Liquidity effects, risk control and BSDEs – LIQUIRISK

From the financial point of view, the purpose of this proposal is to take into consideration liquidity market frictions and risk management constraints in hedging and optimal allocation problems. From the mathematical point of view, this requires new researches on statistics for random processes, theoretical mathematical finance, singular constrained stochastic control policies, highly non linear second order parabolic PDEs, regularity and discrete time approximation of Backward Stochastic Differential Equations (BSDEs).

We decompose the project in three interconnected tasks, concerning respectively the design of optimal hedging policies under liquidity frictions, the characterization of optimal allocation strategies under risk constraints and finally the corresponding BSDE representation and numerical approximation.

The first task is dedicated to the analysis of the impact on hedging strategies of elaborate market frictions that incorporate liquidity effects and include market microstructure, exogenous liquidity regimes, trading price impactor order book dynamics. The consequences induced by the addition of the market frictions will be studied from a theoretical point of view (absence of arbitrage, optimal controls) and from the practical one (parameters estimation, tracking error).

In the second task, we intend to focus on the influence of additional prudential risk management issues on optimal allocation strategies. This leads to the introduction of non-standard optimal control problems connected to stochastic target problems which translate into highly-non linear second order parabolic PDEs on which little is known (e.g. comparison principle, convergence of numerical schemes). Once the corresponding stochastic control problems well understood, we shall investigate the effects of the liquidity frictions modeled in the first task.

In order to propose alternatives to standard resolution technics for PDEs for the resolution of the previous control problems, we intend to use and sometimes introduce in the third task some probabilistic reformulations of such problems in terms of BSDEs. This corresponds to the consideration of additional constraints, delays or irregularities to classical BSDE framework. Finally the design and convergence rates of corresponding purely probabilistic associated numerical schemes is investigated and we introduce in particular new high order schemes for BSDEs. Such formulations enable to consider non-Markovian settings and are also particularly efficient for high-dimensional problems.

To sum up, we first investigate the incorporation of liquidity frictions on hedging policies and we study the impact of prudential risk control restrictions on optimal allocation strategies. These two fields of investigation lead to the consideration of non standard singular stochastic
control problems. We finally discuss alternative BSDE representations and analyze the convergence of associated numerical schemes.

All these problems will be discussed from the mathematical point of view in the most general possible framework and the combination of the whole program will be done on specific problems of interest in practical finance.