current literature, the value function of the insider can be finite when her strategies are restricted to be absolutely continuous and infinite when this restriction is not imposed. This implies that the folk result doesn’t hold for those pricing rules and that they are not consistent with equilibrium. We derive the necessary conditions for a pricing rule to be consistent with equilibrium and prove that, when a pricing rule satisfies these necessary conditions, the insider’s optimal strategy is absolutely continuous, thus obtaining the classical result in a more general setting. This, furthermore, allows us to justify the standard assumption of absolute continuity of insider’s strategies since one can construct a pricing rule satisfying the necessary conditions derived in the paper that yield the same price process as the pricing rules employed in the modern literature when insider’s strategies are absolutely continuous