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Qingshuo Song

Publications and source records attributed to Qingshuo Song.

At least 19 recordsLinked to original sources
arXiv2026

Ergodic McKean-Vlasov Games: Verification Theorems and Linear-Quadratic Applications

This paper investigates two-player ergodic nonzero-sum stochastic differential games with McKean-Vlasov dynamics. We establish a verification theorem connecting solutions of coupled Hamilton-Jacobi-Bellman (HJB) Master equations to Nash equilibria, characterized through an auxiliary control problem defined on the measure space. A key contribution is showing that the value functions are uniquely determined (up to an additive constant) by the uniqueness of the invariant measure of the optimal state process. The theory is applied to Linear-Quadratic-Gaussian (LQG) settings, where explicit solutions to the Master equations are derived by exploiting their polynomial structure in measure variables.

math.OC↗
arXiv2026

Long-Time Behaviors of Stochastic Linear-Quadratic Optimal Control Problems

This paper investigates the asymptotic behavior of the solution to a linear-quadratic stochastic optimal control problems. The so-called probability cell problem is introduced the first time. It serves as the probability interpretation of the well-known cell problem in the homogenization of Hamilton-Jacobi equations. By establishing a connection between this problem and the ergodic cost problem, we reveal the turnpike properties of the linear-quadratic stochastic optimal control problems from various perspectives.

math.OC↗
arXiv2025

On modified Euler methods for McKean-Vlasov stochastic differential equations with super-linear coefficients

We introduce a new class of numerical methods for solving McKean-Vlasov stochastic differential equations, which are relevant in the context of distribution-dependent or mean-field models, under super-linear growth conditions for both the drift and diffusion coefficients. Under certain non-globally Lipschitz conditions, the proposed numerical approaches have half-order convergence in the strong sense to the corresponding system of interacting particles associated with McKean-Vlasov SDEs. By leveraging a result on the propagation of chaos, we establish the full convergence rate of the modified Euler approximations to the solution of the McKean-Vlasov SDEs. Numerical experiments are included to validate the theoretical results.

math.NA↗
arXiv2023

The convergence rate of the equilibrium measure for the hybrid LQG Mean Field Game

In this work, we study the convergence rate of the $N$-player LQG game with a Markov chain common noise towards its asymptotic Mean Field Game. By postulating a Markovian structure via two auxiliary processes for the first and second moments of the Mean Field Game equilibrium and applying the fixed point condition in Mean Field Game, we first provide the characterization of the equilibrium measure in Mean Field Game with a finite-dimensional Riccati system of ODEs. Additionally, with an explicit coupling of the optimal trajectory of the $N$-player game driven by $N$ dimensional Brownian motion and Mean Field Game counterpart driven by one-dimensional Brownian motion, we obtain the convergence rate $O(N^{-1/2})$ with respect to 2-Wasserstein distance.

math.OC↗
arXiv2023

Convergence Rate of LQG Mean Field Games with Common Noise

This paper focuses on exploring the convergence properties of a generic player's trajectory and empirical measures in an N-player Linear-Quadratic-Gaussian Nash game, where Brownian motion serves as the common noise. The study establishes three distinct convergence rates concerning the representative player and empirical measure. To investigate the convergence, the methodology relies on a specific decomposition of the equilibrium path in the N-player game and utilizes the associated Mean Field Game framework.

math.PR↗
arXiv2022

From Mean Field Games To Navier-Stokes Equations

This work establishes the equivalence between Mean Field Game and a class of PDE systems closely related to compressible Navier-Stokes equations. The solvability of the PDE system via the existence of the Nash Equilibrium of the Mean Field Game is provided under a set of conditions.

math.AP↗
arXiv2020

Inverse Gaussian quadrature and finite normal-mixture approximation of the generalized hyperbolic distribution

In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss-Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but it exactly integrates the polynomials of both positive and negative orders. Using the quadrature, the generalized hyperbolic distribution is efficiently approximated as a finite normal variance-mean mixture. Therefore, the expectations under the distribution, such as cumulative distribution function and European option price, are accurately computed as weighted sums of those under normal distributions. The generalized hyperbolic random variates are also sampled in a straightforward manner. The accuracy of the methods is illustrated with numerical examples.

stat.CO↗
arXiv2020

Solving A Class of Mean-Field LQG Problems

In this work, we study a class of mean-field linear quadratic Gaussian (LQG) problems. Under suitable conditions, explicit solutions of the distribution-dependent optimal control problems are obtained. Riccati systems are derived by directly solving the associated master equations. Some extensions on controls with partial observations are also considered.

math.PR↗
arXiv2019

Exit problem as the generalized solution of Dirichlet problem

This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $α$-stable processes, which help verify the solvability of the original HJB equation.

math.PR↗
arXiv2018

The density evolution of the killed Mckean-Vlasov process

The study of the density evolution naturally arises in Mean Field Game theory for the estimation of the density of the large population dynamics. In this paper, we study the density evolution of McKean-Vlasov stochastic differential equations in the presence of an absorbing boundary, where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the unique solution of an integro-differential Fokker-Planck equation with Cauchy-Dirichlet data.

math.PR↗
arXiv2017

Solvability of Dirichlet problem with Integro-differential Operator

This paper studies the solvability of a class of Dirichlet problem associated with non-linear integro-differential operator. The main ingredient is the probabilistic construction of continuous supersolution via the identification of the continuity set of the exit time operators under Skorohod topology.

math.AP↗
arXiv2017

The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent $δ\in (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $δ\ge 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.

math.PR↗
arXiv2016

Weak Convergence of Path-Dependent SDEs in Basket CDS Pricing with Contagion Risk

We investigate the computational aspects of the basket CDS pricing with counterparty risk under a credit contagion model of multinames. This model enables us to capture the systematic volatility increases in the market triggered by a particular bankruptcy. The drawback of this problem is its analytical complication due to its path-dependent functional, which bears a potential failure in its convergence of numerical approximation under standing assumptions. In this paper we find sufficient conditions for the desired convergence of the functionals associated with a class of path-dependent stochastic differential equations. The main ingredient is to identify the weak convergence of the approximated solution to the underlying path-dependent stochastic differential equation.

q-fin.MF↗
arXiv2015

A Strong Limit Theorem for Two-Time-Scale Fucntional Stochastic Differential Equations

This paper focuses on a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional. It develops ergodicity of the fast component and obtains a strong limit theorem for the averaging principle in the spirit of Khasminskii's averaging approach for the slow component.

math.PR↗
arXiv2014

On Singular Control Problems with State Constraints and Regime-Switching: A Viscosity Solution Approach

This paper investigates a singular stochastic control problem for a multi-dimensional regime-switching diffusion process confined in an unbounded domain. The objective is to maximize the total expected discounted rewards from exerting the singular control. Such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments. With the aid of weak dynamic programming principle and an exponential transformation, we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi-variational inequalities. Several examples are analyzed in details to demonstrate the main results.

math.OC↗
arXiv2014

On Optimal Harvesting Problems in Random Environments

This paper investigates the optimal harvesting strategy for a single species living in random environments whose growth is given by a regime-switching diffusion. Harvesting acts as a (stochastic) control on the size of the population. The objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting {\em up to the time of extinction} of the species; the income rate is allowed to be state- and environment-dependent. This is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition. One aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy. Consequently, one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon. This paper introduces a new sufficient condition under which the continuity of the value function for the regime-switching model is established. Further, it is shown that the value function is a viscosity solution of a coupled system of quasi-variational inequalities. The paper also establishes a verification theorem and, based on this theorem, an $\varepsilon$-optimal harvesting strategy is constructed under certain conditions on the model. Two examples are analyzed in detail.

math.OC↗