We introduce a notion of approximate viscosity solution for a class of nonlinear path-dependent PDEs (PPDEs), including the Hamilton-Jacobi-Bellman type equations. Existence, comparaison and stability results are established under fairly general cond
itions. It is also consistent with smooth solutions when the dimension is less or equal to two, or the non-linearity is concave in the second order space derivative. We finally investigate the regularity (in the sense of Dupire) of the solution to the PPDE.
Using Dupires notion of vertical derivative, we provide a functional (path-dependent) extension of the It^os formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated b
y its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty
We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce the number
of historical scenarios that potentially belong to the set of worst scenarios. The number of simulations increases as the number of candidate scenarios is reduced and the distance between them diminishes. For the most naive scheme, we show that the L p-error of the estimator of the Expected Shortfall is bounded by a linear combination of the probabilities of inversion of favorable and unfavorable scenarios at each step, and of the last step Monte Carlo error associated to each scenario. By using concentration inequalities, we then show that, for sub-gamma pricing errors, the probabilities of inversion converge at an exponential rate in the number of simulated paths. We then propose an adaptative version in which the algorithm improves step by step its knowledge on the unknown parameters of interest: mean and variance of the Monte Carlo estimators of the different scenarios. Both schemes can be optimized by using dynamic programming algorithms that can be solved off-line. To our knowledge, these are the first non-asymptotic bounds for such estimators. Our hypotheses are weak enough to allow for the use of estimators for the different scenarios and steps based on the same random variables, which, in practice, reduces considerably the computational effort. First numerical tests are performed.
We prove a robust super-hedging duality result for path-dependent options on assets with jumps, in a continuous time setting. It requires that the collection of martingale measures is rich enough and that the payoff function satisfies some continuity
property. It is a by-product of a quasi-sure version of the optional decomposition theorem, which can also be viewed as a functional version of It{^o}s Lemma, that applies to non-smooth functionals (of c{`a}dl{`a}g processes) which are only concave in space and non-increasing in time, in the sense of Dupire.
We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong re
gularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (2007). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of It{^o}s Lemma for path-dependent functionals that are only C^{0,1} in the sense of Dupire.
We extend the study of [7, 18] to stochastic target problems with general market impacts. Namely, we consider a general abstract model which can be associated to a fully nonlinear parabolic equation. Unlike [7, 18], the equation is not concave and th
e regularization/verification approach of [7] can not be applied. We also relax the gamma constraint of [7]. In place, we need to generalize the a priori estimates of [18] and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the super-hedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefficients are smooth enough. This representation leads to a general dual formulation. We finally provide an asymptotic expansion around a model without impact.
We model the behavior of three agent classes acting dynamically in a limit order book of a financial asset. Namely, we consider market makers (MM), high-frequency trading (HFT) firms, and institutional brokers (IB). Given a prior dynamic of the order
book, similar to the one considered in the Queue-Reactive models [14, 20, 21], the MM and the HFT define their trading strategy by optimizing the expected utility of terminal wealth, while the IB has a prescheduled task to sell or buy many shares of the considered asset. We derive the variational partial differential equations that characterize the value functions of the MM and HFT and explain how almost optimal control can be deduced from them. We then provide a first illustration of the interactions that can take place between these different market participants by simulating the dynamic of an order book in which each of them plays his own (optimal) strategy.
In this paper we study a class of combined regular and singular stochastic control problems that can be expressed as constrained BSDEs. In the Markovian case, this reduces to a characterization through a PDE with gradient constraint. But the BSDE for
mulation makes it possible to move beyond Markovian models and consider path-dependent problems. We also provide an approximation of the original control problem with standard BSDEs that yield a characterization of approximately optimal values and controls.
We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on
this, we propose two new numerical schemes inspired by the branching processes based algorithm of [8]. Our numerical experiments show that approximating the discontinu-ous driver of the associated reaction/diffusion PDE by local polynomials is not efficient, while a simple randomization procedure provides very good results.
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this r
equires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm.
