【摘要】：自2009年Buckdahn,Djehiche,Li和Peng[1]率先引入平均場倒向隨機微分方程(簡記為,MFBSDEs),這類方程就倍受關(guān)注。他們研究了 MFBSDEs和相應(yīng)偏微分方程(簡記為,PDEs)粘性解的關(guān)系。本文主要研究的是一類新型的平均場PDEs的弱解一 Sobolev解。與粘性解不同的是Sobolev解的存在唯一性不需要依賴于比較定理的結(jié)果,故方程的系數(shù)可以依賴于(?)。本文主要研究的方程形式如下:平均場SDE:平均場BSDE:以及新型平均場PDE:第一部分:主要的假設(shè)條件有:假設(shè)3.1:(A1)(i)函數(shù)b和σ關(guān)于(?),x滿足Lipschitz條件。(ii)b(.,0,0)和σ(.,0,0)是F-循序可測連續(xù)函數(shù)且存在常數(shù)l0,使得對任意的0≤t≤T,(?),x(?)R~d(A2)(i)Φ是F(?)B(R)-可測隨機變量,f(·,(?),x,(?),y,(?),z)是F-適應(yīng)的可測過程,對任意的((?),x,(?),y,(?),z)∈ R~d × R~d × R~n × R~n × R~n×d × R~n×d 成立。且 f(t,(?),x,0,0,0,0)∈H_F~2(0,T;R~n)。(ii)f 關(guān)于 (?),x,(?),y,(?),z 滿足 Lipschitz 條件。(iii)f和Φ滿足線性增長條件,也就是說,存在c0,使得a.s.對任意的(?),x∈R~d，|f(t,(?),x,0,0,0,0)| + |Φ((?),x)| ≤c(1+|(?)|+|x|).(iv)G,θ,Ψ,κ:R~d→R~d,∧:R~d → R~d,Γ:R~n×d→R~n×d 的 Lipschitz 連續(xù)函數(shù)。(A3)給定((?))∈Rd × Rn × R×d,氣對任意的s ∈[0,T],(x,y,z)→f(s,(?),x,(?),y,(?),z)∈Cb3,3,3(Rd × Rn × Rn×d,Rn).(A4)b ∈C61,3,3([0,T]× Rd × Rd,Rd)且 σ ∈Cb1,3,3([[0,T]× Rd ×RRd,Rd×d)。同時,我們給出值函數(shù)的定義為u(t,x)=Ytt,x。那么,在假設(shè)3.1下,平均場PDE(3)存在唯一解,且滿足以下關(guān)系式Y(jié)st,x=u(s,Xxt,x),Zst,x= Dxu(s,Xst,x)σ(s,E[θ(Xs0,x0)],Xst,x).借助隨機逆流、等價范數(shù)及測試函數(shù),最終我們可以得到在假設(shè)3.1-(A2),(A4)下,u(t,x)=Ytt,x是平均場 PDE(3)的唯一 Sobolev 解。第二部分:第一,我們研究的是以下假設(shè)4.1條件成立的情況下,帶全局單調(diào)系數(shù)的MFBSDE(2)解的存在唯一性定理。假設(shè)4.1:(H1)對任意固定的(ω,t),f(ω,t,.,.,.,.)連續(xù);(H2)存在一過程ft∈HF2(0,T;R)和一個常數(shù)L0,使得|f(i,(?),(?),y,z)|≤ft + L(|(?)| + |(?)| + |y| + |z|).(H3)存在常數(shù)λ1,λ2 ∈ R,使得對任意的t∈[0,T],yi,(?)i ∈ Rn,z,(?) ∈ Rn×d(i = 1,2),(y1-y2)(f(t,(?),y1,(?),z)-f(t,(?),y2,(?),z))≤λ1(y1-y2)((?))+λ2|y1-y2|2.(H4)存在 L0,使得 P-a.s.對任意的t∈[0,T],y,y ∈ Rn,zi,zi ∈ Rn×d(i = 1,2),|f(t,y,y,z1,z1)-f(t,y,y,z2,z2)|2 ≤ L(|z1-z2|2 + |z1-z2|2).第二,我們研究的是帶局部單調(diào)系數(shù)的MFBSDE(2)解的存在性和唯一性,假定如下條件成立,假設(shè)4.2:(H2,)存在L0和0 ≤ γ ≤ 1,使得|f(t,y,z,y,z)| ≤L(1 + |y|γ +|z|γ + |y|γ + |z|γ).(H3')對任意的N ∈ N,存在常數(shù)λN,λN∈R,使得對任意的t ∈[0,T],yi,yi∈Rn,z,z ∈ Rn×d 滿足"yi|,|yi|,|z|,|z|≤N(i = 1,2),有(y1-y2)(f(t,y1,y1,z,z)-f(t,y2,y2,z,z))≤λN(y1-y2)(y1-y2)+ λN|y1-y2|2.(H4')對任意的N ∈N,存在LN0,使得P-a.s對任任的的f ∈[0,T],y,y∈Rn,zi,zi∈Rn×d滿足|yi|,|yi|,|z|,|z|≤N(i=1,2),成立|f(t,y,y,z1,z1)-f(t,y,y,z2,z2)|2LN(|z1-z2|2 + |z1-z2|2).那么,我們可以得到在假設(shè)4.1-(H1)和假設(shè)4.2成立的情況下,且滿足1 + exp(2L + 2|λN|+2λN-+ +2LNθ-1 + 2)→0,當(dāng)N→∞時,(4)其中θ是一個任意固定的常數(shù),使得0θ1-2α。帶局部單調(diào)系數(shù)的MFBSDE(2)有唯一解(Y,Z)。第三:在前面的結(jié)論成立的情形下,我們可以開始研究相應(yīng)平均場PDE(3)的Sobolev解的存在唯一性。首先,我們可以得到在以下假設(shè)下:假設(shè)4.3:(B1)6,σ 滿足假設(shè) 3.1-(A1),(A4)。(B2)f,Φ 滿足假設(shè) 3.1-(A2)-(i)(iii),以及假設(shè) 3.1-(A2)-(iv)成立,Φ ∈ L2(Rd,ρ(x)dx)。(B3)對任意的0≤t≤T,x1,x2,x1,x2∈Rn,y,y1,y2,y,y1,y2∈Rn,z,z1,z2,z,z1,z2 Rn×d,存在 C0,λ1,A2 ∈R,使得|Φ(x1,x1)-Φ(x2,x2)|2 +|f(t,x1,x1,y,y,z1,Z1)-f(t,x2,x2,y,y,z2,z2)|2C(|x1-x2|2 + |x1-x2|2 + |z1-z2|2 + |z1-z2|2).(y1-y2)(f(t,x1,x1,y1,y1,z,z)-f(t,x2,x2,y2,y2,z,z))≤ λ1(y1-y2)(y1-y2)+ λ2|y1-y2|2.(B4)|f(t,(?),x,(?),y,(?),z)| ≤ |f(t,(?),x,0,0,0,0)| + K(|y| + |y| + |z| + |z|),f(t,x,x,0,0,0,0)∈ L2(Rd,ρ(x)dx)且滿足線性增長。值函數(shù)u(t,x):=Ytt,x是帶全局單調(diào)系數(shù)的平均場PDE(3)的唯一Sobolev解。接著我們也可以得到在局部單調(diào)性的假設(shè)下平均場PDE的Sobolev解的存在唯一性定理的結(jié)論。相應(yīng)的局部單調(diào)性假設(shè),如下:假設(shè)4.4:(B3,)對任意的N ∈ N,存在LN0,λN,λN∈R,使得對x1,x1,x2,x2∈Rd,y1,y1,y2,y2∈Rn,z1,z1,z2,z2∈Rn×d,滿足|y1|,|y1|,|y2|,|y2|,|z1|,|z1|,|z2|,|z2|≤N,成立|Φ(x1,x1)-Φ(x2,x2)|2 + |f(t,x1,x1,y1,y1,z1,z1)-f(t,x2,x2,y1,y1,z2,Z2)|2≤ LN(|x1-x2|2 + |x1-x2|2 + |z1-Z2|2 + |z1-z2|2),(y1-y2)(f(t,x1,x,x1,y1,y1,z1,z1)-f(t,x1,x1,y2,y2,z1,z1))≤ λN(y1-y2)(y1-y2)+ λN|y1-y2|2(B4,)存在K0和0≤γ≤1,使得|f(t,(?),x,(?),y,(?),z)|≤K(1 +|y|γ +|z|γ+ |y|γ +|z|γ),對任意的t,(?),x,(?),y,(?),z.于是,我們就可以得到在假設(shè)4.3-(B1),(B2),假設(shè)4.4和(4)式成立的條件下,帶局部單調(diào)系數(shù)的平均場PDE(3)式存在唯一的Sobolev解。
[Abstract]:In 2009, Buckdahn, Djehiche, Li and Peng[1] took the lead in introducing the mean field backward stochastic differential equations (simple as, MFBSDEs). These equations are concerned. They have studied the relationship between MFBSDEs and the corresponding partial differential equation (Li, PDEs). This paper mainly deals with a new class of Weak Solutions of the mean field PDEs. The existence and uniqueness of the solution of Sobolev is not dependent on the result of the comparison theorem, so the coefficient of the equation can be dependent on (?). The main equations of this paper are as follows: the mean field SDE: mean field BSDE: and the new mean field PDE: first part: the main hypothesis conditions are: the 3.1: (A1) (I) function B and sigma (?), X satisfaction (II) B (., 0,0) and sigma (. 0,0) are F- sequential measurable continuous functions and exist constant l0, making any 0 less t less than T, (?), X (?) R~d (A2) is a measurable random variable. Constructs f importantly Marxism importantly Marxist Marxist Marxist Marxist Marxist Marxist Marxist Marxist societies traditions Marxism traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Gamma: The Lipschitz continuous function of n * D. (A3) Rd * Rn x R x D, gas to any s [0, T]. Under assumption 3.1, the average field PDE (3) has a unique solution and satisfies the following relational expression Yst, x=u (s, Xxt, x), Zst, x= Dxu (s, Xst, x)). With the aid of random currents, equivalent norms and test functions, we can finally get the only solution of average field (3). Second: First, we study the existence and uniqueness theorem of MFBSDE (2) solution with global monotone coefficients under the condition of the following hypothesis 4.1 conditions. 4.1: (H1) is assumed to be arbitrarily fixed (omega, t), f (omega, t,,,.) continuous; (H2) there exists a process FT HF2 (? 0, T; R) and a constant L0. Beings Marxist Marxist traditions Marxist Marxist societies veins veins occasions veins veins veins occasions veins veins veins veins occasions veins veins veins veins veins veins occasions veins traditions souls veins veins veins occasions veins traditions souls veins veins veins veins veins occasions veins veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins traditions souls veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins traditions souls veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins traditions souls veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins it is the same. Z2|2 + |z1-z2|2). Second, we study the existence and uniqueness of the MFBSDE (2) solution with local monotone coefficient, assuming the following conditions are established, assuming that 4.2: (H2,) exists L0 and 0 < < < 1 >, and makes |f (T, y, Z, y, z) less than equal. Derive Rn clauses importantly Marxist traditions Marxist Marxist souls societies veins veins veins occasions veins veins veins veins veins veins occasions veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins Y, y, Z2, Z2) |2LN (|z1-z2|2 + |z1-z2|2). Then, we can get the assumption that 4.1- (H1) and hypothesis 4.2 are set up and satisfy the 1 + exp (2L + 2| lambda [lambda] [[lambda], theta 2) - 0. (4) theta is a arbitrarily fixed constant, which makes 0 theta 1-2 alpha (2) with local monotone coefficient. Third: Third: Third: Third: Third: before third: Third: before third: Third: Third: before third: Third: Third: before We can begin to study the existence and uniqueness of the Sobolev solution of the corresponding mean field PDE (3). First, we can get the following hypothesis: assuming 4.3: (B1) 6, sigma satisfies the hypothesis 3.1- (A1), (B2) f, and the hypothesis is assumed to be 3.1- (A2) - (I). , x1, X2, x1, Rn, Rn, y, Y1, Rn, Rn, Y1, Rn, Rn, Rn, Y1, Rn, Y1 less than 1, 1 and 2 |y1-y2|2. (B4) |f (T, (?), x, y, z), z) and K (x, 0,0,0,0) and satisfy linear growth. The corresponding local monotonicity hypothesis, as follows: assuming 4.4: (B3,) for arbitrary N N, LN0, N, N R. -f (T, X2, X2, X2, X2, X2, Y1, Y1, Y1, Y1, Y1) they are (?). So, we can get the only Sobolev solution of the mean field PDE (3) with local monotone coefficient under the assumption that 4.3- (B1), (B2), hypothesis 4.4 and (4) are established.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O211.63

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