Partial Derivative With Respect To The Measure In Wasserstein Space And Its Applications To General Controlled Mean-field Systems

Mean-field stochaustic differential equations,also known as Mckcan-Vlasov cqua-tions,have their different applications,namely in stochastic mechanics and physics,in quantum mechanics,also in quantum chemistry.J.M.Lasry and P.L.Lions[62]studied equations of mean-field type for the problems in economics and finance,and also for the stochastic differential games.Buckdahn,Djehiche,Li and Peng[16]studied a special type of mean-field problem completely by a stochastic method,and deduced a new type of backward stochastic differential equations.which were called mean-field backward stochastic differential equations.Since then,many scholars have studied the optimal control problem of this type of McKean-Vlasov type stochastic system.In 2013,P.L.Lions[73]was the first to introduce the definition of the differentiability of a function with respect to a probability measure.Inspired by his work,Buckdahn,Li and Ma[1 9]studied a generalized mean-field stochastic control system whose coefficients depend not only on the state process but also on the law of the state process.They applied the second-order Taylor expansion to variational equation proposed by Peng[84]to give the global stochastic maximum principle.In a controlled system,the "participants" need to make decisions based on the information they already had.However,there will often be incomplete information in reality."Participants" cannot observe all the information or not all the information is valid.They can only make decisions based on the partial information they have mastered.Therefore,the stochastic control problems based on partial observations are of great significance.This paper mainly consider two topics:the first one is to investigate,the partial differentiability with respect to the law ? conditioned to its marginal ?2=?(Rd×·)without any assumption of regularity with respect to ?2:the second one is to study the application of partial derivatives with respect to the measure in Wasserstein space to stochastic control problems:stochastic maximum principle for a mean-field controlled system with partial observation.Let us introduce the content and structure of the thesis in detail.Chap ? gives an overview of our topics in Chap ? and Chap ?.Chap ? is devoted to study partial derivatives with respect to the measure in Wasserstein space.More precisely,let(E,?)be an arbitrary measurable space,consider a function f:P2,0?Rd×E)?R defined on the space of probability measlres ? over(Rd×E,B(Rd)(?)?)whose first marginal ?1:=?(·× E)has a finite second order moment.This partial derivative is taken with respect to q(dx,z),where ? has the desintegration?(dxdz)=q(dx,z)?2(dz)with respect to its second marginal ?2(·)=?(Rd×·).Simpli-fying the language,we will speak of the derivative with respect to the law ? conditioned to its second marginal.The novelty of this chapter:We notice that(E.?)is an arbitrary measurable space,such a global differentiability property with respect to ? is rather restrictive for f:P2,0((Rd × E)? R.Our results extend those of the derivative of a function g:P2(Rd)?R over the space of probability measures with finite second order moment by P.L.Lions(see[73]).Chap ? discusses the application of partial derivatives with respect to the measure in Wasserstein space to stochastic control problems,where the controlled state process is driven by a general mean-field stochastic differential equation with partial information.The control set is just supposed to be a measurable space,and the coefficients of the controlled system,i.e.,those of the dynamics as well as of the cost functional,depend on the controlled state process X,the control v,a partial information on X,as well as on the joint law of(X,v).Since the coefficients b and ? depend not only on E[Xt1|FtW1],but also on the joint law P(X,v):there are some technical difficulties to prove the second-order expansion.A new method is employed to prove the relative estimates.The novelty of this chapter:The control set is generalized to an arbitrary measurable set,and the control system is a generalized mean-field stochastic differential equation with partial information.Technical methods are used to deal with the estimation of partial information and distribution terms,which play a crucial role in dealing with the estimates in the Taylor expansion.In particular,for the estimate reflects the specificity of stochastic controlled systems with mean-field dependence,we develop an operator argument,which is totally new and different from the classical case.The Chapters ? and ? of the present paper are based on:R.Buckdahn,Y.Chen,J.Li.Partial derivative with respect to the measure and its application to controlled mean-field systems with partial information.Submitted.This dissertation is composed of the three chapters mentioned above.We now give an overview of the structure and the main conclusions.Chapter ? Introduction;Chapter ? Partial derivative with respect to the law conditioned to its second marginal;Chapter ? Stochastic maximum principle for a controlled system with mean-field term and partial observation.Chapter II:We mainly study the partial differentiability with respect to the law ? conditioned to its marginal ?2=?(Rd ×·)without any assumptiol of regularity with respect to/12.Let(E,?)be an arbitrary measurable space.By P(Rd × E)we denote the set of all probability measures over(Rd×E,B(Rd)× ?),and we put P2,0(Rd × E):={??P(Rd× E):?Rd×E|x|?(dxdz)<+?}.For ??P2,0(Rd× E),we denote by ?1(A):=?(A × E),A?B(Rd),and ?2(B):=?(Rd×B),B??,the marginals of ?.Given a function ?:P2,0(Rd× E)?R and any ? ? P2,0(Rd × E),we are interested in studying the partial differentiability of f at ? with respect to its conditional law q(dx,z)conditioned to ?2,where ?(dxdz)=q(dx,z)?2(dz).Let us write q? for q.Let(?):=f(P(?,v)),(?,v)?L2(F;Rd)× L0(F;E),be the function f lifted to L2(f;Rd)× L0(F;E).We give the definition of partial differentiability of function f as follows,in the sense of the Frechet derivative.Definition 0.1 We say that f:P2,0(Rd × E)?R is partially differentiable with respect to q? at ? ? P2,0(Rd × E),if there is some random variable(?,v)?L2(F;Rd)× L0(F;E)with joint law P(?,v)=?,such that ?'?)(?):=(?)is Frechet differentiable at? as a function over the Hilbert space L2(F;Rd),i.e.,there exists a continuous linear functional(?)? L(L2(F;Rd);R)such that f(P(?+?,v)-?(P(?,v)=(?)+o(|?|L2),for all ?? L2(F;Rd)with |?|L2:=(E[|?|2])1/2? 0.With the help of the Riesz Theorem with the corresponding random variable(?)?L2(F;Rd),we have,for all ??L2(F;Rd)with |?|L2:=(E[|?|2])1/2? 0,f(P(?+?,v)-f(P(?,v))=E[(?)+o(|?|L2).In order to prove the partial differentiability with respect to the law,the following technical result need to be discussed.Lemma 0.2(?,F)is a Radon space,i.e.,any Borel probability measure on this space is inner regular.Equivalent to this property to be a Radon space is the following one:(?,F)satisfies the regular conditional probability property,i.e.,for any probability measure P on(?,F),any measurable space(E,?)and any measurable map v:?? E there exists a regular conditional probability P|v=.:E×F ?[0,1],such thati)P|v=z(·)is a probability on F,for all z?E;ii)P|v=.(A):E?[0,1]is ?-measurable,for all A ?F;iii)P{A|v=z}=P|v=z(A),Pv(dz)-a.s.,for z ? E and A ? F.Let now 0 ? L2(F;Rd)and v be a measurable map over(?,F,P)with values in a measurable space(E,?).Let ??L2(F;Rd)be a standard Gaussian vector which is independent of(?,?v).For any ?>0,we put ??:=?+??.Proposition 0.3 With the above notations,for every ?>0,there is a measurable map H?:Rd ×E?Rd such that,Pv(dz)-a.s.for z ? E,i)PH?(??,v)|v=z=P?|v=z,ii)E[|?-??|2|v=z]? E[|H?(??,v)-??|2|v=z].Extending the fundamental result by P.L.Lions on the differentiability with respect to a probability measure,we have:According to the Lemma 0.2 and the Proposition 0.3,we can characterize the partial derivative with respect to the measure as a Borel function,and the function depends on(?,v)only through the law P(?,v).Theorem 0.4 With the above assumptions and notations we have the existence of a?(dydz)-almost unique measurable function g:Rd×E?Rd such that(?)=g(?,v),P-a.s.Moreover,the function g=g(y,z)depends on(?,v)only through its law?=P(?,v).The above theorem allows to introduce the notation((?)?f)1(?,y,z):=g(y,z),(y,z)?Rd × E,which is the definition of partial derivative of the function f:P2,0(Rd × E)? R at ?with respect to its conditional law q(dy,z).Moreover,we observe that the function is only ?(dydz)-a.s.uniquely defined on Rd × E.Chapter III:We mainly study the optimal control problem of the controlled system with partial information under the mean-field frame.The controlled state process is driven by a general mean-field stochastic differential equation with partial information.The control set is just supposed to be a measurable space,and the coefficients of the controlled system,i.e.,those of the dynamics as well as of the cost functional,depend on the controlled state process X,the control v,a partial information on X,as well as on the joint law of(X,v).We investigate the necessary conditions for optimal control under such control system,namely global stochastic maximum principle.We consider the following optimal control problem with the state equation(?)t?[0,T],The goal of the optimal control problem is to minimize the following cost functional J(v):=E[?(t,P(Xtv,vt),Xtv,E[Xtv|FtW1],vt)dt+?(PXTv,XTv,E[XTv|FTW1])]over the admissible control set Uad:#12 A control u?uad satisfying J(u)=infv?uad J(v)is called an optimal control.In order to obtain stochastic maximum principle,we need to introduce the first-order and second-order adjoint equation as follows#12 where H(t,?,x,y,x,p,q):=pb(t,?,x,y,v)+g?(t,?,x,y,v)-?(t,?,x,y,v).Now we introduce the first order and the second order variational equations.#12 Define the spike variation of the optimal control u as follows:u?(t):=v(t)IE?(t)+u(t)IE?c(t),t ?[0,T].Denote X:= Xu(resp.,X?:=Xu?)is the the controlled state process associated to the optimal control u(resp.,the spike variation u?).The following proposition justifies the correctness of variation equations.Proposition 0.5 Let the Assumption(H3.2.1)holds true.For every p>1,there exists a constant Cp>0 depending on p,such that,for all ? ?(0,1],#12The following two technical estimates play a crucial role in obtaining the second-order expansion of X?.Proposition 0.6 Let ? ? LF2(0,T)be such that E[?t2]?C?,t ?[0,T].Then,there exists a function p:[0,T]× R+?R+such that pt(?)? 0 as ??0 ?t(?)<C?',t ?[0,T],?>0,and|E[?t,Xt1]|??t(?)(?),?>0,t?[0,T],where C?,C?',are constants which depend on ? only through the bounnd of E[?t2],t ?[0,T].Lemma 0.7 Let the Assumptions(H3.2.1),(H3.2.2)and(H3.2.3)hold true.Then,for all p? 2,there exists some ?t(p):R+?R+such that ?t(p)(?)? 0 as ??0,and Pt(p)(?)? Cp,t?[0,T],?>0,for some Cp ? R+,and E[|E[Xt1|FtW1]|p]p??2/p?t(p)(?),t?[0,T],?>0.The following estimate gives the second-order expansion of X?,which is also the key point for the proof of the stochastic maximum principle.Proposition 0.8 Assume that the Assumptions(H3.2.1),(H3.2.2)and(H3.2.3)hold true.Then,for any 1 ? p ?3/2,we have#12 where p(·)is a positive function defined on(0,?),such that p(?)?0 as ??0.Then we can obtain the following maximum principle.Theorem 0.9 Suppose that the Assumptions(H3.2.1),(H3.2.2)and(H3.2.3)hold true,and let(u,X)be an optimal solution of the above control problem.Then,for the both pairs of F-adapted processes(p,q)and(P,Q)solving the first-order and second-order adjoint equations,respectively,and for all v?uad,,a.e.t?[0,T],P-a.s.H(t,P(Xt,vt),Xt,E[Xt | FtW1],vt,pt,qt)-H(t,P(Xt,ut),Xt,E(Xt|FtW1],ut,pt,qt)+1/2Pt|?(t,P(Xt,vt),Xt,E[Xt|Ftw1],vt)-?(t,P(Xt,ut),Xt,E[Xt|FtW1],ut)|2 ?0.Moreover,on the classical Wiener space(?=C([0,T];R2),F=B(?)?Np,P)with W=(W1,W2)as coordinate process,if the coefficients b(t,?,x,y,v),?(t,?,x,y,v)and f(t.?,x,y,v)are continuous with respect top ? P2,0(Rd×E)in the W2,TV(·,·)-metric,for all(t,?,x,y,v),with a continuous modulus ?(?:R+?R+ increasing,continuous,?(0)=0)then,for all v?uad and v?U,dtdP-a.e.,H(t,P(Xt,vt),Xt,E[Xt|FtW1],(?),pt,qt)-H(t,P(xt,ut),Xt,E[Xt|FtW1],ut,pt,qt)+1/2Pt| ?-(t,P(xt,vt),Xt,E[Xt|FtW1],(?))-?(t,P(Xt,ut)Xt,E[Xt|FtW1],ut)|2 ?0.