Backwark Stochastic Differential Equations Generated By Fractional Space-Time Noise And Partially Observed Mean-Field Control Problem

The topic of stochastic partial differential equation(SPDE)is a very dynamically developing research field which enjoys a lot of attention because of its numerous applications,among themes in biology,finance and physics.Pardoux and Peng in their seminal paper([107])introduced the backward doubly stochastic differential equation,and gave a probabilistic interpretation of a class of semilinear stochastic partial differential equations(the generalised Feynman-Kac formula).Stochastic optimal control is one of the important contents of control theory.The system controller intends to design the optimal strategy to minimize the cost or maximize the benefit when the system completes the expected task under the assumption of random noise disturbance in the dynamic system or during the observation process.In fact,however,it is often impossible for the controller to observe all the information when making a decision.Therefore,the study of stochastic control system with only partial observation has a very practical significance.The mean field theory is a theoretical method to describe a system of a large number of particles in which the movement of each particle is affected by the average movement of all the other particles rather than by some other specific particles,which simplifies the calculation in many-body problems.The theory is widely used in mathematics,physics,biology,chemistry and other fields.For example,in mathematics,the research on mean field stochastic differential equations(SDE),mean field backward stochastic differential equations(BSDE)and mean field stochastic control problems develops very dynamically.The extension of this theory to the mean field optimal control problem with partial observation is a challenging subject.In this paper,we consider the probabilistic interpretation of a type of SPDE with spatio-temporal noise and the mean field stochastic control problem.Firstly,we consider a type of SPDE with fractional spatio-temporal weighted noise,and we study the properties of the corresponding BSDE and its explicit solution as well as its probabilistic explanation.Secondly,we study the general stochastic control problem with only partially observed information,the relevant control system has only a weak solution,and the Peng stochastic maximum principle for the derivation of the control is obtained.In what follows we explain the organisation of this thesis.In Chapter 1 we present the main objectives and their research background studied in Chapter 2 to Chapter 3.This chapter is concerned with backward stochastic differential equations whose generator is a weighted fractional Brownian field:where W is a,(d+1)-parameter weighted fractional Brownian field of Hurst parameter H=(H0,H1,…,Hd).It provides a probabilistic interpretation(the generalised Feynman-Kac formula)for certain type of linear stochastic partial differential equations with coloured space-time noise.Conditions on the Hurst parameter H and on the decay rate of the weight are given to ensure the existence and the uniqueness of the solution.Moreover,the explicit expression for both components Y and Z of the solution is given.The novelties of this chapter are the followings:We are the first to study BSDEs whose generators are driven by a fractional noise,which dose not only contain time andspace variables,but beyond that does not have any(inverted)martingale properties.We obtain the proper and explicit expression of the solution of the equation,we determine the continuity coefficient of the solution with respect to time,and we also prove that the BSDE provides a probabilistic explanation for a type of SPDE.The calculations and estimates involved are novel,subtle and quite technical.This Chapter is based on the following paper:Y.Hu,J.Li,C.Mi.BSDEs generated by fractional space-time noise and relatedSPDEs.Applied Mathematics and Computation,450,127979,2023.DOI:10.1016/j.amc.2023.127979 In Chapter 3 we focus on a general type of mean-field stochastic control problems with partial observation,in which the coefficients depend in a non-linear way not onlyon the state process X and its control u,but also on the law of E[Xt|FtY]which is the conditional expectation of the state process X conditioned with respect to the past of observation process Y.We first deduce the well-posedness of the controlled system by showing the weak existence and the uniqueness in law.Then we study Peng’s stochastic maximum principle for our control problem.The necessary optimality condition we get extends Peng’s one with new,non-trivial terms.The novelties of this chapter are the following ones:The coefficients of the system depend nonlinearly on the law of conditional expectations of the state process with respect to the observed process.This has as consequence for the study of Peng’s maximum principle that we get a new type of first and second order variational equations and adjoint backward stochastic differential equations,all with new mean-field terms and with coefficients which are not Lipschitz.For their estimates and for those for the Taylor expansion new techniques have had to be introduced and rather technical results with rather subtle and technical proofs have had to be established.This Chapter is based on the following paper:J.Li,H.Liang,C.Mi.A stochastic maximum principle for partially observed general mean-field control problems with only weak solution.Submitted.https://doi.org/10.48550/arXiv.2109.12316.In Chapter 4 we give a summary and put forward possible future research directions.This dissertation includes four chapters mentioned above.We now give an outline of the structure and the main conclusions:Chapter 1 Introduction;Chapter 2 BSDEs generated by fractional space-time noise and related SPDEs;Chapter 3 A stochastic maximum principle for partially observed general meanfield control problems with only weak solution;Chapter 4 Summary and future works.Chapter 2:In this chapter we investigate the existence and the uniqueness of a solution for a class of linear BSDEs with weighted fractional noise with temporal and spatial parameters.Further,we study the Holder continuity of the solution Y and Z.Finally,we study the related SPDEs and prove the corresponding Feynman-Kac formula.We consider the following(one dimensional)linear backward stochastic differential equation(BSDE for short)with fractional noise generator:where B is a d-dimensional standard Brownian motion independent of W.Define(we shall justify it in the next section)and denote byFt=σ(Bs,0≤s≤t;W(t,x),t≥0,t≥0,x∈Rd)the σ-algebra generated by Brownian motion up to time t≥0 and by W(t,x)for all t and x ∈Rd.Then we have formally the following candidate for the solution pair where DtB is the Malliavin gradient with respect to the Brownian motion B(see next section for the definition and properties).By studying(0.0.14)and(0.0.15),some of the conclusions are summarized and the following theorem,one of the main results of this chapter,is given:Theorem 2.0.1.Suppose ∑i=1 d=1(2Hi-βi)<2 and ξ∈DB1,q is FTB-measurable,for q>2/(2H-1),where H=min{H0,…,Hd}and FtB=σ(Bs,0≤s≤t).Then we have the followilg results:(1)The process {(Yt,Zt),0 ≤t≤T}∈SF2(0,T;R)×MF2(0,T;Rd)formally defined by(0.0.15)is well-defined,and it is the solution to BSDE(0.0.14).Moreover,Z has the following alternative expression:(2)If for all q>2,E|DtB-DsBξ|q≤C|t-s|κq/2 for some κ∈(0,2),then for any a>1 and for any ε>0,we have the following H(?)_lder continuity for Y and Z:(3)If a solution(Y,Z)satisfies the conditions in(2)for some a,k>0,then(Y,Z)is represented by(0.0.15)and,hence,the BSDE(0.0.14)has a unique solution in SF2(0,T;R)× MF2(0,T;Rd).(4)If(Y,Z)∈SF2(0,T;R)×MF2(0,T;Rd)is a solution pair of BSDEs(0.0.14)so that Y and DBY are both in D1,2,then the solution also has the explicit expression(0.0.15)and,hence,the BSDEs(0.0.14)has a unique solution.The organisation of Chapter 2 is briefly described below.In order to study the properties of(Y,Z)defined by(0.0.15)it is first necessary to show that ∫tT W(ds,Bs)is well-defined.So we need the following approximation of(W(t,x))/(dt):where φη and pε are the approximations of Dirac function:Proposition 2.1.1.Let ρ:Rd→R be a continuous function of power decay,i.e.,ρ satisfying 0≤ρ(x)≤C∏i=1 d(1+|xi|)-βi,where βi ∈(0,2)and 2Hi>βi,for all i=1,2,…d,and suppose Then the stochastic integral Vtε,η:=∫tT Wε,η(S,Bs)ds converges in L2(Ω)to a limit denoted by Moreover,conditioning with respect to FB,Vt is a mean-zero Gaussian random variable with varianceFrom Proposition 2.1.1 we want to show ∫tT W(ds,Bs)is exponentially integrable.Proposition 2.1.2.Let ρ:Rd→R be a continuous function satisfying(0.0.17).Then,for all λ∈R,With the help of the above two statements,we can study the properties of(Y,Z)by using approximate methods.Consider the approximating equation for(0.0.14):Then there is solution(Ytε,η,Ztε,η)to the above equation,and it is given by Moreover,we haveTheorem 2.2.1.Suppose ξ∈Lq(Ω)for some q>2 and suppose that(0.0.17)holds.Then we have Yε,ηconverges to Y={Yt,t ∈[0,T]}∈SFp(0,T;R)for all p∈[1,q).Theorem 2.2.2.Denote Suppose that ∑i=1 d(2Hi-βi)<2,and that the FTB-measurable terminal conditionζbelongs to for q>2/(2H-1).Then Zε,η∈ MF2(0,T;Rd)and Zε,η has a limit Z={Zs,s∈[0,T]} in MF2(0,T’Rd).This limit can be written as In order to show that(Y,Z)is the solution of(0.0.14)as defined by(0.0.15)we also had to prove the following theorem:Theorem 2.2.3.Suppose that ∑i=1d(2Hi-βi)<2 and ξ∈DB1,q is FTB-measurable,for q>2/(2H-1),where H=min{H0,...,Hd}.Then,for any t ∈[0,T],we have in L2 sense,as ε,η↓0.The previous statements only prove that(Y,Z)as defined by(0.0.15)is the solution of(0.0.14),that is,the existence of the solution.The following theorem provides the uniqueness of the solution of(0.0.14).Theorem 2.4.1.Suppose that the conditions in Theorem 2.0.1 are satisfied.Let(Y,Z)∈SF2(0,T;R)× MF2(0,T;Rd)be the solution of BSDE(0.0.14)so that Y and DBY are in D1,2.Then the solution has the explicit expression(0.0.15)and,hence,BSDE(0.0.14)has a unique solution.Next,using BSDE(0.0.14)we study the probabilistic interpretation of a class of semilinear SPDE by establishing corresponding Feynmann-Kac formula.Consider the SPDE and the BSDE Theorem 2.5.1.Suppose φ∈C2(Rd).Let {u(t,x):t∈[0,T],x∈Rd} be a random field such that u(t,x)is Ft-measurable for every(t,x),u ∈C([0,T]× Rd,R)a.s.,and let u(t,x)satisfy(0.0.18).Then u{t,x)=Ytt,x,and ▽u(t,x)=Ztt,x,where(Ytt,x,Ztt,x)is the solution of(0.0.19).The following result gives the Feynman-Kac formula for SPDE(0.0.18).Theorem 2.5.2.Suppose the same conditions as in Theorem 2.0.1 and let(Yst,x,Zst,x)be the solution of BSDE(0.0.19).Then u(t,x):=Ytt,x,t ∈[0,T],x∈Rd,is in C([0,T]×Rd,R)and it is the solution of SPDE(0.0.18).Chapter 3:In this chapter we focus on a general type of mean-field stochastic control problems with partial observation,in which the coefficientsdepend in a non-linear way not only on the state process X and its control u but also on the law of conditional expectation E[Xt|FtY]of the state process X conditioned with respect to the past of the observation process Y.In order tostudy Peng’s stochastic maximum principle for our control problem,we establish a new type of first and second order variational equations and adjoint backward stochastic differential equations,all with new mean-field terms and with coefficients which are not Lipschitz.The necessary optimality condition for admissible control adapted with respect to the observation process we get extends Peng’s one with new,non-trivial terms.We give the dynamics of the state and the observation process which we focus on in this paper:where(B1,B2)is an(F,P)-Brownian motion,μtX|Y=P(?){EP[Xt|FtY]}.We first show the well-posedness for the above SDE.The difficulty stems from the fact that in the law μtX|Y the process X is conditioned with respect to Y,and both belong to the solution.To overcome this obstacle we transform(0,0.20)with the help of the Girsanov theorem.This leads to where(B1,Y)is an(F,Q)-Brownian motion,and Q=LTP is a probability measure.Lemma 3.2.1.Under Assumption(H1)equation(0.0.21)possesses a unique strong solution.The existence of a strong solution of the equation(0.0.21)allows to show that(0.0.20)has a weak solution.A strong solution for(0.0.20)cannot be expected because of the difficulty described above.The uniqueness in law for(0.0.20)is established by the following:Lemma 3.2.2.Under Assumption(H1),let(Ωi,Fi,Fi,Pi,(B1,i,B2,i),(Xi,Yi)),i=1,2,be two weak solutions of(0.0.20).Then it holds thatThe well-posedness obtained for both(0.0.20)and(0.0.21)allows to consider the associate control problem.where Pu=LTuQ,μtu=μtXu|Y and Eu[·]:=EPu[·]is the expectation under Pu.Here u is an admissible control process adapted to the observation process Y.As the control state set U is not supposed to be convex,we shall consider Peng’s stochastic maximum principle here.Assume that u ∈ Uad is an optimal control,and consider an arbitrary but fixed v ∈ Uad.For ε>0,we let Eε∈ B([0,T])with Borel measure |Eε|=ε,and we putThe process uε∈ Uad is a so-called spike variation of the optimal control u.Let us define the cost functional:The goal of the control problem is to minimize the cost functional.Consider the following first order variational equations:For the above equation we have the following statements:Lemma 3.3.1.Under the Assumption(H2),(0.0.25)has a unique solution(Y1,ε,K1,ε)∈SF2([0,T],Q)×SF2([0,T],Q).Moreover,Y1,ε,K1,ε,V1,ε∈SFp([0,T],Q)for all p≥1.Lemma 3.3.2.For all k≥1,there exists Ck ∈ R+,such that,Observe that(iv)corresponds to a second order Taylor expansion of(Xε,Lε).However,we have to improve the convergence speed of this approximation,as ε↓0.For the mean field case,the following very subtle technical result is crucial:Lemma 3.3.3.For allθ=(θ1,θ2)∈LF2([0,T),Q;R2)with EQ[|θt1|2+|Ltθt2|2)dt]<+∞,and(θt1,Ltθt2)∈ L2(Ft,Q;R2)for all t ∈[0,T],there exists ρ:[0,T]×R+→R+such that with ρt(ε)→ 0(ε↘ 0),t∈[0,T],and ρt(ε)≤ CEQ[|θt1|2+|Ltθt2|2],ε∈(0,1],t ∈[0,T].To better emphasize the method and make the calculation simpler,it can be assumed without loss of generality that σ=σ(γ,ν),h=h(x,ν),(γ,x,ν)∈P2(R)×R× U.The second order variational equation turns out to be as follows:whereIt has to be emphasised that these conclusions still hold true when the coefficients are general,but the relevant proof will only be more complicated.With arguments which are rather standard but although rather technical,we get the following statement:Lemma 3.3.4.Under Assumption(H2),(0.0.26)has a unique solution(Y2,ε,K2,ε)∈SF2|([0,T],Q)×SF2([0,T],Q).Moreover,Y2,ε,K2,ε ∈ SF∞-([0,T],Q)with SFp([0,T],Q)bounds independent of ε>0,for all p>2.Lemma 3.3.5.For all p>1,there is a constant Cp>0 such that,for all t ∈[0,T],and for all ε>0,The above results allow to improve the second order Taylor expansion given byLemma 3.3.2 as follows:Proposition 3.3.1.For all p>2,there exists a constant Cp∈R+ such that,with ρp(ε)→0,as ε↘ 0.Moreover,The first order adjoint BSDE over our reference probability space(Ω,F,Q)is the following:whereLemma 3.3.7.Under Assumption(H2),BSDE(0.0.27)has a unique strong solution((p1,(q1,q1)),(p2,(q2,q2))).Furthermore,for any p≥2,it holds that((p1,(q1,q1)),(p2,(q2,q2)))∈(SFp([0,T],Q)×(LFp([0,T],Q))2)×(SF2p([0,T],Q)×(LF2p([0,T],Q))2).To simplify the calculation,we make the following assumptions about σ and h.Assumption(H3):where φ:R→R is bounded Lipschitz function.To be able to derive the stochastic maximum principle by rewriting J(uε)-J(u)in terms of the dual relation(see(3.3.30)),we establish a novelly technical method which leads to the following very critical statement:Proposition 3.3.2.Assume(H2)and(H3)hold true.Define for all 0 ≤s ≤t≤T,where φ=Φ,f,respectively,and defineThen the right side of(3.3.21)can be written as:With the HamiltonianH(t,x,l,γ,v,q1,q2):=σ(t,γ,v)q1+h(t,x,y,v)lq2-f(t,x,γ,v),(0.0.28)for(t,x,l,γ,v,q1,q2)∈[0,T]× R×R+ ×P2(R)×U×R×R,and the notations the second adjoint BSDE is the followingNow,finally,with the help of the duality relations(we refer to(3.3.30)),we can obtain our Peng’s stochastic maximum principle.Theorem 3.3.1.Under the Assumptions(H2)and(H3),let u∈Uad be optimal and(X,L)be the associated solution of system(0.0.23).Then,for all v∈U,it holds that for dtdQ-a.e.(t,ω)E[0,T]× Ω,where((p1,(q1,q1)),(p2,(q2,q2)))and(P1,(Q1,1,Q1,2))are the unique solutions to(0.0.27)and(0.0.29),respectively.Let us emphasize that the maximum principle of random R=(Rt)t∈[0,t]and M=(Mt)t∈[0,t]are novel(see(3.3.36)and(3.3.37)).