Minimal Supersolution Research Articles

Optimal liquidation with conditions on minimum price

The classical optimal trading problem is the closure of a position in an asset over a time interval; the trader maximizes the expected revenues under the constraint that the position be closed by terminal time. Since the asset price is stochastic, the liquidation constraint is too restrictive; the trader may want to relax it or slow down/stop trading depending on price behavior. We consider two additional parameters that serve these purposes within the Almgren-Chriss framework: a binary valued process that prescribes when trading takes place and a set that prescribes when full liquidation is required. The permanent price impact parameter enters the problem as the negative part of the terminal cost. A terminal cost that can take negative values implies that the BSDE associated with the value function of the control problem can explode backward in time and that existence results on solutions of BSDE with singular terminal values are not directly applicable. When liquidation costs are quadratic, the problem is convex and, under a general filtration, the minimal supersolution of the BSDE gives the value function and the optimal control. For the non-quadratic case, we assume a stochastic Markovian volatility model. These give PDE/PDE-system representations for the value functions.

Continuity problem for singular BSDE with random terminal time

We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $\xi$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $\xi$ is of the form $\xi$ = g($\Xi$ S) where $\Xi$ is a continuous Markovian diffusion process and S is a hitting time of $\Xi$ and g is a deterministic function 2) terminal conditions of the form $\xi$ = $\infty$ $\times$ 1 {$\tau$ $\le$S} and 3) $\xi$ 2 = $\infty$ $\times$ 1 {$\tau$ >S} where $\tau$ is another stopping time. For general $\xi$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $\infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.

American options in a non-linear incomplete market model with default

We study the superhedging problem for American options with completely irregular payoffs in a non-linear and incomplete market model with default. We give a dual representation of the seller’s (superhedging) price in terms of the value of a non-linear mixed control/stopping problem, involving a suitable set of equivalent probability measures. We characterize the seller’s price process as the minimal supersolution of two types of reflected BSDEs: a constrained one and an optional one. Under some regularity assumptions on the pay-off, we show a duality result for the buyer’s price in terms of the value of a non-linear control/stopping game problem.

Backward stochastic differential equations with non-Markovian singular terminal conditions for general driver and filtration

We consider a class of Backward Stochastic Differential Equations with superlinear driver process f adapted to a filtration supporting at least a d dimensional Brownian motion and a Poisson random measure on Rm∖{0}. We consider the following class of terminal conditions: ξ1=∞⋅1{τ1≤T} where τ1 is any stopping time with a bounded density in a neighborhood of T and ξ2=∞⋅1AT where At, t∈[0,T] is a decreasing sequence of events adapted to the filtration Ft that is continuous in probability at T (equivalently, AT={τ2>T} where τ2 is any stopping time such that P(τ2=T)=0). In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We note that the first exit time from a time varying domain of a d-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density. Therefore such exit times can be used as τ1 and τ2 to define the terminal conditions ξ1 and ξ2. The proof of existence of the density is based on the classical Green’s functions for the associated PDE.

An obstacle problem arising in large exponent limit of power mean curvature flow equation

We study limit behavior for the level-set power mean curvature flow equation as the exponent tends to infinity. Under Lipschitz continuity, quasiconvexity, and coercivity of the initial condition, we show that the limit of the viscosity solutions can be characterized as the minimal supersolution of an obstacle problem involving the 1 1 -Laplacian. Such behavior is closely related to applications of power mean curvature flow in image denoising. We also discuss analogous behavior for other evolution equations with related applications.

Lower semicontinuous obstacles for the porous medium equation

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime m>1. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.

Regularity of BSDEs with a convex constraint on the gains-process

We consider the minimal super-solution of a backward stochastic differential equation with constraint on the gains-process. The terminal condition is given by a function of the terminal value of a forward stochastic differential equation. Under boundedness assumptions on the coefficients, we show that the first component of the solution is Lipschitz in space and 1/2-Holder in time with respect to the initial data of the forward process. Its path is continuous before the time horizon at which its left-limit is given by a face-lifted version of its natural boundary condition. This first component is actually equal to its own face-lift. We only use probabilistic arguments. In particular, our results can be extended to certain non-Markovian settings.

Linear Quadratic Stochastic Control Problems with Stochastic Terminal Constraint

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.

Reflected BSDEs, optimal control and stopping for infinite-dimensional systems

We introduce the notion of mild supersolution for an obstacle problem in an infinite dimensional Hilbert space. The minimal supersolution of this problem is given in terms of a reflected BSDEs in an infinite dimensional Markovian framework. The results are applied to an optimal control and stopping problem.

Dual representation of minimal supersolutions of convex BSDEs

Nous donnons une representation duale des sur-solutions minimales d’equations differentielles stochastiques retrogrades avec des conditions terminales integrables mais non necessairement bornees, et de faibles hypotheses sur le generateur qui peut de plus dependre de la valeur processus de l’equation meme. Reciproquement, nous montrons que toute mesure de risque dynamique satisfaisant une telle representation duale provient d’une EDSR. Nous donnons aussi une condition sous laquelle une sur-solution d’EDSR est en fait une solution.

Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting

We study the existence of a minimal supersolution for backward stochastic differential equations when the terminal data can take the value +∞ with positive probability. We deal with equations on a general filtered probability space and with generators satisfying a general monotonicity assumption. With this minimal supersolution we then solve an optimal stochastic control problem related to portfolio liquidation problems. We generalize the existing results in three directions: firstly there is no assumption on the underlying filtration (except completeness and quasi-left continuity), secondly we relax the terminal liquidation constraint and finally the time horizon can be random.

Stability and Markov property of forward backward minimal supersolutions

We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painleve-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.

Minimal supersolutions of convex BSDEs under constraints

We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form d Z = Δ d t + Γ d W . The generator may depend on the decomposition ( Δ,Γ ) and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in Δ and Γ . We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou’s lemma and L 1 -lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.

Minimal supersolutions of BSDEs under volatility uncertainty

We study the existence of minimal supersolutions of BSDEs under a family of mutually singular probability measures. We consider generators that are jointly lower semicontinuous, positive, and either convex in the control variable and monotone in the value variable, or that fulfill a specific normalization property.

Continuous dependence property of BSDE with constraints

In this paper, we study continuous properties of adapted solutions for backward stochastic differential equations with constraints (CBSDEs in short). Comparing with many existing literatures about this topic, our case is very general in the sense that constraints are formulated by general non-negative real functions. In general case, we proved a continuous property from below and a lower semi-continuous property of the minimal super-solution of CBSDE in its effective domain. Furthermore, in the special convex case, we obtained a continuous property with the help of convex analysis.

Minimal supersolutions of BSDEs with lower semicontinuous generators

Nous etudions des sur-solutions minimales d’equations stochastiques retrogrades. Nous montrons l’existence et l’unicite de telles sur-solutions minimales lorsque le generateur est conjointement semi-continu inferieurement, minore par une fonction affine de la variable de controle et satisfait une condition specifique de normalisation. Le resultat principal est obtenu en utilisant une convergence de semi-martingales.

Minimal supersolutions of convex BSDEs

We study the nonlinear operator of mapping the terminal value $\xi$ to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in $y$, convex in $z$, jointly lower semicontinuous and bounded below by an affine function of the control variable $z$. We show existence, uniqueness, monotone convergence, Fatou's lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.

Incentive compatibility constraints and dynamic programming in continuous time

This paper is devoted to the study of infinite horizon continuous time optimal control problems with incentive compatibility constraints that arise in many economic problems, for instance in defining the second best Pareto optimum for the joint exploitation of a common resource, as in Benhabib and Radner [Benhabib, J., Radner, R., 1992. The joint exploitation of a productive asset: a game theoretic approach. Economic Theory, 2: 155–190]. An incentive compatibility constraint is a constraint on the continuation of the payoff function at every time. We prove that the dynamic programming principle holds, the value function is a viscosity solution of the associated Hamilton–Jacobi–Bellman (HJB) equation, and that it is the minimal supersolution satisfying certain boundary conditions. When the incentive compatibility constraint only depends on the present value of the state variable, we prove existence of optimal strategies, and we show that the problem is equivalent to a state constraints problem in an endogenous state region which depends on the data of the problem. Some economic examples are analyzed.