Stochastic Perron Method Research Articles

Convergence of Optimal Investment Problems in the Vanishing Fixed Cost Limit

We consider an optimal investment problem for an investor facing constant and proportional transaction costs and study the limit as the constant cost tends to zero. Combining the stochastic Perron's method with stability arguments for viscosity solutions, we show that the value function converges to the value function of the problem with purely proportional costs. Moreover, using a Komlos-type argument, we show that forward-convex combinations of the optimal strategies in the problem with constant costs converge to an optimal strategy without a constant cost.

Optimal entry and consumption under habit formation

AbstractThis paper studies a composite problem involving decision-making about the optimal entry time and dynamic consumption afterwards. In Stage 1, the investor has access to full market information subject to some information costs and needs to choose an optimal stopping time to initiate Stage 2; in Stage 2, the investor terminates the costly full information acquisition and starts dynamic investment and consumption under partial observation of free public stock prices. Habit formation preferences are employed, in which past consumption affects the investor’s current decisions. Using the stochastic Perron method, the value function of the composite problem is proved to be the unique viscosity solution of some variational inequalities.

Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

We study the problem of minimizing the discounted probability of exponential Parisian ruin, that is, the discounted probability that an insurer's surplus exhibits an excursion below zero in excess of an exponentially distributed clock. The insurer controls its surplus via reinsurance priced according to the mean-variance premium principle, as in Liang, Liang, and Young (2019). We consider the classical risk model and apply stochastic Perron's method, as introduced by Bayraktar and Sirbu (2012,2013,2014), to show that the minimum discounted probability of exponential Parisian ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with boundary conditions at $\pm \infty$. A major difficulty in proving the comparison principle arises from the discontinuity of the Hamiltonian.

Contract Theory in a VUCA World

In this paper we investigate a Principal-Agent problem with moral hazard under Knightian uncertainty. We extend the seminal framework of Holmström and Milgrom by combining a Stackelberg equilibrium with a worst-case approach. We investigate a general model in the spirit of [J. Cvitanić, D. Possamaï, and N. Touzi, Management Sci., 63 (2016), pp. 3328--3346], [J. Cvitanić, D. Possamaï, and N. Touzi, Finance Stoch., 22 (2018), pp. 1--37]. We show that optimal contracts depend on the output and its quadratic variation by extending [T. Mastrolia and D. Possamaï, J. Optim. Theory Appl., 179 (2018), pp. 452--500], [J. Sung, SSRN Electronic Journal, 1 (2015), 2601174]. We compute the optimal effort of the Agent through the solution to a second order BSDE, and we show that the value of the problem of the Principal is the viscosity solution of a Hamilton--Jacobi--Bellman--Isaacs equation, by using the stochastic Perron's method.

Martingale Optimal Transport with Stopping

We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in [A. M. G. Cox and S. Kallblad, SIAM J. Control Optim., 55 (2017), pp. 3409--3436] allows us to obtain an equivalent infinite dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.

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.

Martingale Optimal Transport with Stopping

We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in [A. M. G. Cox and S. Kallblad. Model-independent bounds for Asian options: a dynamic programming approach. SIAM Journal on Control and Optimization, 55(6):3409–3436, 2017.] allows us to obtain an equivalent infinite-dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.

A General Verification Result for Stochastic Impulse Control Problems

This paper establishes existence of optimal controls for a general stochastic impulse control problem. For this, the value function is characterized as the pointwise minimum of a set of superharmonic functions, as the unique continuous viscosity solution of the quasi-variational inequalities (QVIs), and as the limit of a sequence of iterated optimal stopping problems. A combination of these characterizations is used to construct optimal controls without relying on any regularity of the value function beyond continuity. Our approach is based on the stochastic Perron method and the assumption that the associated QVIs satisfy a comparison principle.

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.

Stochastic Perron for stochastic target games

We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solution (super-solution) of a Hamilton-Jacobi-Bellman (HJB) equation. We then characterize the value function as a viscosity solution to the HJB equation using a comparison result and a byproduct to obtain the dynamic programming principle.

Robust Feedback Switching Control: Dynamic Programming and Viscosity Solutions

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) game switcher versus nature is then formulated as a two-step (robust) optimization problem. We develop the stochastic Perron method in this framework, and prove that it produces a viscosity sub and supersolution to a system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which envelope the value function. Together with a comparison principle, this characterizes the value function of the game as the unique viscosity solution to the HJB equation, and shows as a byproduct the dynamic programming principle for robust feedback switching control problem.

Stochastic Perron's Method for the Probability of Lifetime Ruin Problem Under Transaction Costs

We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming principle (DPP), we characterize the value function as the unique viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.

Verification by Stochastic Perron's Method in stochastic exit time control problems

We apply the Stochastic Perron Method, created by Bayraktar and Sîrbu, to a stochastic exit time control problem. Our main assumption is the validity of the Strong Comparison Result for the related Hamilton–Jacobi–Bellman (HJB) equation. Without relying on Bellman's optimality principle we prove that inside the domain the value function is continuous and coincides with a viscosity solution of the Dirichlet boundary value problem for the HJB equation.

Stochastic Perron's Method for the Probability of Lifetime Ruin Problem Under Transaction Costs

We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming principle (DPP), we characterize the value function as the unique viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.

Stochastic Perron's Method and Elementary Strategies for Zero-Sum Differential Games

We develop here the stochastic Perron method in the framework of two-player zero-sum differential games. We consider the formulation of the game where both players play, symmetrically, feedback strategies as opposed to the Elliott--Kalton formulation prevalent in the literature. The class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic Perron method carry through in a rather simple way. More precisely, we define the game over elementary strategies, which are well motivated by intuition. Within this framework, the stochastic Perron method produces a viscosity subsolution of the upper Isaacs equation dominating the upper value of the game, and a viscosity supersolution of the upper Isaacs equation lying below the upper value of the game. Using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper Isaacs equation. An identical statement holds true for the lower value and the lower Isaacs equation. A version of the dynamic programming principle is obtained as a byproduct. If the Isaacs condition is satisfied, the game has a value over elementary (pure) strategies.

Stochastic Perron's method for optimal control problems with state constraints

We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state $X$ is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function $v$ is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for the Hamilton-Jacobi-Bellman equation. In the case of a smooth domain, under some additional assumptions, these estimates allow to identify $v$ with a unique continuous constrained viscosity solution of this equation.

Stochastic Perron's Method for Hamilton--Jacobi--Bellman Equations

We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton--Jacobi--Bellman equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using the stochastic Perron's method we construct a supersolution lying below the value function and a subsolution dominating it. A comparison argument easily closes the proof. The program has the precise meaning of verification for viscosity solutions, obtaining the DPP as a conclusion. It also immediately follows that the weak and strong formulations of the stochastic control problem have the same value. Using this method we also capture the possible face-lifting phenomenon in a straightforward manner.