We develop novel duality methods for stochastic optimal control problems under two sources of randomness and non-Markovian random coefficients adapted to just one of them. The Hamilton-Jacobi-Bellman (HJB) equation is a second-order backward stochastic partial differential equation. Our duality theory provides an alternative description of the random value function in terms of a suitable pathwise optimal control problem parameterized by the realizations of one of the Brownian motions. This allows us to regain Markovianity using the theory of rough optimal control problems, leading to rough second order HJB equations that we solve in a suitable viscosity sense.

We prove two duality descriptions of the value function for a generic stochastic optimal problem. These descriptions also hold when the diffusion is controlled, a case left open by the literature so far.

We study a signature-driven numerical scheme to solve multi-dimensional linear-quadratic (LQ) stochastic control problems. Using that linear signature functionals are dense in the natural class of admissible controls, we show that our approach turns the original LQ problem into a deterministic convex quadratic polynomial optimisation. To underpin a numerical approach based on truncated signatures, we prove that the problem's value function can be approximated by finite-dimensional polynomial approximations when the truncation levels are chosen sufficiently high. Remarkably, our numerical experiments show very decent accuracy already for small truncation levels. Key tools for our analysis are (i) the algebraic representation of controlled stochastic differential equations and the associated cost function as linear functionals of the path signatures of the driving noise, (ii) the convergence of the truncated linear functionals, and (iii) the density of signature controls.

In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.

We present a multi-agent and mean-field formulation of a game between investors who receive private signals informing their investment decisions and who interact through relative performance concerns. A key tool in our model is a Poisson random measure which drives jumps in both market prices and signal processes and thus captures common and idiosyncratic noise. Upon receiving a jump signal, an investor evaluates not only the signal's implications for stock price movements but also its implications for the signals received by her peers and for their subsequent investment decisions. A crucial aspect of this assessment is the distribution of investor types in the economy. These types determine their risk aversion, performance concerns, and the quality and quantity of their signals. We demonstrate how these factors are reflected in the corresponding HJB equations, characterizing an agent's optimal response to her peers' signal-based strategies. The existence of equilibria in both the multi-agent and mean-field game is established using Schauder's Fixed Point Theorem under suitable conditions on investor characteristics, particularly their signal processes. Finally, we present numerical case studies that illustrate these equilibria from a financial-economic perspective. This allows us to address questions such as how much investors should care about the information known by their peers.

This paper proposes to parameterize open loop controls in stochastic optimal control problems via suitable classes of functionals depending on the driver's path signature, a concept adopted from rough path integration theory. We rigorously prove that these controls are dense in the class of progressively measurable controls and use rough path methods to establish suitable conditions for stability of the controlled dynamics and target functional. These results pave the way for Monte Carlo methods to stochastic optimal control for generic target functionals and dynamics. We discuss the rather versatile numerical algorithms for computing approximately optimal controls and verify their accurateness in benchmark problems from Mathematical Finance.

We consider a stochastic optimal control problem where the controller can anticipate the evolution of the driving noise over some dynamically changing time window. The controlled state dynamics are understood as a rough differential equation. We combine the martingale optimality principle with a functional form of Itô's formula to derive a Hamilton-Jacobi-Bellman (HJB) equation for this problem. This HJB equation is formulated in terms of Dupire's functional derivatives and involves a transport equation arising from the anticipativity of the problem.

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock's price fluctuations. With linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be not only well posed, but it even allows for a closed-form solution. We describe this solution and the resulting problem value for this stochastic control problem with partial observation by solving its convex-analytic dual problem.

We propose a price impact model where changes in prices are purely driven by the order flow in the market. The stochastic price impact of market orders and the arrival rates of limit and market orders are functions of the market liquidity process which reflects the balance of the demand and supply of liquidity. Limit and market orders mutually excite each other so that liquidity is mean reverting. We use the theory of Meyer-$\sigma$-fields to introduce a short-term signal process from which a trader learns about imminent changes in order flow. In this setting, we examine an optimal execution problem and derive the Hamilton--Jacobi--Bellman (HJB) equation for the value function. The HJB equation is solved numerically and we illustrate how the trader uses the signal to enhance the performance of execution problems and to execute speculative strategies.

In this paper we study optimal investment when the investor can peek some time units into the future, but cannot fully take advantage of this knowledge because of quadratic transaction costs. In the Bachelier setting with exponential utility, we give an explicit solution to this control problem with intrinsically infinite-dimensional memory. This is made possible by solving the dual problem where we make use of the theory of Gaussian Volterra integral equations.

This paper presents a new framework for Merton's optimal investment problem which uses the theory of Meyer $\sigma$-fields to allow for signals that possibly warn the investor about impending jumps. With strategies no longer predictable, some care has to be taken to properly define wealth dynamics through stochastic integration. By means of dynamic programming, we solve the problem explicitly for power utilities. In a case study with Gaussian jumps, we find, for instance, that an investor may prefer to disinvest completely even after a mildly positive signal. Our setting also allows us to investigate whether, given the chance, it is better to improve signal quality or quantity and how much extra value can be generated from either choice.

We consider the Bachelier model with information delay where investment decisions can be based only on observations from $H>0$ time units before. Utility indifference prices are studied for vanilla options and we compute their non-trivial scaling limit for vanishing delay when risk aversion is scaled liked $A/H$ for some constant $A$. Using techniques from [7], we develop discrete-time duality for this setting and show how the relaxed form of martingale property introduced by [9] results in the scaling limit taking the form of a volatility control problem with quadratic penalty.

In this paper we study a representation problem first considered in a simpler version by Bank and El Karoui (Ann. Probab. 32 (2004) 1030-1067). A key ingredient to this problem is a random measure mu on the time axis which in the present paper is allowed to have atoms. Such atoms turn out to not only pose serious technical challenges in the proof of the representation theorem, but actually have significant meaning in its applications, for instance, to irreversible investment problems. These applications also suggest to study the problem for processes which are measurable with respect to a Meyer-sigma-field that lies between the predictable and the optional sigma-field. Technically, our proof amounts to a delicate analysis of optimal stopping problems and the corresponding optimal divided stopping times.

We study a continuous-time version of the intermediation model of Grossman and Miller (1988). To wit, we solve for the competitive equilibrium prices at which liquidity takers' demands are absorbed by dealers with quadratic inventory costs, who can in turn gradually transfer these positions to an exogenous open market with finite liquidity. This endogenously leads to transient price impact in the dealer market. Smooth, diffusive, and discrete trades all incur finite but nontrivial liquidity costs, and can arise naturally from the liquidity takers' optimization.

We prove a scaling limit theorem for the super-replication cost of options in a Cox--Ross--Rubinstein binomial model with transient price impact. The correct scaling turns out to keep the market depth parameter constant while resilience over fixed periods of time grows in inverse proportion with the duration between trading times. For vanilla options, the scaling limit is found to coincide with the one obtained by PDE methods in [12] for models with purely temporary price impact. These models are a special case of our framework and so our probabilistic scaling limit argument allows one to expand the scope of the scaling limit result to path-dependent options.

In stochastic control problems delicate issues arise when the controlled system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. We propose to use Meyer-$\sigma$-fields as a flexible tool to model information flow in such situations. The possibilities of this approach are illustrated first in a very simple linear stochastic control problem and then in a fairly general formulation for the singular stochastic control problem of irreversible investment with inventory risk. For the latter, we illustrate in a first case study how different signals on exogenous jumps lead to different optimal controls, interpolating between the predictable and the optional case in a systematic manner.

We study super-replication of contingent claims in markets with fixed transaction costs. This can be viewed as a stochastic impulse control problem with a terminal state constraint. The first result in this paper reveals that in reasonable continuous time financial market models the super-replication price is prohibitively costly and leads to trivial buy-and-hold strategies. Our second result derives nontrivial scaling limits of super-replication prices for binomial models with small fixed costs.

We establish a super-replication duality in a continuous-time financial model where an investor's trades adversely affect bid- and ask-prices for a risky asset and where market resilience drives the resulting spread back towards zero at an exponential rate. Similar to the literature on models with a constant spread, our dual description of super-replication prices involves the construction of suitable absolutely continuous measures with martingales close to the unaffected reference price. A novel feature in our duality is a liquidity weighted $L^2$-norm that enters as a measurement of this closeness and that accounts for strategy dependent spreads. As applications, we establish optimality of buy-and-hold strategies for the super-replication of call options and we prove a verification theorem for utility maximizing investment strategies.

We introduce a price impact model which accounts for finite market depth, tightness and resilience. Its coupled bid- and ask-price dynamics induce convex liquidity costs. We provide existence of an optimal solution to the classical problem of maximizing expected utility from terminal liquidation wealth at a finite planning horizon. In the specific case when market uncertainty is generated by an arithmetic Brownian motion with drift and the investor exhibits constant absolute risk aversion, we show that the resulting singular optimal stochastic control problem readily reduces to a deterministic optimal tracking problem of the optimal frictionless constant Merton portfolio in the presence of convex costs. Rather than studying the associated Hamilton-Jacobi-Bellmann PDE, we exploit convex analytic and calculus of variations techniques allowing us to construct the solution explicitly and to describe the free boundaries of the action- and non-action regions in the underlying state space. As expected, it is optimal to trade towards the frictionless Merton position, taking into account the initial bid-ask spread as well as the optimal liquidation of the accrued position when approaching terminal time. It turns out that this leads to a surprisingly rich phenomenology of possible trajectories for the optimal share holdings.

We study a linear quadratic optimal control problem with stochastic coefficients and a terminal state constraint, which may be in force merely on a set with positive, but not necessarily full probability. Under such a partial terminal constraint, the usual approach via a coupled system of a backward stochastic Riccati equation and a linear backward equation breaks down. As a remedy, we introduce a family of auxiliary problems parametrized by the supersolutions to this Riccati equation alone. The target functional of these problems dominates the original constrained one and allows for an explicit description of both the optimal control policy and the auxiliary problem's value in terms of a suitably constructed optimal signal process. This suggests that, for the minimal supersolution of the Riccati equation, the minimizers of the auxiliary problem coincide with those of the original problem, a conjecture that we see confirmed in all examples understood so far.