Properties Of The Solution Of Backward Stochastic Differential Equation And Applications In Finance

In 1990, Pardoux and Peng [1] introduced the following backward stochastic differential equation:According to their theory, the solution of this kind of BSDE is a pari of adapted process, denoted by(y, z). Due to the connection of this subject with mathematical finance, stochastic control, partial differential equations, stochastic games, stochastic geometry and mathematical economics, many researchers have been working on this subject and related properties of the solutions of BSDEs, among these results, the comparison theorem of BSDEs with respect to y (first introduced by Peng (1991) ) is the more excellent work.An interesting question is how to compare part z of the solution(y, z) of the BSDE(0.1)? What about the properties of z? In fact the ability of comparing part z and knowing the properties of z is very important, because the part z represents the portfolio process in the problem of pricing of contingent claim. Peng connected BSDE with PDE and first investigated part z by the theory of PDE, and he gave a simple and explicit representation of part z by the method of PDE (see[12]). Then Zengjing Chen et al did much work by adopting the representation of part z (see[3], [4]). But all the above work is based on the fact that the generator of BSDE(0.1) g is deterministic. When g is stochastic, the method of investigating part z by PDE isn't much effective.In chapter 1, we try to explore the properties of part z by applying Malliavinderivative in the setting of BSDEs with stochastic generators. We can compare part z, so we shall give some comparison theorems about part z. The main results we have got are theorem 1.3.1, example 1.4.1, example 1.4.2, theorem 1.4.5 and so on. Because theorem 1.3.1 is the most basic one, we would list it in the following:Theorem 1.3.1: Suppose that the parameter (g,￡) of BSDE satisfies the as--sumptions of (H3), (H4) and (H5). Let (yt, Zt;Q 0,a.s. and Dtg(s, ys, zs) > 0, dp x dt, a.s. 0 < t < s < T.Then we have that almost surely for any time t 6 [0, T], Zt > 0; Moreover, if for any t € [0, T\,D￡ > 0, a.s. or Dtg(s, ya, zs) >0,dpx dt a.s.O < t < s < T, then Zt > 0, a.e. t e [0, T].(remark: The assumptions of (H3), (H4) and (H5) would be found in lemma 1.3.1.)Applying the comparison theorems about z, we study the comonotonic theorems of BSDEs given by Zengjing Chen et al. Firstly, we give two counter examples to point out that the comonotonic theorems for part z given by Zengjing Chen et al aren't correct when the generators of BSDEs are stochastic, then we shall give some special conditions under which the comonotonic theorems are correct even the generators of BSDEs are stochastic, and applying the comonotonic theorems, we shall explore the additivity of a class of conditional ^-expectations. Lastly, we shall study which kinds of conditional g- expectations can be represented as Choquet integrals.In the second chapter, we go on studying the properties of part z with this fact of the generators of BSDEs are all deterministic. About this topic, Zengjing Chen et al had done much work using the method of PDE, but the initial states of SDEs they used are real numbeds(see [4]). In this chapter, the initial states of SDEs we would use are random variables, and the method is Malliavin derivative. The generators of BSDEs only satisfy the assumptions of HI and H2. In the last, we would get the results similar to those of the first chapter. The main results we would get are such that lemma 2.2.2, theorem 2.2.1, theorem 2.3.1 and theorem 2.3.2; where the lemma 2.2.2 and theorem 2.2.1 are the basic conclusions, so we would give them in the following:Lemma 2.2.2: Suppose that the generator of BSDE g : [0, T] x R x R1 ~* Rsatisfies the assumptions of (HI) and (H2); ￡ satisfies (H5). Let (yuzt;0 O.a.s. Then we have that almost surely for any time t G [0, T],zt> 0; Moreover, if for any t G [0, T\,D￡ > 0, a.s. then zt > 0, a.s. t e [0,T]. (remark: The assumptions of (HI) and (H2) would be found in page 3.)Let's see the following SDEs, applying them to-construct the terminal .values of BSDEs.X? = C,se[t,T).Yf* = C,se[t,T). where bud {i - 1,2) satisfy assumption A, and C,C e L2(fl,^i,.P; J*)flP1>2-Theorem 2.2.1: Let g{{i = 1,2) satisfy (HI) and (H2), and f = ${X%c),r} = ^C~), where {X?} and {*??*} are defined by SDEs (0.2) and (0.3). $ and * are all continuously differentiable with uniformly bounded derivatives such that ￡ and 77 satisfy (H5). Suppose that (y^,z^) and (yr),z'n) are the solutions of BSDEs with parameters (51,0 and (g2,v)-(1) If $ and ^ are comonotonic, and for any 1 < j < d, D3^ > 0, LP9C, > 0, a.e. 9 € [0,t]; oi(6,XiQ) > 0,<4(d,Y^) > 0 a.e. 6 € [t,T] (or DJC < 0,i^C < 0 a.e. 9 € [0,*]; <^(0,X￡C) < 0,^(^,r^) < 0 a.e. 0 € [t,T]) then we have z\Qzn9> 0, a.e. 9 e [0, T]. (remark: D{Q and c^ are respectively the j - f/i components of De￡ and o"! )(2) Furthermore, if $ and * are strictly comonotonic, and for any 1 < j < d, Die > 0,DiC > 0, a.e. 9 € [O,tJ; o{{6,X?) > 0,<4{d,Yj'*) > 0 a.e. B G [t,T] (or ￡^C < 0,D{C < 0 a.e. 0 G [0,t]; af(^X^) < O,aJ(0,Y^") < 0 a.e. ff G [i,r]), then we have 4 0 ze > 0, a.e. 0 G [0, T]In chapter 3 and chapter 4, we study the applications of the solution of BSDE in finance. In chapter 3, we introduce the generalized stochastic differential utility (GSDU) by the solution of BSDE. We consider the case of an agent, who can't observethe random drift of the stock price process. This problem with partial information can be transformed into a problem with complete information by the method of filtering. That is to say that the random drift of the stock price process is replaced by its expected value conditional on the information given by the stock price history. The optimal consumption and portfolio choice in a complete market is considered, and the method to find the optimal consumption is got. The main knowledge that we use is the backward stochastic differential equation (BSDE). We get-the sufficient and necess_ary conditions to find the optimal consumption. The main results we get generalize those given by Jaksa Cvitanic etc(see [18]), Zengjing Chen etc(see [2]). The main result is followed :Firstly, we apply the following BSDE to define the preference relation on the consumption set C.-dYtc = /(<%, >?, Zt)dt - (Zfi'dWt, Y{ = 0. (0.4)Yo : C -> R, Yoc = E[( f(ca, Ysc, Zcs)ds], JVc1)C2 eC,Cly c2&yoc> >y0C3.Applying the following SDE to define the wealth process{XXiC>ir(t)} and a feasible portfolio choice and consumption strategy set A(x) ?. = x. (0.5)LetA(x) := {(c, tt) : for any initial wealth x > 0, Xx'c^{t) > 0,0 < t < T, a.s.}. Define the value function as following:v(x)= sup yoc. (o.6)()Theorem 3.6.1: Suppose that assumption 1-assumption 6 are satisfied. If c* E C is the optimal consumption for (0.6), then there exists a constant u > 0 such thatexp( f fy(c*(s),Ysc\Zf)ds)fc(c*(t),Ytc',Zf)Mt = uHu0 0 is a constant, and Xj G ￡2(fi, Ft, P\ R)-p : L2(fi, TT, P; R) -> R p(XT) = ￡.[------]. (0.9)r. Li \\l, J-t, r, ti) —? L [il,J-t-,! , K) Pt(AT) = ta\------\j-t . (0.10)Theorem 4.2.1: The dynamic risk measure pt is a dynamic coherent risk measure, and there exists a nonempty probability measures' set ? such thatPt(XT) = esssup{EQv[------|^i], Qv € 0}, VZr G L2(r2,^V, P; i2).Theorem 4.3.1: VXT G L2(Q,fT,P;R), where there exist functions 6(i,x), and a(t, x) satisfying assumption A and belonging to C3, such that XT is the value of the solution {Xt} of the following SDE at time T:dXs = 6(5, Xs)ds + a{s, Xs)dWs, X0 = x,se [0, T]. If a(s,Xs) > 0, a.e. s G [0,T], thenPt(XT) = ￡;Q[-^|^],p(Xr) = EqI-ZZ).where §\^ := exp{-i?2r - uWT).Theorem 4.3.2: Suppose XXT are the values of the solutions {XI} of the following SDEs at time T -. Xi0 = z\s e [0,21,1 = 1,2. (0.11)If a1 (s,X])a2{s,X2S) > 0, a.e. s G [0, T], thenChapter 5 is different from the four chapters above. We study the limit properties of probability measures on the topological semigroup. The study of probability measures' convolution powers on topological groups (or semigroups) is essentially the study of product of I.I.D random variables. Because of it's extensive implication in the aspects such as Markov's chains in abstract space, random walks, analysis of matrix etc, many authors have attached importance to the problem. The classical result about the problem is first obtained by Kawada-It 6 on compact groups. Sazonov and Collins make deep study about the result. They use the limit set about probability maesure's support set to depict weak convergence about the sequences of convolution powers of probability measures (see [47) Theorem 4.8). Recently, professor Liu jin e, Xu Kan and I have made progress in the same problem, and have obtained some new results ( see [48-51], [55-58]). In this paper, we shall study the equivalent conditions about weak convergence about the sequences of convolution powers of probability measures. The result we shall get covers the classical result made by Kawada-It 6 on compact groups. As far as the topological structure is concerned, there are radical difference between compactness and local compactness, thus our result is more valuable. The main result is the following theorem:Theorem 5.2.1: Suppose S is a locally compact group, jx € P{S), {nn} is tight, A(/x) is the set of weak limits of {/zn, n = 1,2, ? ? ?}. F containing S^ is the smallest closed subgroup. Let A = A2 is the identity of A(/i). Then the following conditions are equivalent.(1) {nn} is convergent;(2) lim^n ^ ;(3) limn5'/tn = li(5) Sj, doesn't been contained any proper N coset of F, where AT is a closed regular subgroup of F;(6) Sp doesn't been contained any proper S\ coset of F;(7) A is a Haar measure of F, i.e. Vx 6 F,VB e B(S), we have that \iBx~1) = Xix^B) = X(B) and Sx = F.