Stochastic Target Problems Research Articles

Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality:(⋆)∫|Stu0−Stv0|φ0dx≤∫|u0−v0|Gtφ0dx,∀φ0≥0,∀u0,∀v0, where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying (⋆). This amounts to finding an optimal weighted L1 contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation∂tφ=supξ⁡{F′(ξ)⋅Dφ+tr(A(ξ)D2φ)}. Since weighted L1 contraction results are mainly used for possibly nonintegrable L∞ solutions u, the natural spaces behind this duality are L∞ for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions φ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called Lint∞. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].

A stochastic target problem for branching diffusion processes

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain-based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. Next, we show that the value function can be simplified into a novel function with the use of a finite-dimensional argument through a concept known as the branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.

A Numerical Scheme for the Quantile Hedging Problem

We consider numerical approximations to the quantile hedging price of a European claim in a nonlinear market with Markovian dynamics. We study an equivalent stochastic target problem with the conditional probability of success as a new state variable, in addition to the asset value process. We propose numerical approximations based on piecewise constant policy time stepping coupled with novel finite difference schemes. We prove convergence in the monotone case combining backward stochastic differential equation arguments with the Barles and Jakobsen and Barles and Souganidis approaches for nonlinear PDEs. The difficulties compared to the classical setting consist in the construction of monotone schemes under degeneracy due to the perfectly correlated joint process, the unboundedness of the control variable, and the effect of the boundaries in the probability variable on the analysis. We extend the method to a class of nonmonotone schemes using higher order interpolation and prove convergence for linear drivers. In a numerical section, we illustrate the performance of our schemes by considering an example in a financial market with imperfections, and show that a standard nonmonotone scheme produces financially counterintuitive solutions.

A Level-Set Approach for Stochastic Optimal Control Problems Under Controlled-Loss Constraints

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton–Jacobi–Bellman equation usually calls for strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in the absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then solve the latter by a level-set approach. With this approach, state constraints are managed through an exact penalization technique.

Quenched Mass Transport of Particles Toward a Target

We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. This kind of problems is motivated by limiting behavior of interacting particles systems with applications in, for example, agricultural crop management. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost surely transported toward a given target, along the paths of a stochastic differential equation. Our results extend those of Soner and Touzi, Journal of the European Mathematical Society (2002) to our setting.

Second-Order Stochastic Target Problems with Generalized Market Impact

We extend the study of [B. Bouchard, G. Loeper, and Y. Zou, SIAM J. Control Optim., 55 (2017), pp. 3319--3348; G. Loeper, Ann. Appl. Probab., 28 (2018), pp. 2664--2726] stochastic target problems w...

PORTFOLIO OPTIMIZATION UNDER A QUANTILE HEDGING CONSTRAINT

We study a problem of portfolio optimization under a European quantile hedging constraint. More precisely, we consider a class of Markovian optimal stochastic control problems in which two controlled processes must meet a probabilistic shortfall constraint at some terminal date. We denote by [Formula: see text] the corresponding value function. Following the arguments introduced in the literature on stochastic target problems, we convert this problem into a state constraint one in which the constraint is defined by means of an auxiliary value function [Formula: see text] characterizing the reachable set. This set is therefore not given a priori but is naturally integrated in [Formula: see text] solving, in a viscosity sense, a nonlinear parabolic partial differential equation (PDE). Relying on the existing literature, we derive, in the interior of the domain, a Hamilton–Jacobi–Bellman characterization of [Formula: see text]. However, [Formula: see text] involves an additional controlled state variable coming from the diffusion of the probability of reaching the target and belonging to the compact set [Formula: see text]. This leads to nontrivial boundaries for [Formula: see text] that must be discussed. Our main result is thus the characterization of [Formula: see text] at those boundaries. We also provide examples for which comparison results exist for the PDE solved by [Formula: see text] on the interior of the domain.

On the Controller-Stopper Problems with Controlled Jumps

We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth (Ann Probab 29(3):1111–1127, 2001), Karatzas and Zamfirescu (Ann Probab 36(4):1495–1527, 2008) and Karatzas and Zamfirescu (Appl Math Optim 53(2):163–184, 2006) when the volatility of the state process is controlled as in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems [of Soner and Touzi (SIAM J Control Optim 41(2):404–424, 2002; J Eur Math Soc 4(3):201–236, 2002)] with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sîrbu (SIAM J Control Optim 51(6):4274–4294, 2013) but not the dynamic programming principle, which is difficult to prove directly for games.

A Comparison Principle for PDEs Arising in Approximate Hedging Problems: Application to Bermudan Options

In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard, Elie and Touzi in [1] and is known to solve an Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in [2]. [1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et al. Bsdes with weak terminal condition. The Annals of Probability, 43(2):572-604,2015.

Book Review: Optimal stochastic control, stochastic target problems, and backward SDE

Book Review: Optimal stochastic control, stochastic target problems, and backward SDE

Stochastic Perron for Stochastic Target Problems

In this paper, we adapt stochastic Perron's method to analyze stochastic target problems in a jump diffusion setup, where the controls are unbounded. Since classical control problems can be analyzed under the framework of stochastic target problems (with unbounded controls), we use our results to generalize the results of Bayraktar and S?rbu (SIAM J Control Optim 51(6):4274---4294, 2013) to problems with controlled jumps.

Differential Games in $$L^{\infty }$$

The Isaacs equations for differential games in \({L^{\infty }}\) first derived in Barron (Nonlinear Anal 14:971–989, 1990) are reformulated so that the Hamiltonians are continuous and result in a simpler problem to analyze numerically. Relaxed differential games in \({L^{\infty }}\) are considered. \(L^{\infty }\) differential games with time and state independent dynamics and convex or quasiconvex terminal data are solved explicitly using a type of Hopf–Lax formula. The stochastic differential game in \({L^{\infty }}\) connected to stochastic target problems is also discussed.

State-Constrained Stochastic Optimal Control Problems via Reachability Approach

This paper deals with a class of stochastic optimal control problems (SOCPs) in the presence of state constraints. It is well known that for such problems the value function is, in general, discontinuous, and its characterization by a Hamilton--Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system. Here, we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions. To this end, the SOCP is first translated into a state-constrained stochastic target problem. Then a level-set approach is used to describe the backward reachable sets of the new target problem. It turns out that these backward reachable sets describe the value function. The main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalization. However, the target problem involves a new state variable and a new control variable that is u...

BSDEs with weak terminal condition

We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[\Psi(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $\Psi$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in Follmer and Leukert, and in Bouchard, Elie and Touzi. Finally, we observe a surprisingly strong connection between BSDEs with weak terminal condition and 2nd order BSDEs in the quasi linear case.

Dual formulation of second order target problems

This paper provides a new formulation of second order stochastic target problems introduced in [SIAM J. Control Optim. 48 (2009) 2344-2365] by modifying the reference probability so as to allow for different scales. This new ingredient enables us to prove a dual formulation of the target problem as the supremum of the solutions of standard backward stochastic differential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial differential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.

Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation

We consider a singular version with state constraints of the stochastic target problems studied in Soner and Touzi (SIAM J. Control Optim. 41:404–424, 2002; J. Eur. Math. Soc. 4:201–236, 2002) and more recently Bouchard et al. (SIAM J. Control Optim. 48:3123–3150, 2009), among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly fits the market models with proportional transaction costs and the order book liquidation issues. Our main result is a direct PDE characterization of the associated pricing function. As an example application, we discuss the valuation of VWAP-guaranteed-type book liquidation contracts, for a general class of risk functions.

A Stochastic Target Approach for P&L Matching Problems

Within a Brownian diffusion Markovian framework, we provide a direct PDE characterization of the minimal initial endowment required so that the terminal wealth of a financial agent (possibly diminished by the payoff of a random claim) can match a set of constraints in probability. Such constraints should be interpreted as a rough description of a targeted profit and loss (P&L) distribution. This allows us to give a price to options under a P&L constraint, or to provide a description of the discrete P&L profiles that can be achieved given an initial capital. This approach provides an alternative to the standard utility indifference (or marginal) pricing rules that is better adapted to market practices. From the mathematical point of view, this is an extension of the stochastic target problem under controlled loss, studied in Bouchard, Touzi, and Elie [Bouchard B, Touzi N, Elie R (2009) Stochastic target problems with controlled loss. SIAM J. Control Optim. 48(5):3123–3150], to the case of multiple constraints. Although the associated Hamilton-Jacobi-Bellman operator is fully discontinuous, and the terminal condition is irregular, we are able to construct a numerical scheme that converges at any continuity points of the pricing function.

Optimal control versus stochastic target problems: An equivalence result

Within a general abstract framework, we show that any optimal control problem in standard form can be translated into a stochastic target problem as defined in Soner and Touzi (2002) [5], whenever the underlying filtered probability space admits a suitable martingale representation property. This provides a unified way of treating these two classes of stochastic control problems. As an illustration, we show, within a jump diffusion framework, how the Hamilton–Jacobi–Bellman equations associated to an optimal control problem in standard form can be easily retrieved from the partial differential equations associated to its stochastic target counterpart.

Stochastic Target Problems with Controlled Loss in Jump Diffusion Models

In this paper, we consider a mixed-diffusion version of the stochastic target problem introduced in [B. Bouchard, R. Elie, and N. Touzi, SIAM J. Control Optim., 48 (2009), pp. 3123–3150]. This consists in finding the minimum initial value of a controlled process which guarantees to reach a controlled stochastic target with a given level of expected loss. It can be converted into a standard stochastic target problem by increasing both the state space and the dimension of the control. In our mixed-diffusion setting, the main difficulty comes from the presence of jumps, which leads to the introduction of a new kind of control that takes values in an unbounded set of measurable maps. This has a nontrivial technical impact on the formulation and derivation of the associated partial differential equations.

Martingale Representation Theorem for the G-Expectation

This paper considers the nonlinear theory of G-martingales as introduced by Peng in [16, 17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in [20] for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility.