Huyên Pham

Bio: Huyên Pham is an academic researcher from Paris Diderot University. The author has contributed to research in topics: Stochastic control & Stochastic differential equation. The author has an hindex of 26, co-authored 89 publications receiving 1675 citations. Previous affiliations of Huyên Pham include ENSAE ParisTech & Centre national de la recherche scientifique.

A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization

Abstract: We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte-Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the error of the scheme is provided, as well as numerical tests on the problem of superreplication of option with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [7].

Optimal quantization methods for nonlinear filtering with discrete-time observations

Abstract: We develop an optimal quantization approach for numerically solving nonlinear filtering problems associated with discrete-time or continuous-time state processes and discrete-time observations. Two quantization methods are discussed: a marginal quantization and a Markovian quantization of the signal process. The approximate filters are explicitly solved by a finite-dimensional forward procedure. A posteriori error bounds are stated, and we show that the approximate error terms are minimal at some specific grids that may be computed off-line by a stochastic gradient method based on Monte Carlo simulations. Some numerical experiments are carried out: the convergence of the approximate filter as the accuracy of the quantization increases and its stability when the latent process is mixing are emphasized.

Deep backward schemes for high-dimensional nonlinear PDEs

Abstract: We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minima as it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

Optimal Portfolio Liquidation with Execution Cost and Risk

Abstract: We study the optimal portfolio liquidation problem over a finite horizon in a limit order book with bid-ask spread and temporary market price impact penalizing speedy execution trades. We use a continuous-time modeling framework, but in contrast with previous related papers (see, e.g., [L. C. G. Rogers and S. Singh, Math. Finance, 20 (2010), pp. 597-615] and [A. Schied and T. Schoneborn, Finance Stoch., 13 (2009), pp. 181-204]), we do not assume continuous-time trading strategies. We consider instead real trading that occur in discrete time, and this is formulated as an impulse control problem under a solvency constraint, including the lag variable tracking the time interval between trades. A first important result of our paper is to prove rigorously that nearly optimal execution strategies in this context actually lead to a finite number of trades with strictly increasing trading times, and this holds true without assuming ad hoc any fixed transaction fee. Next, we derive the dynamic programming quasi-variational inequality satisfied by the value function in the sense of constrained viscosity solutions. We also introduce a family of value functions which converges to our value function and is characterized as the unique constrained viscosity solutions of an approximation of our dynamic programming equation. This convergence result is useful for numerical purpose but is postponed until a companion paper [F. Guilbaud, M. Mnif, and H. Pham, Numerical Methods for an Optimal Order Execution Problem, preprint, 2010].

Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE

Abstract: We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of Forward Backward Stochastic Differential Equation (FBSDE) with a simulatable forward process For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

Abstract: Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.