Related papers: Martingale Inequalities and Deterministic Counterp…
Optimality conditions in the form of a variational inequality are proved for a class of constrained optimal control problems of stochastic differential equations. The cost function and the inequality constraints are functions of the…
We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an…
This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from…
We propose martingale consumption as a natural, desirable consumption pattern for any given (proportional) investment strategy. The idea is to always adjust current consumption so as to achieve level expected future consumption under the…
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…
In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and…
Variational inequalities are a universal optimization paradigm that is interesting in itself, but also incorporates classical minimization and saddle point problems. Modern realities encourage to consider stochastic formulations of…
This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a num\'eraire. It is shown that the presence of arbitrarily small…
With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time,…
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of…
While many questions in robust finance can be posed in the martingale optimal transport framework or its weak extension, others like the subreplication price of VIX futures, the robust pricing of American options or the construction of…
We consider the problem of optimal consumption from labor income and investment in a general incomplete semimartingale market. The economic agent cannot borrow against future income, so the total wealth is required to be positive at (all or…
We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and…
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly…
Variational inequalities are a universal optimization paradigm that incorporate classical minimization and saddle point problems. Nowadays more and more tasks require to consider stochastic formulations of optimization problems. In this…
We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more…
Given the marginal distribution information of the underlying asset price at two future times $T_1$ and $T_2$, we consider the problem of determining a model-free upper bound on the price of a class of American options that must be…
In this article we present a general framework for non-concave robust stochastic control problems under model uncertainty in a discrete time finite horizon setting. Our framework allows to consider a variety of different path-dependent…
We consider the pricing of derivatives in a setting with trading restrictions, but without any probabilistic assumptions on the underlying model, in discrete and continuous time. In particular, we assume that European put or call options…
We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…