We consider the classical Merton problem of lifetime consumption-portfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in...

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in [12], and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main...

This paper analyzes the interactions between competitive (wholesale) spot, retail, and forward markets and vertical integration in electricity markets. We develop an equilibrium model with producers, retailers, and traders to study and quantify the impact of forward markets and vertical...

We consider a financial market with liquidity cost as in Cetin, Jarrow and Protter [3] where the supply function S"(s; ) depends on a parameter " 0 with S0(s; ) = s corresponding to the perfect liquid situation. Using the PDE characterization of Cetin, Soner and Touzi [6] of the super-hedging...

This paper analyzes the interactions between vertical integration and (wholesale) spot, forward and retail markets in risk management. We develop an equilibrium model that fits electricity markets well. We point out that vertical integration and forward hedging are two separate levers for demand...

In the context of the multi-dimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. Similar to the one-dimensional derivation in our accompanying paper [42], the asymptotic expansion is...

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by [9,11]. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more...

We define a coherent risk measures as set-valued maps satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner, Delbaen, Eber and Heath (1998). We then discuss the aggregation issue, i.e. the passage from valued random...

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod...