Mihai Sîrbu

Stochastic Perron's method and verification without smoothness using viscosity comparison: The linear case

We introduce a stochastic version of the classical Perron’s method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the stochastic differential equation. The unique viscosity solution is actually equal to the expected payoff. This amounts to a verification result (Ito’s Lemma) for non-smooth viscosity solutions of the linear parabolic equation. This is the first step in a larger program to prove verification for viscosity solutions and the Dynamic Programming Principle for stochastic control problems and games.

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 Perrons 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.

Null controllability of an infinite dimensional SDE with state- and control-dependent noise

Abstract We consider a controlled stochastic linear differential equation with state- and control-dependent noise in a Hilbert space H . We investigate the relation between the null controllability of the equation and the existence of the solution of “singular” Riccati operator equations. Moreover, for a fixed interval of time, the null controllability is characterized in terms of the dual state. Examples of stochastic PDEs are also considered.

Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

We revisit the optimal investment and consumption model of Davis and Norman [Math. Oper. Res., 15 (1990), pp. 676--713] and Shreve and Soner [Ann. Appl. Probab., 4 (1994), pp. 609--692], following a shadow-price approach similar to that of Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20 (2010), pp. 1341--1358]. Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton--Jacobi--Bellman equation for this singular stochastic control problem to a nonstandard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.

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 hol...

Stochastic Perron's method and verification without smoothness using viscosity comparison: Obstacle problems and Dynkin games

We adapt the Stochastic Perron’s method in [1] to the case of double obstacle problems associated to Dynkin games. We construct, symmetrically, a viscosity sub-solution which dominates the upper value of the game and a viscosity super-solution lying below the lower value of the game. If the double obstacle problem satisfies the viscosity comparison property, then the game has a value which is equal to the unique and continuous viscosity solution. In addition, the optimal strategies of the two players are equal to the first hitting times of the two stopping regions, as expected. The (single) obstacle problem associated to optimal stopping can be viewed as a very particular case. This is the first instance of a non-linear problem where the Stochastic Perron’s method can be applied successfully.

Optimal Investment with High-watermark Performance Fee

We consider the problem of optimal investment and consumption when the investment opportunity is represented by a hedge fund charging proportional fees on profit. The value of the fund evolves as a geometric Brownian motion and the performance of the investment and consumption strategy is measured using discounted power utility from consumption on infinite horizon. The resulting stochastic control problem is solved using dynamic programming arguments. We show by analytical methods that the associated Hamilton-Jacobi-Bellman equation has a smooth solution and then obtain the existence and representation of the optimal control in feedback form using verification arguments.

In which financial markets do mutual fund theorems hold true

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are: (i)Let N be the wealth process of the numéraire portfolio (i.e., the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the MFT holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black–Merton–Scholes markets as well as the work of Chamberlain in the framework of Brownian filtrations (Chamberlain in Econometrica 56:1283–1300, 1988).Conversely, under a supplementary weak completeness assumption, we show that the validity of the MFT for general utility functions implies the replicability property for options on the numéraire portfolio described above.(ii)If for a given class of utility functions (i.e., investors) the MFT holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.

Optimal investment on finite horizon with random discrete order flow in illiquid markets

We study the problem of optimal portfolio selection in an illiquid market with discrete order flow. In this market, bids and offers are not available at any time but trading occurs more frequently near a terminal horizon. The investor can observe and trade the risky asset only at exogenous random times corresponding to the order flow given by an inhomogenous Poisson process. By using a direct dynamic programming approach, we first derive and solve the fixed point dynamic programming equation satisfied by the value function, and then perform a verification argument which provides the existence and characterization of optimal trading strategies. We prove the convergence of the optimal performance, when the deterministic intensity of the order flow approaches infinity at any time, to the optimal expected utility for an investor trading continuously in a perfectly liquid market model with no-short sale constraints.

On Martingale Problems with Continuous-Time Mixing and Values of Zero-Sum Games without the Isaacs Condition

We consider a zero-sum stochastic differential game over elementary mixed feedback strategies. These are strategies based only on the knowledge of the past state, randomized continuously in time from a sampling distribution which is kept constant in between some stopping rules. Once both players choose such strategies, the state equation admits a unique solution in the sense of the martingale problem of Stroock and Varadhan. We show that the game defined over martingale solutions has a value, which is the unique continuous viscosity solution of the randomized Isaacs equation.