Research Statement - Carleton University

4 Future Research Plan
My goal is to be a research mathematician and an educator, and to make life-time commitment
to mathematical research and training the next generation of mathematicians, scientists, and engineers.
My ultimate plan is to make substantial contributions to the research of stochastic systems
and game theory. In the next few years, my research plan is outline below.
4.1 Further Topics in Stochastic Hybrid Systems
The study of stochastic hybrid systems is still in an early stage. A number of problems remain
open. Obtaining convergence rates for other numerical methods like Milstein scheme (see [28, 31])
for regime-switching diffusion processes will have many significant applications, especially in the
more challenging case that the switching component is diffusion-dependent. Next, concerning the
degenerated switching diffusion, can we obtain a strong invariance principle? It appears that the
desired result will be more difficult to obtain compared to the case of traditional diffusion process
(see [16]) since one needs to deal with the Markov dependent component when estimating the
moments. Also, it will be useful and natural to consider the regime-switching model for the system
of many interacting particles in a random environment. Although some stability results in this
direction when the number of particles is finite was obtained in [42, 44], there are still a lot of
open questions on the asymptotic behavior of such systems when the number of particles becomes
large. Attempts in finding answers to these questions present new challenges, which will lead to
many substantial discoveries. Hence, my immediate future work will be devoted to the above
problems. In addition, I will also focus on applications of the theoretical results of hybrid systems
to real world problems including risk theory, financial engineering, stochastic approximation and
stochastic optimization and control for stochastic hybrid systems.
4.2 Mean-Field Game Theory and Stochastic McKean-Vlasov Equations
In the past decade, the mean field games theory has been investigated intensively with many
applications in wireless communication systems, biological science, and economics. However, this
still is a relative new area with many open questions. In the future, I would like to focus on
this topic. In particular, I am interested in the general problems in the mean field games theory
with a large population of minor player and a major player which generalize the LQG cases (see
[18, 19, 20]), numerical solutions for mean-field game and control problems (see [21]), and stochastic
mean-field systems (see [9, 13]). In what follows, two specific problems will be described in more
detail.
Mean-Field Game Problem. In order to obtain low complexity strategies so that each player
only needs to know its own state information, the consistent mean field approximation (see [18,
19, 20]) is a powerful approach. For the general mean field game problems with a major player
and a large population of minor players, this approach leads to the importance of investigating the
limiting behavior of the empirical measure obtained by stochastic interacting particles described
by the following equations
(
dx0(t) = f0 t, x0(t), ε N ) (
x(t) dt + σ0 t, x0(t), ε N )
x(t) dW0(t), (4.1)
dxi(t) = f ( t, x0(t), xi(t), ε N )
N∑
x(t) dt + σ ( t, x0(t), xi(t), ε N x(t) , j) dWj(t), 1 ≤ i ≤ N (4.2)
j=1
where {Wi, 0 ≤ i ≤ N} are n dimensional independent standard Brownian motions, f0, σ0 : R ×
R n × P → R n , f : R × R 2n × P → R n , σ : R × R 2n × P × {1, 2, ..., N} → R n , P is the set of
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