Research Statement - Carleton University

RESEARCH STATEMENT
Son Luu Nguyen
My primary research interests lie in the areas of regime-switching systems and game theory
with a large population.
Regime-switching diffusions, also known as hybrid diffusions, have two components. One of
them is the continuous state component as in the usual diffusion, and the other is a switching
component represented by a pure jump process. In recent years, it has been recognized that the
traditional formulation of differential equation models is often inadequate for treating complex
systems in many real-world applications. Due to the inherent complexity, the underlying systems
often involve characters that cannot be represented by the usual dynamic systems, but rather
display interactive behavior of continuous and discrete dynamics. As a result, effort has been
directed to finding alternative models. Along this line, regime-switching diffusions have drawn
increasing attention. The models have been used in such applications as option pricing [29], jump
linear systems in automatic control [30], hierarchical decision making in production planning [38],
estimation in hybrid systems [41], stock liquidation [43], and competitive Lotka-Volterra model in
random environments [47] among others.
With the aforementioned motivations, my research to date mainly focuses on the study of hybrid
systems. One part of my research is on the pathwise convergence rates for numerical solutions
of regime-switching stochastic differential equations. Our results are already interesting for the
traditional stochastic differential equations since they present a new angle of ascertaining the convergence
rates. Another part of my research is on asymptotic properties of Markov-modulated
random sequences which can be considered as discrete-time counterpart of switching diffusion systems.
We obtain the weak convergence and a strong invariance principle for the scaled processes
of a two-component process under suitable conditions.
Recently, I also have interests in mean field game theory with mixed players. In the context of
noncooperative game theory, large population models have been well studied in economics [22, 10],
social science [40], biological science [2], and engineering [12]. In these areas, of particular interest
is the class of games in which each player interacts with the average effect of many others and
individually has negligible effect on the overall population. Such an interaction pattern may be
modeled by mean field coupling, and it has naturally arisen in economics [22, 10], engineering
[19], and public health research [5]. The class of games we consider involves a major player and
many minor players. The major player has a significant role in affecting others, but each minor
player possesses only a weak influence. This kind of interaction modeling is motivated by many
socio-economic problems.
For large population dynamic games, a central issue is the development of low complexity
solutions so that each player may implement a strategy based on local information. We consider
a mean field linear-quadratic-Gaussian (LQG) game with a major player and a large number of
minor players parametrized by a continuum set and construct the set of decentralized strategies
which has an ε-Nash equilibrium property when applied to the large but finite population model.
In the following sections, these lines of research are described in more detail. In addition, future
directions of research are outlined.
1 Convergence of Numerical Solutions of Stochastic Differential Equations
Because of the nonlinearity, closed-form solutions of many important stochastic differential equations
are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. Due to
its simplicity, Euler-Maruyama (E-M) numerical method has been investigated intensively by many
authors and for different modes of convergence. For example, the pointwise strong and pointwise
1

weak convergence can be found in [28]. The pathwise strong convergence is studied in [14]. In [6],
the pathwise weak convergence is considered for a general class of numerical methods including
E-M method, but the rate is not obtained.
In our works in [36] and [37], we investigate a new approach to provide the rate of pathwise weak
convergence (i.e. the convergence of the distributions of the whole trajectories of the numerical
solutions) in the sense of strong invariance principle (see [8]) for the E-M numerical solutions of
SDEs and regime-switching SDEs. Different from the well-known results in numerical solutions of
stochastic differential equations, in lieu of the usually employed Brownian motion increments in
the algorithm, an easily implementable sequence of independent and identically distributed (i.i.d.)
random variables is used. Being easier to implement compared to the construction of Brownian
increments, such an i.i.d. sequence is preferable in the actual computation. To establish the
convergence of the algorithms, using either the Brownian increments or i.i.d. random variables do
not make much difference. Nevertheless, the analysis becomes much more difficult for our study
of rates of convergence because one has to deal with the difference in the almost sure sense of the
Brownian increments and the i.i.d. sequence.
Next, we will further describe our recent work [36, 37] concerning convergence rate for E-M
numerical solutions of SDEs and regime-switching SDEs.
1.1 Convergence Rate for Numerical Solutions of SDEs
We begin with the SDE
dX(t) = f(X(t))dt + σ(X(t))dW (t), X(0) = x0, (1.1)
where W (·) is a standard one-dimensional Brownian motion, and f(·) and σ(·) are real-valued
functions satisfying suitable conditions. A simple numerical scheme for obtaining approximate
solutions to (1.1) is the E-M method using Brownian increments: for k ≥ 0 and step size ε,
x ε k+1 = xε k + εf(xε k ) + σ(xε k )( W ((k + 1)ε) − W (kε) ) , x ε 0 = x0.
In [36], we consider a similar scheme called weak E-M numerical method: for k ≥ 0 and step size ε,
x ε k+1 = xε k + εf(xε k ) + √ εσ(x ε k )ξk+1, x ε 0 = x0, (1.2)
where {ξn} is a sequence of independent and identically distributed (i.i.d.) random variables with
mean 0 and covariance 1.
One of the main advantages of this weak method is that it allows us to use simple random
variables {ξn} instead of Brownian increments. However, the solution X(t) to (1.1) may be defined
in a probability space different from that of the sequence {xε k , k ≥ 0} defined in (1.2).
Assumptions. Assume that f(·) and σ(·) are real-valued Lipschitz functions, and {ξn} is a
sequence of bounded i.i.d. random variables with zero mean and variance 1.
Main Result. We show that one can define in the same probability space as the Brownian motion
W (t) a sequence of i.i.d random variables { ˜ ξn} which have the same distribution as that of {ξn}
such that for any 0 < λ < 1/4,
sup
0≤t≤T
ε
X(t) − ˜x (t) λ
= O(ε ) a.s.
where ˜x ε (t) = ˜x ε n for nε ≤ t < (n + 1)ε, and ˜x ε n+1 = ˜xε n + εf(˜x ε n) + √ εσ(˜x ε n) ˜ ξn+1, ˜x ε 0 = x0.
Our approach is based on the Skorohod embedding theorem (see [15]) and the strong invariance
principles. The rate we obtained appear to be sharp. That is, if we use the Skorohod embedding,
it appears that the rate cannot be further improved; see Kiefer [27].
2

1.2 Convergence Rate for Numerical Solutions of Regime-Switching SDEs
In [37], we extend the result in previous section to the following regime-switching diffusion system
and
dX(t) = f(X(t), α(t))dt + σ(X(t), α(t))dW (t), (1.3)
X(0) = x0, α(0) = α0
P ( α(t + ∆t) = j|α(t) = i, α(s), x(s), s ≤ t ) = qij∆t + o(∆t), i, j ∈ M, i ̸= j (1.4)
where W (·) is a d-dimensional standard Brownian motion defined in (Ω, F, P ), x ∈ R d , M =
{1, 2, ..., m}, f(·, ·) : R d × M ↦→ R d , and σ(·, ·) : R d × M ↦→ R d×d are appropriate functions,
and Q = (qij) ∈ R m0×m0 is the generator of the Markov chain satisfying qij ≥ 0 for i ̸= j and
∑ m0
j=1 qij = 0 for each i ∈ M.
To approximate the d-dimensional standard Brownian motion W (·), we use a sequence of i.i.d.
random vectors {ξn, n ≥ 0} taking values in R d . Therefore, to approximate the solution to (1.3)
we propose the following algorithm
x ε n+1 = x ε n + εf(x ε n, α ε n) + √ εσ(x ε n, α ε n)ξn, ε > 0, n = 0, 1, ..., (1.5)
where {ξn, n ≥ 0} is a sequence of R d −valued i.i.d. random vectors and {α ε n, n ≥ 0} is a Markov
chain independent of {ξn, n ≥ 0} with transition probability matrix P = exp(εQ).
Assumptions. We assume the following conditions hold.
(A1) For some K > 0, the functions f and σ satisfy
|f(x, α) − f(y, α)| ≤ K|x − y|, |σ(x, α) − σ(y, α)| ≤ K|x − y| for x, y ∈ R d , α ∈ M.
(A2) {ξn, n ≥ 0} is a sequence of R d -valued i.i.d. random vectors independent of {α ε n, n ≥ 0} such
that Eξ0 = 0 and Eξ0ξ ′ 0 = I. Furthermore, the moment generating function of ξ0 exists, that
is
E[exp⟨h, ξ0⟩] < ∞ for |h| ≤ t0,
where t0 > 0, h ∈ R r , and ⟨h, ξ0⟩ denotes the inner product of h and ξ0.
Main Result. We show that one can construct in the same probability space as the Brownian
motion W (t) a Markov chain {˜α ε n, n ≥ 1} with transition probability matrix P = exp(εQ) and a
sequence of i.i.d. random vectors { ˜ ξn, n ≥ 1} which are independent of {˜α ε n, n ≥ 1} and have the
same distribution as that of {αε n, n ≥ 1} such that for any 0 < λ < 1/4,
sup
0≤t≤T
ε
X(t) − ˜x (t) λ
= O(ε ) a.s.
where ˜x ε (t) = ˜x ε n for nε ≤ t < (n + 1)ε, and ˜x ε n+1 = ˜xε n + εf(˜x ε n, ˜α ε n) + √ εσ(˜x ε n, , ˜α ε n) ˜ ξn+1, ˜x ε 0 = x0.
2 Asymptotic Properties for Markov-Modulated Sequences
My recent research in [34, 35] has been encompassing asymptotic properties of the process {X(k, αk)},
where αk is a Markov chain taking values in a finite set M, and for each α ∈ M, {X(k, α)} is the
primary random sequence. Such processes belong to the class of regime-switching models and may
be viewed as the discrete-time counterpart of the switching diffusion systems. The motivation of
our study stems from a wide variety of applications in actuarial science, communication networks,
3

production planning, manufacturing, and financial engineering. Due to the increasing complexity of
the real-world scenarios, one is often forced to deal with large-scale systems. Due to the uncertainty
of the random environment, there is a growing interest in modeling, analysis, and optimization of
large-scale systems using a regime-switching model in addition to the usual dynamic systems.
For the random sequences under consideration, there are two main issues. Firstly, not much
structure of the sequence {X(k, αk)} is known. Secondly, for a large |M|, there are |M| sequences
of {X(k, α)} with α ∈ M to be dealt with and the complexity becomes an important issue.
Among the large number of states of a Markov chain, a typical situation is that transitions
among some of the states are changing rapidly, whereas others are varying slowly. The state space
can often be split into smaller subspaces such that within each subspace the transitions are about
the same rate, and from one subspace to another, the transitions happen relatively rarely. More
precisely, suppose the state space M admits the representation M = M1 ∪ · · · ∪ Ml0
so that Mi,
i = 1, . . . , l0, can be considered as subspaces, where the Mi’s are not isolated. Such a model is
known to have nearly completely decomposable structure [7, 39] in the literature. A systematic
study of the related Markovian models has been taken recently [45].
Using two-time scales in the formulation, we introduce a small parameter ε > 0 into the transition
probabilities so as to highlight the different rates of transitions. Then, we can write the
sequence as X(k, α ε k ). We assume for each α, {X(k, α)} is mixing, and {αε k
} is a discrete Markov
chain with nearly completely decomposable structure. Due to the lack of structure of the multiple
random sequences, the problem is difficult to solve. Using the idea of aggregation, we lump the
states of the Markov chain in each Mi into one state for i = 1, . . . , l0 to get an aggregate process αε k .
Corresponding to this, we consider a new sequence X(k, αε k ). Effectively, we use a single sequence
{X(k, i)} as a representative for the sequences {X(k, α)} for all α ∈ Mi.
We aim to demonstrate that the new sequence leads to a limit under suitable interpolations.
The limit process is a switching diffusion with a well-defined operator. Denote the original state
space and the state space of the limit by M and M, respectively. If |M| ≫ |M|, a substantial
reduction of computational complexity will be achieved when one treats control and optimization
problems.
In the literature, treating discrete sequences, under suitable scaling, one may obtain a diffusion
limit, which is the classical weak sense functional central limit theorem known as Donsker’s
invariance principle. One may also be interested in obtaining the rates of convergence of the scaled
sequence commonly referred to as strong invariance principle results. In what follows, our effort to
obtain asymptotic properties of the underlying processes under suitable scaling will be described
in more detail.
2.1 Weak Convergence
For ε > 0, let αε k be a time-homogeneous Markov chain on a probability space (Ω, F, P ) with state
space M and transition matrix Pε = P + εQ, where P = (pij) is a transition probability matrix
and Q = (qij) is a generator of a continuous-time Markov chain (i.e., pij ≥ 0 and ∑m j=1 pij = 1;
qij ≥ 0 for i ̸= j and ∑m j=1 qij = 0 for each i). Assume that the state space M can be written as
M = {s11, . . . , s1m1 } ∪ · · · ∪ {sl01, . . . , sl0ml 0 } = M1 ∪ · · · ∪ Ml0 ,
with m0 = m1 + · · · + ml0 and P = diag[P 1 , . . . , P l0 ], where the subspace Mi consists of recurrent
states belonging to the ith ergodic class and the transition probability matrix P i is irreducible and
aperiodic, i = 1, . . . , l0.
) the
Let pε k be the probability vector, pε k = (P (αε k = sij)) ∈ R1×m0 , and νi = (νi1, . . . , νimi
stationary distribution corresponding to the transition matrix P i . Assume that pε 0 = p0 and define
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an aggregated process α ε k of αε k by
α ε k = i if αε k ∈ Mi, i = 1, . . . , l0, α ε (t) = α ε k
for t ∈ [εk, ε(k + 1)).
It is shown in [45] that the aggregated process α ε (·) converges weakly to α(·), which is a continuoustime
Markov chain generated by Q = (q ij) = diag(ν1, . . . , νl0 )Qdiag(1lm1 , . . . , 1lml 0 ) where 1ll =
(1, . . . , 1) ′ ∈ R l×1 .
For each sij ∈ M, let {X(k, sij)} be a wide-sense stationary sequence of R d -valued random
vectors on (Ω, F, P ). Denote X(n, i) = ∑ mi
j=1 νijX(n, sij) for i = 1, ..., l0.
Assumptions. Assume that the sequence {(X(k, s11), . . . , X(k, sl0ml )) : k ∈ Z} is independent of
0
the Markov process {αε k }, and is ϕ-mixing with mixing measure denoted by ϕ(·). Moreover, assume
that there exist δ > 0 and a constant C that does not depend on k, i such that
∞∑
n=0
ϕ(n) δ
1+δ < ∞, EX(k, sij) = 0, E|X(k, sij)| 2(1+δ) ≤ C, k ≥ 0, sij ∈ M.
Main Results. We obtain the following results:
(i) Denote
z ε (t) = √ ε
⌊t/ε⌋−1
∑
n=0
[ ε
X(n, αn) − X(n, α ε n) ] = √ ⌊t/ε⌋−1 ∑
ε
n=0
l0∑ mi ∑
X(n, sij)[1 {αε − ν n=sij} ij 1 {αε n=i}].
i=1 j=1
Then (z ε (·), α ε (·)) converges weakly to (z(·), α(·)) which is the unique solution to the martingale
problem with operator L determined by the initial layer term in the asymptotic expansion of ( ) k;
Pε
see [45].
(ii) Denote zε (t) = √ ε ∑⌊t/ε⌋−1 n=0 X(n, αεn). Then (zε (·), αε (·)) converges weakly to (z(·), α(·))
which is the unique solution to the martingale problem with operator given by
Lf(x, i) = 1
2
d∑
d∑
j1=1 j2=1
where A(i) = (a j1j2 (i)) = EX(0, i)X ′ (0, i) + ∑ ∞
k=1
Qf(x, ·)(i) = ∑ l0
j=1 q ijf(x, j).
2.2 Strong Invariance Principle
a j1j2 ∂
(i) 2f(x, i)
∂xj1∂x + Qf(x, ·)(i), i = 1, . . . , l0,
j2
[
EX(k, i)X ′ (0, i) + EX(0, i)X ′ ]
(k, i)
Let ε > 0 and αε k be a time-homogeneous Markov chain on (Ω, F, P ) with state space M =
{1, 2, . . . , m} and transition matrix
Pε = P + εQ, (2.1)
where P = (pij) is a transition probability matrix and Q = (qij) is a generator of a continuous-time
Markov chain. Suppose that P is irreducible and aperiodic (i.e. l0 = 1 or M = M1) with the
stationary distribution denoted by ν = (ν1, . . . , νm) ∈ R1×m . Denote pε k = (P (αε k = 1), . . . , P (αε k =
m)) and assume that the initial probability p ε 0 is independent of ε, i.e., pε 0
= p0.
Let {(X(k, 1), X(k, 2), . . . , X(k, m)) : k ∈ Z} be an Rm-valued wide-sense stationary ϕ-mixing
sequence on (Ω, F, P ) which is independent of the Markov process {αε k }. Denote
z ε (t) = √ ⌊t/ε⌋−1 ∑
ε
n=0
m∑
i=1
5
X(k, i)[1 {α ε n =i} − νi].
and

Assumptions. Assume that the sequence {X(k, i) : k ∈ Z, i ∈ M} and its the mixing measure
function ϕ(·) satisfy
EX(k, i) = 0, E|X(k, i)| 4 4
−
< ∞, ϕ(n) < Cn 3 (1+β) for some constants C and β > 0.
Main Result. We show that one can construct on a probability space a one-dimensional Brownian
motion W (t) with EW (t) = 0 and E[W (t)] 2 = σ 2 t where σ can be determined explicitly, and
stochastic processes ˜z ε (t) having the same distributions as those of z ε (t) such that
for some θ > 0.
sup
0≤t≤T
ε
˜z (t) − W (t) θ
= o(ε ) a.s.
3 LQG Mixed Games with Continuum-Parametrized Minor Players
We consider a mean field LQG game model with a major player and a large number of minor
players which are parametrized by a continuum set. Large population stochastic dynamic games
with mean field coupling have experienced intense investigation in the past decade; see, e.g., [19, 20,
23, 24, 25, 26]. To obtain low complexity strategies, consistent mean field approximations provide
a powerful approach, and in the resulting solution, each agent only needs to know its own state
information and the aggregate effect of the overall population which may be pre-computed off-line.
One may further establish an ε-Nash equilibrium property for the set of control strategies [20].
The LQG model in [18] shows that the presence of the major player causes an interesting
phenomenon called the lack of sufficient statistics. More specifically, in order to obtain asymptotic
equilibrium strategies, the major player cannot simply use a strategy as a function of its current
state and time; for a minor player, it cannot simply use the current states of the major player and
itself. To overcome this lack of sufficient statistics for decision, the system dynamics are augmented
by adding a new state, which approximates the mean field and is driven by the major player’s state.
This additional state enters the obtained decentralized strategy of each player and it captures the
past history of the major player.
A crucial modeling assumption in [18] is that the minor players are from a finite number of
classes labelled by a set {1, . . . , K}, where players in each class share the same set of parameters
in their dynamics and costs. The size of the additional state introduced in [18] depends on the
number of classes so that it provides sub-mean field approximations for different classes, and this
approach becomes invalid when the dynamic parameters are from an infinite set.
In [32], we consider a population of minor players parametrized by an infinite set such as
a continuum, and seek a different approach for mean field approximations. Due to the linear
quadratic structure of the game with a finite number of players, it is plausible to assume that the
limiting mean field is a Gaussian process and may be represented by using the driving noise of the
major player. Eventually, we will justify this argument by showing the consistency of the mean field
approximation. Given the above representation of the limiting mean field, we may approximate
the original problems of the major player and a typical minor player by stochastic control problems
with random coefficients in the dynamics and costs [3]. This further enables the use of powerful
tools from the theory of backward stochastic differential equations [3]. Also, the Gaussian property
of various processes involved will play an important role and we exploit this to develop kernel
representation to reduce the analysis to function spaces [17].
Mean Field Games with Major and Minor Players. Consider the LQG game with a major
player A0 and a population of minor players {Ai, 1 ≤ i ≤ N}. At time t ≥ 0, the states of the
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players A0 and Ai are, respectively, denoted by x0(t) and xi(t), 1 ≤ i ≤ N. Let (Ω, F, Ft, t ≥ 0, P )
be the underlying filtration. The dynamics of the N + 1 players are given by a system of linear
stochastic differential equations (SDEs)
dx0(t) = [ A0x0(t) + B0u0(t) + F0x (N) (t) ] dt + D0dW0(t), (3.1)
dxi(t) = [ A(θi)xi(t) + B(θi)ui(t) + F (θi)x (N) (t) ] dt + D(θi)dWi(t), (3.2)
where x (N) = 1/N ∑ N
i=1 xi is the mean field term. The initial states {xj(0), 0 ≤ j ≤ N} are measurable
with respect to F0 and have finite second moments. The states x0, xi and controls u0, ui
are, respectively, n and n1 dimensional vectors. The noise processes W0 and Wi are n2 dimensional
independent standard Brownian motions adapted to (Ft)t≥0, which are also independent of
{xj(0), 0 ≤ j ≤ N}. The vector θi ∈ R d is a parameter in the dynamics of player Ai.
For 0 ≤ j ≤ N, denote u−j = (u0, ..., uj−1, uj+1, ..., uN). For positive semi-definite matrix
M ≥ 0, we may write the quadratic form z T Mz as |z| 2 M . The cost function for A0 is given by
∫ T [
J0(u0, u−0) = E x0(t) − χ0(x
0
(N) (t)) 2 Q0 + uT0 (t)R0u0(t) ] dt, (3.3)
where χ0(x (N) (t)) = H0x (N) (t) + η0, Q0 ≥ 0 and R0 > 0. The cost function for Ai, 1 ≤ i ≤ N, is
∫ T [
Ji(ui, u−i) = E xi(t) − χ(x0(t), x
0
(N) (t)) 2 Q + uT i (t)Rui(t) ] dt, (3.4)
where χ(x0(t), x (N) (t)) = Hx0(t) + ˆ Hx (N) (t) + η, Q ≥ 0 and R > 0. The component Hx0 in
the coupling term χ indicates the strong influence of the major player on each minor player. By
contrast, the mutual impact of two minor players is insignificant. We assume that all matrix
or vector parameters (A0, B0, . . . , η, . . . ) in (3.1)-(3.4) are deterministic and have compatible
dimensions.
Assumptions. We introduce the following assumptions:
(A1) The initial states {xj(0), 0 ≤ j ≤ N}, are independent, and there exists a constant C independent
of N such that sup 0≤j≤N E|xj(0)| 2 ≤ C.
(A2) There exists a distribution function F(θ, x) on Rd+n such that the sequence of empirical
distribution functions FN(θ, x) = 1 ∑N N i=1 1 {θi≤θ,Exi(0)≤x}, N ≥ 1, where each inequality holds
componentwise, converges to F(θ, x) weakly, i.e., for any bounded and continuous function
h(θ, x) on Rd+n ,
∫
lim
N→∞ Rd+n ∫
h(θ, x)dFN(θ, x) =
Rd+n h(θ, x)dF(θ, x).
(A3) A(·), B(·), F (·) and D(·) are continuous matrix-valued functions of θ ∈ Θ, where Θ is a
compact subset of R d .
Main Results. We approximate the mean field generated by the minor players by a kernel
representation using the Brownian motion of the major player.
Solve local optimal control problems for both the major player and a representative minor player
via backward stochastic differential equations.
Construct a set of decentralized control strategies based on consistent mean field approximations
and show that these strategies have an ε-Nash equilibrium property.
7

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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all probability measures on Rn , and εN x(t) = (1/N) ∑N i=1 δxi(t). This is an interesting extension of
traditional stochastic interacting particle system considered in [9] with a major particle.
In [9] (see also [13]), several significant results for the traditional stochastic interacting particle
system are presented. In particular, the limiting process of the sequence {εN x(t) } is characterized
as a weak solution to a stochastic McKean-Vlasov equation. For the above system, we also hope
to characterize the limit of the coupled sequence {(x0(t), ε N x(t)
)} as a weak solution to a coupled
stochastic McKean-Vlasov equation by using martingale problem approach. Moreover, conditions
for the existence and uniqueness for the weak solution, the strong solution, and the regular property
of the paths of the solution to the desired equation may be determined by considering the martingale
problems in a suitable rigged Hilbert space. This study is expected to bring new insights to
stochastic mean field control and game problems with mixed players in our far future plan.
Numerical Solution for Mean Field Stochastic System. Let consider the celebrated mean
field stochastic differential equation (MFSDE) in which the state process involves as follow
(
dX(t) = f t, X(t), Eφ ( X(t) )) (
dt + σ t, X(t), Eφ ( X(t) ))
dW (t) (4.3)
where W (t) is a multi-dimensional Brownian motion defined on a probability space (Ω, F, P ).
The studies of MFSDE are of great theoretical values due to its interesting state law dependence
structure. On the other hand, the MFSDE also possesses broad-range real applications. Such largepopulation
interacting system has already been extensively investigated in many different fields,
such as engineering and socioeconomic science, game theory, finance, etc. Since the MFSDEs are
well-defined and widely-used systems, it is of importance to study the numerical approximations
of X(t) and its probability law µt. In [1] and [4], stochastic particle methods were proposed to
construct approximations for the distribution function Vt of µt. These methods, however, require to
generate too many stochastic processes if one expects to get an accurate approximation. Therefore,
we intend to investigate another method to find a numerical approximation of Vt and X(t) with
reduced computational effort. It is expected that by combining both the MFSDE of X(t) and the
Fokker-Planck-Kolmogorov equation of Vt, we will be able to obtain the desired method.
I am keen on continuing these and other inter-disciplinary projects in the directions described
above. One of my primary goal at this stage is to keep broadening my horizons to as many areas
of mathematics as possible.
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School of of Mathematics & Statistics, Carleton University, 1125 Colonel By
Drive, Ottawa, ON, Canada, K1S 5B6
Email: snguyen@math.carleton.ca
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