Viscosity solutions for controlled McKean--Vlasov jump-diffusions
Matteo Burzoni Vincenzo Ignazio A. Max Reppen H. Mete Soner















Viscosity solutions for
controlled McKean–Vlasov jump-diffusions

    Matteo BurzoniMathematical Institute, University of Oxford. Partly supported by the Hooke Research Fellowship from the University of Oxford.1[4]
    Vincenzo IgnazioDepartment of Mathematics, ETH Zürich.1[4]
A. Max ReppenDepartment of Operations Research and Financial Engineering, Princeton University. Supported by the Swiss National Science Foundation grant SNF 181815.
H. Mete Soner[2]1Partly supported by the Swiss National Science Foundation grant SNF 200020_172815.1Partly supported by the ETH Foundation and the Swiss Finance Institute.

    December 30, 2023
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Burzoni, Ignazio, Reppen & SonerControlled McKean–Vlasov jump-diffusions



We study a class of non linear
integro-differential equations on the 
Wasserstein space related to 
the optimal control of McKean–Vlasov jump-diffusions.
We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space
and prove a comparison theorem for these solutions.
We also show that the value function is the unique viscosity solution.






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§ INTRODUCTION


The main goal of this paper
is to develop a viscosity theory for
integro-differential  equations on the 
Wasserstein space related to 
the optimal control of McKean–Vlasov jump-diffusions.
These control problems
 are motivated
by the mean field games theory
developed by Lasry & Lions <cit.> 
(see also the videos of the College de France
lectures of Lions <cit.>) and
by Huang, Caines & Malhamé <cit.>.
Although the mean-field games and McKean–Vlasov
control problems are related, there are subtle differences between 
these problems,
and a thorough introduction is 
given by Carmona, Delarue & Lachapelle <cit.>.
Indeed, for both problems the master equations  share many common
properties as initially derived by Bensoussan, Freshe & Yam
<cit.>.
We refer to the videos of Lions <cit.>, the lecture notes of Cardaliguet <cit.>
and the exhaustive book of Carmona & Delarue <cit.> for more 
information on both problems and also for further references.

The state space of these problems is 
the set of probability measures, and in most
applications the Wasserstein space of 
probability measures with finite second moments are used.
Since this space is not linear, one encounters 
some difficulties in differentiation
and Lions <cit.>
observed that one can naturally lift functions defined on
the Wasserstein space to functions on an appropriate ^2 space,
which allows for standard differentiation
and more importantly an immediate
use of Itô's calculus.
This approach is then used 
by Cardaliguet, Delarue, Lasry & Lions
<cit.> to obtain the regularity
of the solutions to the master equation
of a mean-field game.  This very strong regularity 
result  implies in particular for a classical interpretation 
of the master equations on the Wasserstein space.  On the other hand,
in the absence of such strong regularity,
one needs to develop the notion of viscosity solutions 
for McKean–Vlasov control problems.
Pham & Wei <cit.> initiated this study
using Lions' lifting for controlled diffusion processes.
Bandini, Cosso, Fuhrman & Pham <cit.>
further developed this theory 
for the dynamic programming equations for the 
partially observed systems which also have the same structure.  
An important advantage of this
approach to viscosity theory, in addition to the Hilbert structure
of ^2, is its ability to
utilize the existing results for viscosity solutions
on Hilbert spaces <cit.>.  An intrinsic 
approach to viscosity solutions without lifting 
could also have advantages and Wu & Zhang <cit.>
studies this approach for diffusion process
using the techniques developed for path-dependent
viscosity solution <cit.>.

Our goal is to develop  a viscosity theory for
jump-diffusion processes.  For the standard control
problems, the corresponding 
dynamic programming equations 
contain a non-local integral terms
related to the infinitesimal generator of
the jump-Markov processes.  Still
these equations have maximum principle
and  a viscosity theory is appropriate.
Starting from <cit.>
definitions, stability and comparison results for nonlinear integro-differential
equations of this type have
been developed.  We refer to more recent paper
by Barles & Imbert <cit.> for more information.

The jump terms in these equations introduce
several new aspects.  In particular,  
for the McKean–Vlasov control problems,
the operator appearing in the dynamic programming equations
do not act on the Lions derivative (i.e., the derivative in the ^2
space of the lifted function)
but rather on the standard (sometimes called linear) derivative 
on the Wasserstein space.
Indeed, when all functions are smooth, 
 it is immediate that
the Lions derivative is an ^2 function
and it is equal to the space derivative of the linear 
derivative (see  Section 5.4 in <cit.>).  For the diffusion problems,
only the space derivatives of 
the linear derivative appear in the 
dynamic programming equation and therefore 
one can simply replace them by the Lions derivative.
For the integro-differential equations however,
one needs to recover the linear
derivative from the Lions derivative 
even to state the equations.  
Unfortunately the required regularity (to 
immediately connect these two derivatives)
is not
readily available when one is working 
in the viscosity structure. 

We choose
to work directly on the Wasserstein space with the  
linear derivative
to develop an intrinsic theory.
Although this approach has several
advantages, the dynamic programming equations
on the Wasserstein space are not as well 
studied 
as the lifted equation on the ^2 spaces
and parts of the viscosity theory has to be 
revisited.
Indeed, we  first provide 
appropriate definitions of viscosity sub and super-solutions
for a class of integro-differential 
equations in this space.  
We then show that the value function is a viscosity
solution in this sense.  Several properties 
of the dynamics is used to construct 
the framework that is appropriate for this
problem. In particular, we consider the 
equation only on the subset of the 
measures that have exponential moments.

One of the main contributions of this paper
is a comparison result for the viscosity solutions 
on the Wasserstein space.  
An important ingredient of our approach is a
 distance like function d
given for two probability measures μ, ν by,

    d(μ,ν)= ∑_j=1^∞ c_j ⟨μ-ν,f_j⟩^2,

where the countable set {f_j}_j ∈
is carefully constructed to have
several important invariance type properties.
In the 
standard doubling-variables argument,
we penalize the two
points using d.  Then the subtle properties of 
f_j 
allow us to estimate its linear derivative of d
by itself.  

The paper is organized as follows.  We first introduce
a class of optimal control problems of McKean–Vlasov
type in the next section.  A guiding example
for this class is a model of technological innovation
<cit.>.  We discuss this problem
in Section <ref>.  The natural state space 
for this study is the subset of the Wasserstein space
of measures with exponential moments and under mild
assumptions, the corresponding dynamical system 
lives in this space.  In Section <ref> we
define this space, prove its functional analytic properties 
and show its connection to the 
controlled dynamics. In Section <ref> we give the 
definition of a viscosity solution and in Section <ref>
show that the
value function is a viscosity solution.  Section <ref> provides
the construction 
of the functions f_j and the comparison result.  We prove several technical results
in the Appendix.








Notation.
For a random variable X, defined on a probability space 
(Ω,,P), 
we denote by (X) the distribution of X under P.
We denote by () the space of probability measures 
on  and by  the linear space of countably additive measures.
For any μ∈() and for any integrable function f:→, we 
use the standard compact notation ⟨μ,f⟩:=∫_f(x)μ( x).
If f is smooth, f^(i) denotes the i-th order derivative of f with f^(0)=f.
We endow the space of probability measures ()  with the weak^* topology σ((),_b()), where _b() is the space of continuous and bounded functions on .
We denote by μ_n→μ the σ((),_b())-convergence of μ_n to μ, i.e.,
⟨μ_n,f⟩ converges to ⟨μ,f⟩ for every f ∈_b().



§ THE OPTIMAL CONTROL PROBLEM AND THE ASSUMPTION



Let (Ω,,(_s)_s∈[0,T],P) 
be a given filtered probability space supporting the 
following class of controlled McKean–Vlasov 
stochastic differential equations (SDEs) with initial condition
 (X^α_t)= μ∈() and

    X^α_s=b(s,(X^α_s),α_s)s
    +σ(s,(X^α_s),α_s)  W_s + J_s,
       s >t,

where J_s is a purely discontinuous process with controlled intensity λ(s,(X^α_s),α_s) and jump size given by an independent random variable ξ with distribution γ∈().
The class of admissible controls  is 
the set of all measurable deterministic functions of time
 with values in a prescribed measurable space A.
The value function is then given by

    V(t,μ):=inf_α∈[∫_t^T L(s,(X^α_s),α_s)s
    +G((X^α_T))],

with given functions L and G.
The optimal control problem consists of 
finding the value V and a minimizer (if it exists).

We close this section
by stating a set of conditions  assumed
to hold throughout the paper and they will not always be stated
explicitly later on.


There exist constants C_0,κ_0,δ>0 such that the coefficients b,σ,λ,L:[0,T]×()× A→ satisfy:


  * 
for any μ∈(), a∈ A, s∈[0,T],

    |b(s,μ,a)|+|σ(s,μ,a)|+|λ(s,μ,a)|≤ C_0


  * 
there exists a finite set ⊂ such that for any μ,μ'∈(), a∈ A, t,s∈[0,T]

    |b(t,μ,a)-b(s,μ',a)|+|σ(t,μ,a)-σ(s,μ',a)|   ≤κ_0(|t-s|+∑_i∈|⟨μ - μ',x^i⟩|),
    
    |λ(t,μ,a)-λ(s,μ',a)|   ≤κ_0(|t-s|∑_i∈|⟨μ - μ',x^i⟩|).


  *  γ has δ-exponential moment:

    ∫_exp(δ|x|)γ( x)<∞.


  * 
    L is of the form L_1(t,μ, a)+ L_2(a) ⟨μ,L_3(·)⟩, where
    L_1:[0,T]×()× A→ is continuous in (t,μ), 
    uniformly in a, L_2 : A → with sup_a ∈ A L_2(a) < ∞, 
    and L_3:→ satisfies |L_3(x)x|≤ C_0exp(δ |x|) for every x∈.  
    The terminal cost G is continuous.



In what follows, the constants C_0,κ_0,δ>0 are 
always as in the above assumption.



§ STATE SPACE AND DYNAMIC PROGRAMMING



Since the Brownian motion has 
exponential moments, Assumption <ref>,
in particular <ref>, 
implies that the solutions of the state equation (<ref>)
has also exponential moments.  Therefore it is natural
to study the optimal control problem
in := [0,T) ×, where 
is the subset of probability measures with δ-exponential moments,
i.e., 

    μ∈  ⇔  ⟨μ,exp(δ|·|)⟩ = ∫_exp(δ|x|) μ( x)<∞,

where δ is as in <ref>.
Our first result is the well-posedness of the problem 
and its straightforward proof 
is given in Appendix <ref>.
 Under Assumption <ref>, the SDE (<ref>) has a unique solution 
for any α∈. 


The described McKean–Vlasov control problem 
is deterministic and therefore, it is classical that  
the dynamic programming principle (DPP)  holds <cit.>, 

    V(t,μ)= inf_α∈[∫_t^θ L(s,(X^α_s),α_s)s 
    +V(u,(X^α_u))],   ∀θ∈ [t,T].



We need several definitions to formally state the corresponding
dynamic programming equation.  



For φ:()→, when exists, the linear derivative of φ 
at μ∈() is a function 
φ:()×→ such that for every μ,μ'∈(),

    φ(μ)-φ(μ')=∫_0^1∫_φ(λμ+(1-λ)μ',x) (μ-μ')(dx)  λ.

When φ:[0,T]×()→, with an abuse
of notation, we denote the linear derivative with respect to
the μ-variable still by φ:[0,T]×()×→.


 This derivative was used by Fleming & Viot <cit.> to
study a martingale problem  in populations dynamics.
Also  recently Cuchiero, Larsson & Svaluto-Ferro <cit.> 
provided several of its properties in the  context of polynomial diffusions.


Consider the linear function φ(μ)=⟨μ,f⟩
with some f:→.  It is 
immediate that φ(μ,x)=f(x) for any (μ,x)∈()×.
Moreover, suppose that φ:→ is Frechet differentiable and such that Dφ:→ can be represented as Dφ[μ]=⟨μ,f⟩, for some f:→. Then f=φ, namely, Dφ[μ]=⟨μ,φ⟩.

By the chain rule,  the linear derivative of φ(μ)=F(⟨μ,f⟩)
with some smooth function F is equal to 
φ(μ,x)=F^' (⟨μ,f⟩)f(x).


For a given input function v=v(t,μ,x), the operator 
^a,μ_t acting on the x-variable is given by,

    ^a,μ_t[v](x)   :=      b(t,μ,a)∂v/∂x(t,μ,x)  
    +1/2σ^2(t,μ,a) ∂^2 v/∂x^2(t,μ,x)

               +λ(t,μ,a)∫_(v(t,μ,x+y)-v(t,μ,x))γ(dy).


Using the above definitions, 
classical considerations starting from (<ref>)
formally lead to the following 
dynamic programming equation:

    -∂_t V(t,μ) + sup_a ∈ AH^a(t,μ, V)=0,

where,

    H^a(t,μ, v):=- L(t,μ,a)-⟨μ, ^a,μ_t[v]⟩.


Indeed, as in the finite-dimensional 
optimal control theory,
if the value function is smooth
and cylindrical (i.e., if 
V has the form

    V(t,μ)=F(t,⟨μ,f_1⟩,…,⟨μ,f_n⟩)

for some smooth functions F and f_1,…,f_n,
then it is possible to derive (<ref>) rigorously.
Importantly, in this case, the classical Itô's Formula 
can be applied to  V(u,(X^α_u))=
F(u,[f_1(X^α_u)],…,[f_n(X^α_u)]) 
for any control α and time u.
However, this assumption on the value
function is not expected to hold and also is not needed.
In Section <ref>, 
we prove that the value function is the unique viscosity solution to (<ref>)
even when it is neither smooth nor cylindrical.





§ A MODEL OF TECHNOLOGICAL INNOVATION



We briefly present here an example of a McKean–Vlasov control 
problem where the underlying process is a jump-diffusion.
The controlled equations represent a model of knowledge
diffusion which appeared in the macroeconomic literature in the area of 
search-theoretic models of technological change, e.g. <cit.>[We thank Rama Cont for bringing this paper to our attention.],<cit.>.
With controls α=(α^1,α^2), a social planner 
aims at promoting technological innovation in the society by controlling the process

    X^α_s=b([X^α_s],α^1_s)s+σ W_s +  J_s,

where X_0∼μ_0 and J_s is a purely discontinuous process 
with controlled intensity λ([X^α_s],α^2_s) and jump size given by a non-negative independent random variable ξ with distribution γ.
The value exp(X^α_s) represents the efficiency of the production of a continuum of consumption goods (technological frontier),
and the initial (logarithmic) efficiency is represented by the distribution μ_0.
The aim is to maximize the average efficiency of the production of goods in order to foster the growth of the economy:

    maximize!  [∫_0^T (1 - α^2_s) exp(X^α_s)-(α^1_s)^2s],

where α=(α^1,α^2) is chosen from an
appropriate class of deterministic processes.

The social planner can promote innovation by issuing research funds (exercising the control α^1). On the other hand, she can promote exchange of ideas by setting up meetings at a controlled Poisson rate.
Meetings have the effect of inducing a non-negative  jump in the technological frontier, according to a random variable with distribution γ.
The functions λ and b are bounded since meetings cannot happen too frequently and research funds have a limited impact on the technological frontier.
These functions also depends on the distribution of X^α through its mean.
This aspect represents the positive feedback effect of a productive economy: if the average productivity is higher, technological improvements and meetings happen spontaneously at a higher rate.
Finally, the random Brownian component incorporate fluctuations in the efficiency of the production due to external contingent factors.
This model satisfies Assumption <ref> under some 
appropriate regularity conditions on the parameters and initial distribution.

We refer to <cit.> for further
examples of problems where the 
controlled process is only a diffusion without jump terms.




§ Σ-COMPACTNESS OF THE STATE SPACE



Recall that
:= [0,T)×, and   
is the set of probability measures μ satisfying 
⟨μ,exp(δ|·|)⟩<∞,
where δ is as in <ref>.
We endow this space with the subspace topology induced by , 
i.e., weak^* convergence.
We use the product topology on 
: = [0,T]×,
and emphasize that  is not the topological closure of , but simply includes the final time.

The space 
has a  suitable σ-compact structure 
which is compatible
with the McKean–Vlasov dynamics.
This representation of 
 is instrumental to obtain uniform integrability of the viscosity test 
functions as well as some continuity properties of the Hamiltonian. We continue
by constructing this structure.

For δ as in <ref>, set

    e_δ(x):= exp(δ [√(x^2+1) -1]),    x ∈.

We note that  e_δ is twice continuously differentiable and

    exp(δ [|x|-1]) ≤ e_δ(x) ≤exp(δ|x|) ≤ e^δ e_δ(x),
      ∀ x ∈.

 For  N ∈
 and C_0,δ
as in Assumption <ref>, let

    := {(t,μ) ∈[0,T)×()|⟨μ,e_δ⟩≤
     Ne^K^*t },

where 

    K^*= K^*(C_0,δ):=
    δ C_0/2(2+C_0+δ C_0) +C_0(∫_e^δ |x|γ( x)-1).

The exact definition
of K^* is not important for the functional analytic properties
of  but is used centrally  
in the next lemma to prove an
invariance property.
  
It is clear that  
= [0,T)× = ∪_N=1^∞
and = ∪_N=1^∞, where

    := {(t,μ)∈[0,T]×()|⟨μ,e_δ⟩≤
     Ne^K^*t }.

 We also use the following notation for a constant  b>0,
 
    b := {μ∈()|⟨μ,e_δ⟩≤ b }.

 
The following lemma shows that for each N,
 and thus also , remains invariant under the controlled dynamics (<ref>)
for any control.
In particular, this means that for any given initial law μ∈,
we may restrict the dynamic programming equation (<ref>) 
to .


Under Assumption <ref>, for any N ∈,
the set  is invariant for the SDE (<ref>), namely,

    (t,μ)∈⇒ (u,(X^t,μ,α_u))∈  ∀(u,α)∈ [t,T]×,

where (X^t,μ,α_u)_u∈[t,T] is the solution to (<ref>) with initial condition 
(X^t,μ,α_t)=μ.


Set φ(x):=√(x^2+1) -1 so that

    e_δ(x)=  e^δφ(x),    x ∈.

It is clear that φ is twice continuously differentiable and
both  |φ'| and φ” >0 
are bounded by 1.

Fix (t,μ)∈ and α∈.
For u∈[t,T],
set Y_u:=e^δφ(X_u),
μ_u:=(X_u),
where X_u:=X^t,μ,α_u.
In particular, μ_t = μ for any control α
and by  Itô's Formula,

    Y_u   =       Y_t+∫_t^u b(s,μ_s,α_s)
     δφ'(X_s) Y_ss 

              +1/2∫_t^uσ^2(s,μ_s,α_s)
     [δφ”(X_s)+δ^2(φ'(X_s))^2]Y_ss

              +∫_t^uσ(s,μ_s,α_s)
     δφ'(X_s)Y_s W_s+∑_0≤s≤tΔY_s.

In view of Assumption <ref>, the stochastic integral in the above formula is a local martingale.
We  take expectation on both side up to a localizing sequence of stopping times 
{τ_n}_n.
We also use Assumption <ref>, to estimate
the expectation of the second and third term of the above sum 
are bounded by

    C_1[∫_t^u Y_s∧τ_ns],

where C_1:=δ C_0/2(2+C_0+δ C_0) and C_0 is as in Assumption <ref>.

We next estimate  e_J:=[∑_0≤ s≤ t∧τ_nΔ Y_s].
First observe that for any x,y ∈,
|φ(y+x) -φ(y)| ≤ |x|.
We then estimate by using Assumption <ref>, 

    e_J   =∫_t^u∧τ_nλ(s,μ_s,α_s) 
    ∫_ e^δφ(X_s∧τ_n+x)
    -e^δφ(X_s∧τ_n)γ( x)s
       ≤ C_0∫_t^u∧τ_nY_s∧τ_n∫_(e^δ |x|-1)γ( x)s
       ≤ C_2∫_t^uY_s∧τ_ns,

where C_2:=C_0 (∫_e^δ|x|γ( x)-1).
These and Fubini's Theorem imply that

    [Y_u∧τ_n] ≤[Y_t]+K^*  ∫_t^u[Y_s∧τ_n]s,

where K^* is as in (<ref>). 
By Gronwall's Lemma and Fatou's Lemma, 

    [Y_u]≤ e^K^*(u-t)[Y_t]
    =e^K^*(u-t) ⟨μ, e^δφ⟩= 
    e^K^*(u-t)⟨μ, e_δ⟩.

As (t,μ) ∈, ⟨μ, e_δ⟩≤ N e^K^*t.  Hence,

    [Y_u]≤ e^K^*(u-t)⟨μ, e_δ⟩≤ e^K^*u.



We provide the proof of the following simple result for
completeness.


For N ∈,
 is a compact subset of [0,T]×().


For and b>0 and R sufficiently large,

    sup_μ∈bμ([-R,R]^c) ≤sup_μ∈b∫_|x|≥ Re_δ(x)/|x|μ( x)≤b/R.

and the last term converges to 0 as R→∞.
Hence b is tight and by Prokhorov's Theorem,
it is relatively compact.
We next show that it is also closed.
Consider a sequence {μ_n}_n∈⊂b such that μ_n→μ.
Set f_m(x):=e_δ(x) ∧ m.  
 Since f_m∈_b(), f_m ≤ e_δ(x) and μ_n ∈b, 

    ⟨μ, f_m⟩ = lim_n →∞⟨μ_n, f_m⟩≤ b,   ∀ m>0.

By monotone convergence theorem,

    ⟨μ, e_δ⟩ = lim_m →∞⟨μ, f_m⟩≤ b.

Hence, μ∈b and consequently, b is compact.

For every N,  is a subset of [0,T]×N e^K^*t, hence, it is relatively compact.
Consider a sequence {(t_n,μ_n)}_n∈⊂ 
such that (t_n,μ_n)→(t,μ).
Proceeding exactly as above, we can show that

    ⟨μ, e_δ⟩ = 
    lim_n →∞lim_m →∞⟨μ_n, f_m⟩≤ N e^K^* t.

Hence, (t,μ)∈ and consequently,  is compact.


We close this section by recalling a well-known result;  
see <cit.>.
Suppose μ, ν∈.  Then,

    μ=ν  ⇔  ⟨μ-ν,x^j⟩ =0,   ∀ j=1,2,…




§ VISCOSITY SOLUTIONS AND TEST FUNCTIONS



In this section,
we define viscosity sub and super-solutions
to the dynamic programming equation  eq:DPE.
As it is standard in the viscosity theory, 
one has to first specify the class of test functions.
We continue by this selection.


A cylindrical function is a map of the form (t, μ) ↦ F(t,⟨μ,f⟩) 
for some function f:→ and  F:[0,T]×→.
This function is called cylindrical polynomial if f is a polynomial
and F is continuously differentiable.


The above class is not  large
enough and we extend it to
its linear span.



For E ⊆, a viscosity test function on E is a
function of the form 

    φ(t,μ)=∑_j=1^∞φ_j(t,μ),
       (t,μ) ∈ E,

where φ_j are cylindrical polynomials
and for every N ∈,

    lim_M→∞∑_j=M^∞sup_(t,μ) ∈ E ∑_i=0^(φ_j)| ⟨μ, (φ_j)^(i)⟩| =0.

We let _E be the set of all viscosity test functions on E.


 Lemma <ref> below shows that
for a cylindrical polynomial φ, 
⟨μ, (φ)^(i)⟩
is uniformly bounded on (t,μ,a) ∈× A
for every i=0,…,  (φ).
Therefore, they are test functions on every .



There are several other choices for test functions.  
In particular, we could even restrict F to be 
quadratic or extended to be more general
with some integrability properties.
They all would yield equivalent definitions
and we do no pursue this equivalence here.

When μ_t is the law of
a stochastic process X_t and
φ is  a cylindrical function, 
φ(t,μ_t)
=F(t,⟨μ_t,f⟩)=F(t,[f(X_t)]).
Then, one can employ the standard Itô formula;
see Proposition <ref> below.




 For E ⊆ and (t,μ) ∈ E with t<T, the
 superjet of u at (t,μ) is given by,
 
    E u(t,μ):={(∂_t φ(t,μ), 
     φ(t,μ,·))|φ∈_E,  
     (u-φ)(t,μ) = max_E (u-φ)}.

The subjet of u at (t,μ) 
is defined as E u(t,μ):=-E(-u)(t,μ).



On a subspace E ⊆,
the (sequential) upper semicontinuous envelope of u on E
is defined by[As  is first countable, semicontinuity coincides with sequential semicontinuity.]

    u^*_E(t,μ):=lim sup_E  (t',μ')→(t,μ)u(t,μ),

where the lim sup is taken over all sequences in E converging to (t, μ).
The lower semicontinuous envelope u_*^E is defined analogously.


We use the compact notations

    u^*:= u^*_,   
    u_*:= u_*^,    
    u^*_N:= u^*_,   
    u_*^N:= u_*^.

We note that as opposed to the finite-dimensional cases,
when u is not continuous,
the dependence of u^*_N and u_*^N on N is non-trivial.  
This emanates from the 
fact that the interiors of all  are empty.

To simplify the notation, we write H = sup_a ∈ A H^a.

 
We say that a function u:→ is a
viscosity sub-solution of (<ref>) on , if for every (t,μ)∈,

    -π_t + H(t,μ,π_μ)   ≤ 0    ∀(π_t,π_μ)∈ u^*_N(t,μ).

We say that a function v:→ is a
viscosity super-solution
of (<ref>) on , if for every (t,μ)∈,

    -π_t + H(t,μ,π_μ)   ≥ 0    ∀(π_t,π_μ)∈ u_*^N(t,μ).

A viscosity solution of (<ref>) is a 
function on  that is both a sub-solution and a super-solution 
of (<ref>) on , for every N∈. 


We continue with several  technical results.
Ultimately, we want to show some continuity properties of H.



We say that g has 
δ-subexponential growth if 
|g(x)x|≤Ĉ e^δ|x| for some Ĉ>0 and every x∈.

Note that any polynomial has 
δ-subexponential growth.


Let δ>0 be as in <ref>.
For any g with δ-subexponential growth,

    sup_μ∈b⟨μ,|g|⟩<∞,
      and  lim_R →∞sup_μ∈b∫_|x|≥ R|g(x)|μ( x)= 0.

Moreover, there is a constant C, depending
only on the constants appearing in Assumption <ref>,
such that for any cylindrical polynomial φ and N ∈,

    sup_a ∈ A, (t,μ) ∈|
    ⟨μ, ^a,μ_t[φ]⟩| ≤
    C sup_(t,μ) ∈∑_i=0^(φ)| ⟨μ, (φ)^(i)⟩| <∞.



The estimate sup_μ∈b⟨μ,|g|⟩<∞
follows directly from Assumption <ref> and the definition of b.
Indeed, since g has δ-subexponential growth,

    |g(x)x|≤Ĉexp(δ |x|) ≤Ĉ e^δ e_δ(x)=: C̃ e_δ(x),
       x ∈.

Hence, for R≥ 1,

    sup_μ∈b∫_|x|≥ R|g(x)|μ( x) 
       ≤C̃sup_μ∈b∫_|x|≥ Re_δ(x)/|x|μ( x)
       ≤C̃/Rsup_μ∈b∫_|x|≥ Re_δ(x)μ( x)
       ≤bC̃/R.


Let f be a polynomial. Then,

    ⟨μ, ^a,μ_t[f]⟩=
        b(t,μ,a) ⟨μ,f^'⟩   +1/2σ^2(t,μ,a) ⟨μ,f^''⟩
       +λ(t,μ,a)⟨μ,∫_(f(x+y)-f(x))γ(dy)⟩.

We rewrite the last term by Taylor expansion of the polynomial f as follows,

    ⟨μ,∫_(f(x+y)-f(x))γ(dy)⟩=
    ∑_i=1^(f)⟨μ,f^(i)⟩/i!∫_y^iγ(dy).

The above equations, together with Assumption <ref> and the fact that all derivatives of f have δ-subexponential growth, imply (<ref>).
The result for a cylindrical polynomial follows similarly.





For every φ∈_, (t,μ) ∈ 
and α∈, 

    φ(u,μ_u)
    = φ(t,μ)
    + ∫_t^u [∂_t φ(s,μ_s)+
    ⟨μ_s, ^α_s,μ_s_s[D_mφ]⟩] s,
       u ∈ [t,T],

where μ_s=(X^t,μ,α_s) and
 (X^t,μ,α_s)_s∈[t,T] is the solution to (<ref>) with initial
distribution μ.
Moreover, the map (t, μ) ↦ H(t, μ, φ) 
is continuous on . 


Fix φ∈_, 
(t,μ) ∈ and α∈
and let μ_s be as in the statement.
In view of Lemma <ref>, μ_s ∈
for all s ∈ [t,T].

Let first φ(μ)=⟨μ,f⟩ with f polynomial, so that D_mφ=f and ⟨μ_s, f ⟩ =  f(X^t,μ,α_s). By stochastic calculus

    ⟨μ_u, f ⟩ = ⟨μ, f ⟩ 
    +∫_t^u ⟨μ_s, ^α_s,μ_s_s[f]⟩s.

Moreover, this derivative is uniformly bounded on  by the 
previous lemma.
Now consider a 
cylindrical polynomial φ(t,μ)=F(t,⟨μ,f⟩).
By calculus,

    φ(u,μ_u) = φ(t,μ) 
    +∫_t^u [∂_t φ(s,μ_s) 
    +F_x(s,⟨μ_s,f⟩) ⟨μ_s, ^α_s,μ_s_s[f]⟩] s.

Since D_mφ(s,μ)= F_x(s,⟨μ,s⟩) f, 
the above proves (<ref>)
for cylindrical polynomials.
For a general φ∈_, (<ref>)
follows directly from above, the condition
(<ref>) and the fact that μ_s ∈
for all s ∈ [t,T].

We now show continuity of H.
Since all derivatives of f have δ-subexponential growth,
Lemma <ref> and the fact that φ is a 
smooth function imply ⟨μ, (φ)^(i)⟩ 
is continuous on every , for any i∈.
In particular the uniform continuity of 
(t, μ) ↦⟨μ, ^a,μ_t[φ(t,μ)]⟩ 
follows from <ref> and <ref> and for L it is assumption <ref>.
Hence, H(t,μ, φ) is continuous  for all cylindrical polynomials.
This continuity extends directly to all functions of the type
φ^M:=∑_j=1^M F_j (⟨μ,f_j⟩).

Now consider a general
test function 
φ=∑_j=1^∞ F_j (⟨μ,f_j⟩) and 
for M ∈ set φ^M:=∑_j=1^M F_j (⟨μ,f_j⟩).
Since φ∈_,
it satisfies (<ref>).  This together with
(<ref>) imply that

    lim_M→∞sup_a ∈ A, (t,μ) ∈∑_j=M^∞|⟨μ, ^a,μ_t[φ_j]⟩| =0.

The above uniform limit enables us
to conclude that 
 H(t,μ, φ^M)
converges uniformly to H(t,μ, φ)
as M tends to infinity.
Hence, H(t,μ, φ) is also
continuous.





§ VALUE FUNCTION

 
 
In this section we show that 
the value function V is a viscosity solution to (<ref>).
We start with two technical lemmata.



 For every N∈, (t_0, μ_0) ∈, there exists a viscosity test function
 ϕ∈_
 such that ϕ(t,μ) ≥ 0, with equality only in (t_0, μ_0),
 and
 
    (ϕ(t_0,μ_0),∂_t ϕ(t_0, μ_0), ϕ(t_0, μ_0, ·)) = (0, 0, 0).

 In particular, in the definition of viscosity sub and super-solutions, 
 without loss of generality, 
we may assume that the extrema are strict.



Fix (t_0, μ_0) ∈ and set

    ϕ(t,μ)=ϕ(t,μ;t_0,μ_0):=(t-t_0)^2+∑_j=1^∞1/(j+1) 2^j⟨μ-μ_0,x^j⟩^2.

By (<ref>)
ϕ(t,μ)>0 when (t,μ) ≠ (t_0,μ_0).
For any j∈, let φ_j(μ)=1/(j+1) 2^j⟨μ-μ_0,x^j⟩^2 and observe that

    sup_(t,μ) ∈∑_i=0^(φ_j)| ⟨μ, (φ_j)^(i)⟩| ≤1/2^jK_N,

for some constant K_N which only depend on .
It follows that ϕ satisfies (<ref>). 
It is clear that ϕ has all the claimed properties.





    For each N, V, V^*_N and V_*^N are bounded on .


Let (t,μ)∈.
From Lemma <ref>,  is invariant for (<ref>) and recall that  is compact.
 Assumption <ref> and Lemma <ref> implies that |L|+|g| is uniformly bounded on  by a constant K_N.
It follows that |V(t,μ)|≤ (1+T)K_N on .



The proof of the next result is standard; <cit.>.



Assume (<ref>) holds.
For any N∈, the value function
V is both a viscosity sub and a super-solution to (<ref>) on
 and

    V_N^*(T,·)=V_*^N(T,·)=G   on N e^K^*T.




Fix N ∈ and note that both envelopes V^*_N, V_*^N
 are finite by Lemma <ref>.
 


V^*_N is a viscosity sub-solution for t<T.

Suppose that for φ∈_
and  (t,μ)∈,

    0=(V^*_N-φ)(t,μ)
    =max_ (V^*_N-φ).

Let (t_n,μ_n) be a sequence in  such that 
(t_n,μ_n,V(t_n,μ_n))→ (t,μ,V^*_N(t,μ)).
Fix a ∈ A and
let (X^t_n,μ_n,a_s)_s∈[t_n,T] denote the solution to  (<ref>) with constant control 
a and distribution μ_n at the initial time t_n.
For ease of notation, we set μ^n,a_s:=(X^t_n,μ_n,a_s).
We use the  dynamic programming (<ref>)
with θ_n:=t_n+h for  0<h< T-t,
to obtain

    V(t_n,μ_n) ≤∫_t_n^θ_n L(s,μ^n,a_s,a) s + V(θ_n,μ^n,a_θ_n)
    ≤∫_t_n^θ_n L(s,μ^n,a_s,a) s + φ(θ_n,μ^n,a_θ_n).

We pass to the limit 
to arrive at

    V^*_N(t,μ)=φ(t,μ) 
    ≤∫_t^t+h L(s,μ^a_s,a) s + φ(t+h,μ^a_t+h),

where μ^a_s is the distribution
of the solution to (<ref>) with initial data μ at time t
and constant control a.  We now use (<ref>)
 to obtain

    0≤∫_t^t+h [ 
    ∂_t φ(s,μ^a_s) -H^a(s,μ^a_s,D_mφ)] s.

Since this holds for every h>0 and a ∈ A, we conclude that

    - ∂_t φ(t,μ) +sup_a∈ A H^a(t,μ,D_mφ) ≤0.

 


V_*^N is a viscosity super-solution for t<T. 
Suppose that there exists (t,μ)∈ and 
φ∈_ 
such that 

    0=(V_*^N-φ)(t,μ)
    =min_ (V^*_N-φ).

In view of Lemma <ref>,
without loss of generality we assume that
above minimum is strict.  Towards
a counterposition assume that

    -∂_tφ(t,μ) + H(t,μ,φ)< 0.

By the continuity of H proved in Proposition <ref>,
there exists 
a neighbourhood B of (t,μ)  such that

    -∂_tφ(t,μ) 
    -⟨μ, ^a,μ_t[D_mφ]⟩≤ 
    L(t,μ,a),  ∀ (t,μ)∈ B_N:=B∩, ∀ a∈ A.

Let (t_n,μ_n) be a sequence in  such that 
(t_n,μ_n,V(t_n,μ_n))→ (t,μ,V_*^N(t,μ)).
It is clear that for all large n, (t_n,μ_n)∈ B_N.
Fix an arbitrary control α∈ and 
let (X^t_n,μ_n,α_s)_s∈[t_n,T] denote the solution to  (<ref>) with  distribution μ_n at the initial time t_n.
For ease of notation, we set μ^n,α_s:=(X^t_n,μ_n,α_s).
Consider the deterministic  times

    θ_n:=inf{s≥ t_n:(s,μ_s^n,α)∉ B_N }∧ T.


By (<ref>),

    φ(t_n,μ_n)   =φ(θ_n,μ_θ_n^n,α)
    -∫_t_n^θ_n[ ∂_tφ(s,μ_s^n,α)
    +⟨μ^n,α_s, ^α_s,μ^n,α_s_s[D_mφ]⟩]s
       ≤φ(θ_n,μ_θ_n^n,α)+∫_t_n^θ_n L(s,μ_s^n,α,α)s.

Since ∖B_N=∖B is compact and 
V_*^N-φ 
has a strict minimum at (t,μ), 
there exists η>0, 
independent of α, such that 
φ≤ V_*^N-η≤ V-η on ∖B.
Hence, the above inequality implies that

    φ(t_n,μ_n)≤ V(θ_n,μ_θ_n^n,α)
    +∫_t_n^θ_n L(s,μ_s^n,α,α)s-η

Since the (φ-V)(t_n,μ_n)→ 0, 
for n large enough,

    V(t_n,μ_n)   
    ≤∫_t_n^θ_n L(s,μ_s^n,α,α)s
    +V(θ_n,μ_θ_n^n,α)-η/2.

As the above inequality holds 
with η>0 independent of α∈,
it is in contradiction with (<ref>).  Hence,
V_*^N is a viscosity super-solution to (<ref>).
 


V^*_N= G on N e^K^*T.
Consider a sequence (t_n,μ_n)→(T,μ)
such that V^*_N(T,μ)=lim_n→∞ V(t_n,μ_n).
By Assumption <ref>, the uniform continuity of L_1 
implies ∫_t_n^T L_1(s,μ_s^n,α,α_s)→ 0.
Also, by Lemma <ref>,
the integral 
∫_t_n^T L_2(α_s)⟨μ_s^n,α,L_3⟩≤C̃(T-t_n) 
converges to zero.
We next show that μ_T^n,α→μ.
By the compactness of , there exists μ̂∈ 
such that μ_T^n,α→μ̂ (up to a subsequence).
Itô's Formula and Lemma <ref> imply that 
|⟨μ_T^n,α-μ_n,x^j⟩|→ 0 for every j∈.
This implies that μ̂=μ.
Hence, for an arbitrary α∈, we have,

    V^*_N(T,μ)=lim_n→∞ V(t_n,μ_n)≤lim_n→∞[ ∫_t_n^T L(s,μ_s^n,α,α_s)+G(μ_T^n,α)]= 
    G(μ).

As V^*_N(T,μ) ≥ V(T,μ)=G(μ), we conclude that
V^*_N(T,μ)=G(μ).
 


V_*^N=G on N e^K^*T.
Again consider (t_n,μ_n)→(T,μ)
satisfying
V_*^N(T,μ)=lim_n→∞ V(t_n,μ_n).
As in the previous step 
∫_t_n^T L(s,μ_s^α,n,α_s)→ 0 uniformly in α and 
G(μ_T^n,α)→ G(μ), as n→∞.
For any n∈, choose α^n∈ so that 
V(t_n,μ_n)≥∫_t_n^T L(s,μ_s^n,α^n,α^n_s)+G(μ_T^n,α^n)-1/n.
This implies that

    V_*^N(T,μ)=lim_n→∞ V(t_n,μ_n)
    ≥lim_n→∞[∫_t_n^T L(s,μ_s^n,α^n,α^n_s)
    +G(μ_T^n,α^n)]= G(μ).







§ A COMPARISON RESULT
 

The following is the main comparison result.


Let u be an u.s.c. sub-solution to (<ref>) 
on  and v a l.s.c. super-solution to (<ref>) 
on , satisfying u(T,μ)≤ v(T,μ) for any (T,μ) ∈.
Then u≤ v on .


The following corollary is the unique characterization of the value function.
Recall, for any function u, we use the notation u^*
to denote the upper semicontinuous envelope of u restricted to 
and  u_* is  the lower semicontinuous envelope of u restricted to .


    
    The value function V is the unique  viscosity solution to (<ref>)
    on  satisfying V^*(T,μ)
    =V_*(T,μ)=G for (T,μ) ∈. Moreover, 
    V restricted to  is continuous, i.e., V^*=V_*.


We apply the above comparison result
to V _N^*, V^N_*
and use Theorem <ref> to conclude that 
the sub-solution V^*_N is less than 
the super-solution V^N_*.  Since the
opposite inequality is immediate from their definitions,
 V^*_N=V^N_*=:V_N. In view of Lemma <ref>
 proved in the Appendix, this implies that
 V^*=V_*=V.

Let v be a viscosity solution to (<ref>) and 
v^*(T,μ)=v_*(T,μ)=G for (T,μ) ∈.  
Since v_* ≤ v_*^N ≤ v^*_N ≤ v^*,  we also have
v^*_N(T,μ)=v_*^N(T,μ)=G for (T,μ) ∈.
Then, the comparison result implies that
v^*_N ≤ V^N_* =V_N= V_N^* ≤ v^N_*≤ v^*_N.
Hence, v^*_N=v^N_*=V_N.  This proves the 
uniqueness.


The remainder of this section is devoted to the proof of Theorem <ref>.
We begin by constructing a specific class of polynomials that is central to the comparison proof.
For any polynomial f, (f) is the degree of f.



We say that a set of polynomials χ has the -property
if it satisfies


  * for any g∈χ,  g^(i)∈χ for all i=0,…,(g);

  * for any g∈χ,  ∑_i=1^(g)m_ig^(i)∈χ with m_i:=1/i!∫_y^iγ(dy).

Let Σ be the  collection of all sets of polynomials that has the (*)-property.


Set

    χ(f):= ⋂_χ∈Σ, f ∈χχ.

One can directly show that 
χ(f) has the (*)-property
and hence it is the smallest set
of polynomials with the (*)-property that
includes f. It is also clear that for every g ∈χ(f), χ(g) ⊂χ(f).


The followings are few examples of the above sets.

    χ(x)= {   0,1,m_1,x}
    χ(x^2)={   0,2,2m_1,2m_1^2,2x,2m_1x+2m_2,x^2}
    χ(x^3)={   0,6,6m_1,6m_1^2,6m_1^3, 6x,6m_1x+6m_2,6m_1^2x+12m_1m_2,
    3x^2,
        3m_1x^2+6m_2x+6m_3, x^3}.
 



For any polynomial f, χ(f)  is finite.
 

We show this by induction on the degree of the polynomial.
Indeed if (f)=0, χ(f)={f,0} and hence is finite.
Towards an induction proof, assume 
that we have shown that χ(h) is finite
for every polynomial h with h≤ n
for some integer n ≥ 0.
Let f be a polynomial with (f)=n+1.
Set ĝ:=∑_i=1^(f)m_if^(i).
Then, (ĝ)= n and consequently by
our assumption χ(ĝ) is finite.
Moreover,

    χ(f)={f}∪χ(ĝ)∪⋃_i=1^(f)χ(f^(i)).

As (f^(i)) ≤ n for every i≥ 1,
χ(f^(i)) are finite by the induction hypothesis and therefore,
χ(f) is also finite.


Set Θ:=∪_j=1^∞χ(x^j).
Then, Θ contains all monomials {x^j}_j=1^∞,
it is countable and
χ(f) ⊂Θ for every f ∈Θ.  
Let {f_j}_j=1^∞ be an enumeration of Θ.

The definition of b and Lemma <ref> imply that

    s_j(b):=1+sup_μ∈b⟨μ,f_j⟩^2 <∞,  ∀ j=1,2,….

As χ(f) ⊂Θ for every f ∈Θ, 
there exists a finite index set I_j satisfying,

    χ(f_j)={f_i | i∈ I_j}  
    j=1,2,…

Fix b>0 and set

    c_j(b):=(∑_k∈ I_j 2^k)^-1(∑_k∈ I_j s_k(b))^-2.

Since  f_j ∈χ(f_j), j∈ I_j and  therefore, c_j(b)≤ 2^-j. 
Hence, ∑_j=1^∞ c_j(b) ≤ 1.
Also, for each  i ∈ I_j, f_i ∈χ(f_j) and consequently,
χ(f_i)⊂χ(f_j).  This implies that I_i⊂ I_j.
Moreover, s_j(b) ≥ 1.  Hence,
the definition  (<ref>)  implies that

    c_j(b)≤ c_i(b),    ∀ i∈ I_j.

Finally, observe that, by the definitions of s_j(b) and c_j(b),

    ∑_j=1^∞c_j(b)⟨μ,f_j⟩^2≤ 1,   ∀μ∈b.






Fix N ∈.

To simplify the notation
we write c_j for c_j(Ne^K^*T).  In particular,
for any (t,μ) ∈, μ∈ Ne^K^*t⊂ Ne^K^*T
and therefore, by (<ref>)

    sup_(t,μ) ∈∑_j=1^∞c_j⟨μ,f_j⟩^2≤ 1.


Towards a counterposition, 
suppose that sup_(u-v)>0.
Since u-v is u.s.c. and  is weak^* compact, the maximum

    ℓ:= max_(t,μ)∈( (u-v)(t,μ) - 2 η (T-t) )

is achieved and ℓ>0 for all sufficiently small η∈(0,η_0].


Doubling of variables.

Recall  Θ={f_j}_j=1^∞ and the constants {c_j}  in (<ref>)
with b=Ne^K^*T.
For n∈, ε>0,  η∈( 0,η_0] set

    ϕ_ε(t,μ,s,ν)   :=u(t,μ)-v(s,ν) -1/εμ-ν - β_η, ε(t, s),
    β_η, ε(t, s)     :=   η (T - t + T - s) + 1/ε(t-s)^2.

By our assumptions, ϕ_ε 
admits a maximizer (t^*_ε,μ^*_ε, s^*_ε, ν^*_ε)
satisfying,

    ϕ_ε(t^*_ε,μ^*_ε, s^*_ε, ν^*_ε)
    =max_ϕ_ε≥ℓ >0.




Since  is compact and 
u is u.s.c., M:= max_ u ∈.
As v is l.s.c., similarly
 m:= min_ v ∈.
In view of (<ref>),

    0≤ζ_ϵ/ε:= 1/ε[ μ^*_ε-ν^*_ε 
     + (t^*_ε - s^*_ε)^2 ]≤ M-m-ℓ=:C<∞.


As  is compact, there exist subsequences 
{(t^*_ε_i,μ^*_ε_i),(s^*_ε_i,ν^*_ε_i)}_i∈ 
such that μ^*_ε_i and ν^*_ε_i 
converge to μ^* and ν^* respectively, and 
t^*_ε_i and s^*_ε_i both converge to t^*.

ν^*=μ^*.  Since ζ_ϵ converges to zero, 
⟨μ^*_ε-ν^*_ε, f_j⟩ converges to zero for each j.  
As Θ={f_j}_j=1^∞ contains all the monomials,
lim_ε→ 0⟨μ^*_ε-ν^*_ε,x^j⟩=0 for any j∈.
In view of Lemma <ref>, the map
μ↦⟨μ,x^j⟩ is continuous on .  Hence,

    ⟨μ^*-ν^*, x^j⟩ =
     lim_i ⟨μ^*_ε_i-ν^*_ε_i, x^j⟩ =0,   
     j=1,2,…

By (<ref>), we conclude that ν^*=μ^*.


t^* < T.
Towards a counterposition, assume that
t^*=T.  Since by hypothesis (u-v)(T, ·)  ≤ 0, v is l.s.c., and u is u.s.c.,

    0≥ (u-v)(T,μ^*)    ≥lim sup_i
    u(t_ε_i,μ^*_ε_i)-
    v(s_ε_i,ν^*_ε_i)
       ≥lim sup_i ϕ_ε_i(t^*_ε_i,μ^*_ε_i, s^*_ε_i, ν^*_ε_i)
    ≥ℓ >0.


We claim that lim sup_i→∞ζ_ε_i/ε_i=0.
Indeed,

    ℓ   ≥ϕ_ϵ(t^*,μ^*,t^*,μ^*)
       =u(t^*,μ^*)-v(t^*,μ^*) - 2 η (T-t^*) 
       ≥lim sup_i →∞( u(t^*_ε_i,μ^*_ε_i)-v(s^*_ε_i,ν^*_ε_i) - η (T-t^*_ε_i + T-s^*_ε_i))
       ≥ℓ +lim sup_i →∞1/ε_i( μ_ε_i^*-ν_ε_i^* + (t^*_ε_i - s^*_ε_i)^2 )
       =ℓ +lim sup_i →∞ζ_ε_i/ε_i.

Hence we conclude that

    lim sup_i→∞ζ_ε_i/ε_i=0.



Initial Estimate.

Let {μ^*_ε}, {ν^*_ε} 
as in Step <ref> and set

    π^*_ε(·)
    := 2/εμ_ε^*-ν_ε^*(·).

Note that 

    π^*_ε(·)=φ_1(μ_ε^*,·)=-φ_2(ν_ε^*,·),

where φ_1(μ):= 1/εμ-ν_ε^*, respectively, φ_2(μ):= 1/εμ_ε^*-μ.
One can directly verify that φ_1 and φ_2 are test functions on , i.e., φ_1, φ_2 ∈_.
We thus have,

    (∂_t β_η, ε(t^*_ε, s^*_ε), π^*_ε)
    ∈1+u(t^*_ε, μ^*_ε),   
    (-∂_s β_η, ε(t^*_ε, s^*_ε), π^*_ε) 
    ∈1-v(s^*_ε, ν^*_ε).

Then, by the viscosity properties of u and v,

    -∂_t β_η, ε(t^*_ε, s^*_ε)   
    + H(t^*_ε,μ^*_ε,π^*_ε) ≤ 0,   ∂_s β_η, ε(t^*_ε, s^*_ε) 
     + H(s^*_ε,ν^*_ε,π^*_ε)≥0.

We combine and use the definition of β_η, ε to arrive at

    0 < 2 η   ≤  
    H(s^*_ε,ν^*_ε, π^*_ε)
    - H(t^*_ε,μ^*_ε, π^*_ε)
        =  sup_a ∈ A H^a(s^*_ε,ν^*_ε, π^*_ε)
    -  sup_a ∈ A H^a(t^*_ε,μ^*_ε, π^*_ε)
       ≤sup_a ∈ A(H^a(s^*_ε,ν^*_ε, π^*_ε)
    -H^a(t^*_ε,μ^*_ε, π^*_ε))
    =:sup_a ∈ AI^a.

Moreover,

    I^a       :=
        L(t^*_ε,μ^*_ε,a) - L(s^*_ε, ν^*_ε, a) 
    +⟨μ^*_ε, ^a,μ^*_ε_t^*_ε[π^*_ε]⟩
         - ⟨ν^*_ε, ^a,ν^*_ε_s^*_ε[π^*_ε]⟩
       =
         L(t^*_ε,μ^*_ε,a) - L(s^*_ε, ν^*_ε, a)_I_1^a
        + ⟨μ^*_ε - ν^*_ε,
        ^a,μ^*_ε_t^*_ε[π^*_ε]⟩_I_2^a
        + ⟨ν^*_ε, 
        ^a,μ^*_ε_t^*_ε[π^*_ε] 
    - ^a,ν^*_ε_s^*_ε[π^*_ε]⟩_I_3^a.

By Assumption <ref> and Lemma <ref>,
 lim_ε→ 0sup_a ∈ AI^a_1 → 0.
 
 Estimate of I_2.
We rewrite the second term as,

    I_2 := sup_a ∈ A I_2^a ≤sup_a ∈ A2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, 
     ^a, μ^*_ε_t^*_ε[ f_j]⟩ |
     ≤ I_2^b+I_2^σ +I_2^γ,

related to the three terms
appearing in the generator ^a, μ^*_ε_t^*_ε, which appear explicitly below.

By construction, for every j∈, 
there exists an index k_1(j) such that f_j^' = f_k_1(j).
Also, as f_j^'=f_k_1(j)∈χ(f_j), χ(f_k_1(j)) ⊂χ(f_j),
and consequently, I_k_1(j)⊂ I_j.
Therefore, the definition
(<ref>) yields that c_j ≤ c_k_1(j).
We now directly estimate using these and <ref> to obtain,

    I_2^b
           = sup_a ∈ A2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, b(t^*_ε, μ^*_ε,a) f_j^'⟩| 
       ≤ C 2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, f_j^'⟩ | 
       ≤ C 2/ε( ∑_j=1^∞ c_j ⟨μ_ε^*-ν_ε^*, f_j⟩^2
         + ∑_j=1^∞ c_k_1(j)⟨μ_ε^*-ν_ε^*, f_k_1(j)⟩^2 )
       ≤ C 4/ε∑_j=1^∞ c_j ⟨μ_ε^*-ν_ε^*, f_j⟩^2,

which converges to 0, by (<ref>).

We estimate I^σ_2 similarly.
Indeed, for every j∈, there exists an index k_2(j) such that 
f_j^'' = f_k_2(j) and c_j ≤ c_k_2(j).
Then, 

    I_2^σ   = sup_a ∈ A2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, 
        σ(t^*_ε, μ^*_ε,a) f_j^''⟩| 
       ≤ C 2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, f_j^''⟩ | 
       ≤ C 2/ε( μ_ε^*-ν_ε^*
         + ∑_j=1^∞ c_k_2(j)⟨μ_ε^*-ν_ε^*, f_k_2(j)⟩^2 ),

which also converges to 0, by (<ref>).

We analyse I_2^γ next. By the Taylor expansion of f_j,

    g_j(x)    :=∫_[f_j(x+y)-f_j(x)]γ(dy)
       =∑_i=1^(f_j)f_j^(i)(x)/i!∫_y^iγ(dy) 
     =∑_i=1^(f_j)m_if_j^(i)(x).

Again, by the construction of {f_j}, for all j∈, 
there exists k_λ(j) such that g_j = f_k_λ(j) and c_j ≤ c_k_λ(j).
Hence,

    I_2^λ   = sup_a ∈ A2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, λ(t^*_ε, μ^*_ε,a) g_j⟩| 
       ≤ C 2/ε∑_j=1^∞ c_j |⟨μ_ε^*-ν_ε^*, f_j⟩⟨μ_ε^*-ν_ε^*, g_j⟩ | 
       ≤ C 2/ε( μ_ε^*-ν_ε^*
         + ∑_j=1^∞ c_k_λ(j)⟨μ_ε^*-ν_ε^*, f_k_λ(j)⟩^2 ).

As this quantity also vanishes as ε→ 0, 
we conclude that I_2 → 0 as ε goes to zero.


Estimating I_3.
As in the previous step, we write  

    I_3= sup_a ∈ A⟨ν^*_ε, 
        ^a,μ^*_ε_t^*_ε[π^*_ε] 
        - ^a,ν^*_ε_s^*_ε[π^*_ε]⟩≤ I_3^b+I_3^σ +I_3^γ
  
related to the three terms appearing in the generator.
Since the estimates of each term is very similar to each other,
we provide the details of only the first one.


By <ref>, there exists C_1 such that 

    (b(t^*_ε,μ^*_ε,a)
    -b(s^*_ε,ν^*_ε,a))^2
    ≤ C_1(t^*_ε-s^*_ε)^2
    +C_1μ^*_ε-ν^*_ε.

It follows,

    | I^b_3|    ≤sup_a ∈ A2/ε∑_j=1^∞ c_j
     | ⟨μ_ε^*-ν_ε^*, f_j⟩⟨ν_ε^*, (b(t^*_ε,μ^*_ε,a)
    -b(s^*_ε,ν^*_ε,a)) f_j^'⟩|
       ≤2/εμ^*_ε-ν^*_ε
       
     + 2C_1/ε((t^*_ε-s^*_ε)^2
     +μ^*_ε-ν^*_ε) [f'_j]ν^*_ε .

Note that  by  (<ref>),

    [f'_j]ν^*_ε≤[f_j]ν^*_ε≤ 1.

  Hence,
 
    |I^b_3|  ≤4C_1/ε((t^*_ε-s^*_ε)^2
    +μ^*_ε-ν^*_ε).

In view of  (<ref>), 
we conclude that I^b_3 goes to zero as ε→ 0.
Repeating the same argument for I^σ_3 and I^λ_3, we conclude that I_3
also converges to zero.

Conclusion  In Step <ref> we have shown that

    0<2 η≤sup_a ∈ A I^a =I_1+I_2+I_3.
  
In the preceding steps
we have shown that each of the three terms converge to zero as 
ε tends to zero. 
Clearly this contradicts with the fact that η>0.












§ SOLUTIONS OF CONTROLLED MCKEAN–VLASOV SDES

For completeness,  we provide here an existence result for the McKean–Vlasov SDE (<ref>).

Using the functions and coefficients of Section <ref>,
we  fix b>0 and  start by proving
functional analytic properties of b.
Set

    d(μ,ν;b):=∑_j=1^∞ c_j(b) |⟨μ-ν,f_j⟩|,
      μ,ν∈b.



A sequence {μ_n}_n∈ in  b converges weakly
 to μ∈b if and only if lim_n →∞(μ_n,μ;b)= 0. 


As Θ contains all monomials,
in view of (<ref>), 
(μ,ν;b)=0 if and only if μ=ν,
and one can then directly
verify that  is a metric on b. 
Moreover, since ∑_j=1^∞ c_j(b) ≤ 1, 
by (<ref>),  ≤ 1 on b. 
Suppose μ_n→μ as n→∞. By dominated convergence,

    lim_n→∞(μ_n,μ;b)=∑_j=1^∞ c_j(b) lim_n→∞ |⟨μ_n-μ,f_j⟩|=0,

where the last equality follows from Lemma <ref>.
Now suppose (μ_n,μ;b)→ 0 as n→∞. 
Since b is compact, the sequence {μ_n}
has limit points and since d is a metric, we conclude that
it can only have one limit point μ.


We next fix t ∈ [0,T] and consider the space

    _t(b):={μ̅=(μ_s)_s∈[t,T]|μ_s∈b,  ∀ s∈[t,T]
    },

and the function

    T(μ,ν;b)=sup_t≤ s≤ T(μ_s,ν_s;b).

It is straightforward to see that T is a metric on _t(b).

(_t(b),T) is a complete metric space.


Let {μ̅^n}_n∈ be a Cauchy sequence. In particular {μ_s^n}_n∈ 
is a Cauchy sequence in (b,) for any s∈[t,T] 
and by Lemma <ref>, there exists μ_s∈b 
such that μ^n_s→μ_s as n→∞. 
We claim that μ̅:=(μ_s)_s∈[t,T] is the limit of {μ̅^n}_n∈.
Indeed, for ε >0, there is n̅ such that
(μ^n_s,μ^m_s;b)≤ε 
for every n,m≥n̅ and s∈[t,T]. 
By letting m tend to infinity and by using the previous lemma, 
we conclude that (μ^n_s,μ_s;b)≤ε for any s∈ [t,T]. 
The result follows after  taking the supremum over s∈ [t,T]. 


This structure allows us to study the McKean–Vlasov equation (<ref>).
For similar results we refer to the book of Carmona & Delarue <cit.>
and the references therein.

 Under Assumption <ref>, for any 
(t,μ) ∈ and
control α∈, the equation (<ref>) 
with initial data X_t ∼μ
has a unique solution.


Fix (t,μ) ∈
and control α∈.
There is N∈ such that μ∈Ne^K^*t. 
Let := _t(Ne^K^*(T-t))
and c_j:= c_j((Ne^K^*(T-t)).

For any μ̅=(μ_s)_s∈[t,T]∈, set 

    X^μ̅_s
    :=∫_t^s b(r,μ_r,α_r)dr+∫_t^s σ(r,μ_r,α_r)  W_r
    +∑_t≤ r≤ sΔ J_r,

with distribution μ at time t,
and

    Φ:→,  μ̅↦Φ(μ̅):=((X^μ̅_s))_s∈[t,T].

Recall that the set  in Lemma <ref> is invariant for (<ref>).
Therefore,  Φ(μ̅) ∈.
Moreover, the law of any solution to (<ref>) is a fixed point of Φ.

To simplify the notation for μ̅,μ̅'∈, 
let ν̅=Φ(μ̅),ν̅'=Φ(μ̅').
Consider now f_j∈Θ.  We now apply  Itô's Formula 
to arrive at,


    f_j(X^μ̅_s)   = X^μ̅_t+∫_t^s b(r,μ_r,α_r) f_j'(X^α_r)dr
       +1/2∫_t^sσ^2(r,μ_r,α_r)f_j”(X^α_r) dr
       +∫_t^sσ(r,μ_r,α_r)f_j'(X^α_r)  W_r+∑_t≤ r≤ sf_j(Δ J_r).

From Assumption <ref>, the stochastic integral in the above formula is a local martingale.
Denote by {τ_n}_n∈ a localizing sequence and take expectation on both sides. Recalling that α is deterministic, we obtain:

    [f_j(X^μ̅_s∧τ_n)]   = X^μ̅_t+∫_t^s b(r,μ_r,α_r) [f_j'(X^α_r)1_t≤ r≤τ_n]dr
       +1/2∫_t^sσ^2(r,μ_r,α_r)[f_j”(X^α_r)1_t≤ r≤τ_n] dr+[∑_t≤ r≤ s∧τ_nf_j(Δ J_r)].

By dominated convergence, the equality pass to the limit as n→∞.
For ease of notation, denote Δ b(r):=b(r,μ_r,α_r)-b(r,μ'_r,α_r) and similarly Δσ^2(r) and Δλ(r).
From  ⟨ν_s,f_j⟩=[f_j(X^μ̅_s)], we deduce

     ⟨ν_s-ν'_s,f_j⟩   =       ∫_t^s Δb(r) ⟨ν,f_j'⟩dr+∫_t^s b(r,μ'_r,α_r) ⟨ν-ν',f_j'⟩dr

              +1/2∫_t^s Δσ^2(r) ⟨ν,f_j”⟩dr+1/2∫_t^s σ^2(r,μ'_r,α_r) ⟨ν-ν',f_j”⟩dr

              + ∫_t^s Δλ(r) ⟨ν,g_j⟩dr+∫_t^s λ(r,μ'_r,α_r) ⟨ν-ν',g_j⟩dr,

where g_j=∑_i=1^(f_j)m_if_j^(i) with m_i:=1/i!∫_y^iγ(dy).
Recall now that the collection of coefficients {c_j}_j∈ satisfies (<ref>), so that,

    c_j≤ k_j:=min{c_j_1,c_j_2,c_j_g},

where c_j_1, c_j_2 and c_j_g are the coefficients of f'_j, f”_j and g_j respectively.
We can therefore multiply by k_j both sides of the above equality to get, using also Assumption <ref> and <ref>,

     c_j|⟨ν_s-ν'_s,f_j⟩|   ≤        k_j|⟨ν_s-ν'_s,f_j⟩|

        ≤        C̅ ∫_t^s (μ_r,μ'_r) (c_j_1|⟨ν,f_j'⟩|+c_j_2|⟨ν,f_j”⟩| +c_g_j|⟨ν,g_j⟩|)dr

              +∫_t^s c_j_1|⟨ν-ν',f_j'⟩| + c_j_2|⟨ν-ν',f_j”⟩| dr+∫_t^s c_g_j|⟨ν-ν',g_j⟩| dr,

for some constant C̅ which depends only on the coefficients of (<ref>).
By summing up over j∈ and recalling (<ref>), we obtain

    (ν_s,ν'_s)≤ 3C̅(∫_t^s(μ_r,μ'_r)+∫_t^s(ν_r,ν'_r)).

Using Gronwall's Lemma, we obtain

    s(Φ(μ̅),Φ(μ̅'))≤ e^3C̅s∫_t^sr(μ̅,μ̅'),

for any t≤ s≤ T.
Denoting now C(s):=e^3C̅s and Φ^k the composition of k times the map Φ, it can be verified, by induction,

    T(Φ^k(μ̅),Φ^k(μ̅'))≤C(T)^kT^k/k!T(μ̅,μ̅').

For k large enough Φ^k is a contraction on , which is a complete metric space 
in view of Lemma <ref>.
Thus, the map Φ admits a unique fixed point.




§ SEMICONTINUOUS ENVELOPES



In this section, we show that
the semicontinuous envelopes
defined on  converge to the envelopes
defined on .


    Let (E, τ) be a topological space and (E_N, τ_N)_N ∈ 
    a sequence of topological spaces with (E_N)_N ∈ increasing to E, i.e.,
    ∪_n∈ E_N = E and E_N⊂ E_N+1 for any N. Let τ_N the subspace topology induced by τ.
    Denote by u^* : E →∪{∞} the upper semicontinuous envelope 
    on (E, τ) and by u_N^* : E_N →∪{∞} the upper semicontinuous envelope on (E_N, τ_N).
    Then, lim_N →∞ u_N^* = u^*.
    Similarly, if u^N_* is the lower semicontinuous envelope on (E_N, τ_N), then lim_N →∞ u^N_* = u_*


    Consider the following representations of the semicontinuous envelopes.
    Let U(μ) be the collection of τ-neighborhoods of μ.
    Then, since E_N is endowed with the subspace topology,
    for any N ∈,
    
        u^*(μ)    = inf_W ∈U(μ) sup_W u,          for μ∈E, 

        u_N^*(μ)    = inf_W ∈U(μ) sup_W ∩E_N u,        
          for μ∈E_N.
    
    Clearly u_N^* ≤ u_N+1^* ≤ u^*.
	Suppose first u^*(μ)<∞.
	For W ∈ U(μ), choose a sequence μ_n
    such that sup_W u ≤ u(μ_n) + 1/n.
    Let M : → be a function such that μ_n ∈ E_M(n).
    Without loss of generality, we may choose M to be strictly increasing.
    Thus,
    
    sup_nsup_W ∩ E_M(n) u = sup_W u.

    Since above holds for every W ∈ U(μ),
 lim_N →∞ u_N^*(μ) = u^*(μ).
    If u^*(μ)=∞, we repeat the same argument 
    with a sequence μ_n such that u(μ_n)>n.



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