## About

38

Publications

2,740

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353

Citations

Citations since 2017

Introduction

Additional affiliations

July 2021 - July 2021

**University of Calgary**

Position

- Professor (Associate)

May 2017 - June 2021

January 2016 - May 2017

## Publications

Publications (38)

In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs a group of interacting particles, which perform a minimization over one v...

This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases, the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton–Jaco...

This paper is concerned with the large particle limit for the consensus‐based optimization (CBO), which was postulated in the pioneering works by Carrillo, Pinnau, Totzeck and many others. In order to solve this open problem, we adapt a compactness argument by first proving the tightness of the empirical measures {μN}N≥2$$ {\left\{{\mu}^N\right\}}_...

In this paper, a Neumann problem for the backward stochastic partial differential equation (BSPDE) with singular terminal condition is studied, which characterizes the value function for a constrained stochastic control problem (also called optimal liquidation problem) in target zone models. The existence and the uniqueness of strong solutions to s...

In this paper we provide a rigorous convergence analysis for the renowned Particle Swarm Optimization method using tools from stochastic calculus and the analysis of partial differential equations. Based on a time-continuous formulation of the particle dynamics as a system of stochastic differential equations, we establish the convergence to a glob...

We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward–backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Com...

In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods. Such problems arise in many situations of contemporary interest in machine learning and signal processing. After a brief overview of metaheuristic methods...

This paper is concerned with the large particle limit for the consensus-based optimization (CBO), which was postulated in the pioneering works [6,28]. In order to solve this open problem, we adapt a compactness argument by first proving the tightness of the empirical measures $\{\mu^N\}_{N\geq 2}$ associated to the particle system and then verifyin...

Recently a continuous description of the particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in arXiv:2012.05613 where the authors formally showed the link between PSO and the consensus based optimization (CBO) through zero-inertia limit. This paper is devoted to solving this...

In this paper, we propose and study a stochastic aggregation-diffusion equation of the Keller-Segel (KS) type for modeling the chemotaxis in dimensions
d
=
2
,
3
. Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idi...

We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Com...

In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solution is proved for general nonline...

In this paper, we propose and study the stochastic path-dependent Hamilton-Jacobi-Bellman (SPHJB) equation that arises naturally from the optimal stochastic control problem of stochastic differential equations with path-dependence and measurable randomness. Both the notions of viscosity solution and classical solution are proposed, and the value fu...

This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependent and random coefficients. As opposed to the {deterministic} path-dependent cases, the value function turns out to be a random field on the path spaces and it is characterized with a stochastic path-dependent Hamilton-Ja...

In this paper, we study a class of zero-sum two-player stochastic differential games with the controlled stochastic differential equations and the payoff/cost functionals of recursive type. As opposed to the pioneering work by Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp.~293--314] and the seminal work by Buckdahn and Li [SIAM J. C...

We study the optimal liquidation problems in target zone models using dynamic programming methods. Such control problems allow for stochastic differential equations with reflections and random coefficients. The value function is characterized with a Neumann problem of backward stochastic partial differential equations (BSPDEs) with singular termina...

In this paper, we propose and study a stochastic aggregation-diffusion equation of the Keller-Segel (KS) type for modeling the chemotaxis in dimensions $d=2,3$. Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncr...

This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the ass...

We solve the optimal control problem of a one-dimensional reflected stochastic differential equation, whose coefficients can be path dependent. The value function of this problem is characterized by a backward stochastic partial differential equation (BSPDE) with Neumann boundary conditions. We prove the existence and uniqueness of sufficiently reg...

This paper is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the assoc...

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model conv...

In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity...

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum princi...

In this paper, we are concerned with possibly degenerate stochastic partial
differential equations (SPDEs). An $L^2$-theory is introduced, from which we
derive the H\"ormander theorem with an analytical approach. With the method of
De Giorgi iteration, we obtain the maximum principle which states the $L^p$
($p\geq 2$) estimates for the time-space u...

A H\"ormander type theorem is established for It\^o processes and related
backward stochastic partial differential equations (BSPDEs). A short
self-contained proof is also given for the $L^2$-theory of degenerate BSPDEs,
in which an estimate on directional derivatives is obtained.

This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation
with controlled leading coefficients, which is a type of fully nonlinear
backward stochastic partial differential equation (BSPDE for short). In order
to formulate the weak solution for such kind of BSPDEs, the classical potential
theory is generalized in the backward stoc...

We study a constrained optimal control problem with possibly degenerate
coefficients arising in models of optimal portfolio liquidation under market
impact. The coefficients can be random in which case the value function is
described by a degenerate backward stochastic partial differential equation
(BSPDE) with singular terminal condition. For this...

We consider a stochastic model for the dynamics of the two-sided limit order
book (LOB). For the joint dynamics of best bid and ask prices and the standing
buy and sell volume densities, we derive a functional limit theorem, which
states that our LOB model converges to a continuous-time limit when the order
arrival rates tend to infinity, the impac...

In this paper, we are concerned with backward doubly stochastic differential
evolutionary systems (BDSDESs for short). By using a variational approach based
on the monotone operator theory, we prove the existence and uniqueness of the
solutions for BDSDESs. We also establish an It\^o formula for the Banach
space-valued BDSDESs.

We establish existence and regularity results for a class of backward
stochastic partial differential equations with singular terminal condition. The
equation describes the value function of a non-Markovian stochastic control
optimal problem in which the terminal state of the controlled process is
prespecified. The analysis of such control problems...

This paper is concerned with the quasi-linear reflected backward stochastic
partial differential equation (RBSPDE for short). Basing on the theory of
backward stochastic partial differential equation and the parabolic capacity
and potential, we first associate the RBSPDE to a variational problem, and via
the penalization method, we prove the existe...

A coupled forward-backward stochastic differential system (FBSDS) is
formulated in spaces of fields for the incompressible Navier-Stokes equation in
the whole space. It is shown to have a unique local solution, and further if
either the Reynolds number is small or the dimension of the forward stochastic
differential equation is equal to two, it can...

In the paper, we consider a special coupled forward-backward stochastic
differential system (FBSDS) which is associated to the viscous
incompressible Navier-Stokes equation and provides a probabilistic
solution to the latter via the Feynman-Kac formula. With a probabilistic
method, we first prove the existence and uniqueness of the solution to
the...

The paper is concerned with the existence and uniqueness of a strong solution
to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear
forcing, driven by a Brownian motion. We use the spectral approximation and the
truncation and variational techniques. The methodology features an interactive
analysis on basis of the regularit...

In this paper we are concerned with the maximum principle for quasi-linear
backward stochastic partial differential equations (BSPDEs for short) of
parabolic type. We first prove the existence and uniqueness of the weak
solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral
boundary. Then using the De Giorgi iteration schem...

This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An L
p
-theory is given for the Cauchy problem of BSPDEs, separately for the case of p∈(1,2] and for the case of p∈(2,∞). A comparison theorem is also addressed.