# Abstracts

**Duality in optimal investment with convex frictions**

*Ari-Pekka Perkkiö (Aalto University, Finland)*

Wednesday June 4, 11:30-12:00 | session 4.9 | Transaction Costs | room H

We study the optimal investment problem with portfolio constraints and nonlinear trading costs where we optimize over adapted left continuous processes of bounded variation. We apply the conjugate duality framework to the problem so that the parameters enter the problem as pairs of random variables and optional random measures that describe the amount of cash to be delivered at the terminal time and the quantities of assets to be delivered over time. We derive a dual representation for the optimal value function and we give conditions for the existence of solutions. In particular, for markets with proportional transaction costs without portfolio constraints, we recover well-known dual expressions in terms of consistent price systems.

**Strong supermartingales and portfolio optimisation under transaction costs**

*Christoph Czichowsky (London School of Economics and Political Science, UK)*

Wednesday June 4, 12:00-12:30 | session 4.9 | Transaction Costs | room H

In this talk, we develop a dynamic duality theory for portfolio optimisation under proportional transaction costs with cadlag price processes. In particular, we provide examples that illustrate the new effects arising from the combination of the transaction costs and jumps of the price process.

**Pricing and hedging contingent claims with liquidity costs and market impact**

*Grégoire Loeper (Ecole Centrale Paris, France)*

Wednesday June 4, 12:30-13:00 | session 4.9 | Transaction Costs | room H

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterize its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.