Research About Backward Stochastic Differential Equations And The Related Questions

The study of Backward Stochastic Differential Equations (BSDEs for short) stems from the research about stochastic control in [4] in 1973. BSDEs hadn't been studied intensely in many directions until the generalized BSDEs were proposed by Pardoux, Peng [66] where the existence and uniqueness result was proved. Since then it has become a powerful tool not only in the fundamental theories (see [20], [46], [60] and so on) but also in applications in finance (see [33], [34], [95] and so on). An important application in mathematical finance is option pricing which received many attentions (see [5], [27], [51] and so on ). From [32], [77], we can see the pricing of options under some conditions can be translated into solving a parameterized BSDE, that is a Forward Backward Stochastic Differential Equation (FBSDE for short). Therefore in Chapter 1 we will consider the theory about FBSDEs. It is well known that European option and American option are the most typical options and in [32], it is easy to see the price of European option can be obtained by solving some kind of BSDE given the European contingent claimζsettled at time T. As we all know, in the complete financial market a given contingent claim can be replicated by a self-financing trading strategy , but in the incomplete financial market the idea of a perfect hedge is limited in scope, i.e. derivative typically will carry an intrinsic risk which can not be hedged away completely. At this stage we are going to focus on controlling the shortfall, that is, minimizing the probability P(C_t > 0) that some shortfall occurs. Then in Chapter 2, we will consider the properties of g-expectation, g-probability inducted by BSDEs. Correspondingly, for American option [65] shows us pricing an American option is in fact an optimal stopping problem which will be explored in Chapter 3. Finally in Chapter 4, we study the constrained portfolios in financial markets by BSDEs.In a word, the dissertation is developed based on BSDEs and option pricing. Main results are as follows:1. We prove the existence and uniqueness of an adapted solution of infinite horizon FBSDE with absorption coefficients by successive approximation method, which extends the conditions of solvability of FBSDEs.2. The law invariance, additivity, 2-alternating and sub 2-alternating of non-additive measure are important research problems in mathematical finance and mathematical economics. And we study the law invariance of g-expectation, the additivity and convexity of g-probability.3. High contact principle is one of the important problems in economic circle, and we explore the optimal stopping problem inducted by a kind of BSDEs and a class of general optimal stopping problems in which reward functions depend on initial point and derive the corresponding high contact principle.4. Based on work of Chen [12], the properties of Z in BSDEs and their applications in incomplete financial market are developed.This dissertation consists of four chapters, whose main contents are described as follows:In Chapter one we consider the solvability of following forward-backward stochastic differential equations:where (x_s, y_s, z_s) take values in R^m×R¹Ã—R^d and b,σ, c, h, are given functions with appropriate dimensions; Our aim is to find the unknown {F_t}_t≥0^- adapted processes (x(·),y(·),z(·)) which satisfy the above forward-backward stochastic differential equations, on [0,∞), P-almost surety. In this chapter we are interested in this above infinite horizon forward-backward stochastic differential equations. Instead of monotonicity conditions required in the previous work, we consider absorption conditions which were introduced in Preidlin [35]. As an example illustrated, FBSDEs with absorption coefficients are different from those with monotonicity coefficients. By using the successive approximation method, we prove the uniqueness and existence of the solution of FBSDE.Theorem 1.2.3 Suppose the conditions H1-H3 hold. We consider forward-backward stochastic differential equation (1.2). Then, (?) u⁽ⁿ⁾(t, x) exists and this convergence is uniform on the set {x (?) R^m, 0≤t≤∞}. We also have (x_s^(n),t,x)_{t≤s<∞} converges with probability 1 on [0,∞).This is a successive approximations theorem. Next are the theorems about the uniqueness and existence of the solution of FBSDE.Theorem 1.3.2 Under the assumptions of Lemma 2.1, equation (1.1) has a solution (x(·),y(·),z(·)) where x_s = (?)x_s⁽ⁿ⁾,y_s = (?)u^<sup>n+1))(s,x_s⁽ⁿ⁾) with probability 1 on [t,∞); z = (?) z⁽ⁿ⁾ in M²(0,∞; R^d).Theorem 1.3.3 There exists a unique function u(s,x) and a unique pair of processes (x_s, z_s) such that the triple (x_s, u(s, x_s), z_s) is a solution of equation (1.1).Chapter two includes three main contents. Firstly, we study the law invariance of g-expectation. When we define the law invariance of g-expectation, there are two points deserving attention. One point is that the law invariance is defined on the set of all random variables with the forms of Markov processes owing to the applied perspective in risk management. The other point is that we define law invariance of g-expectation in the sense thatζ-ηandζ-_gη, respectively. We prove thatε_g [·] is law invariant in the sense thatζ-ηif and only if g is a trivial function. Also the proof is developed in two ways. Theorem 2.2.8 (The y-independent case) For the g-expexctation, then the following conclusions are equivalent:(i)ε_g[·] is law invariant;(ii)g(z,s)≡0,(?)(z,s)∈R×[0,T].Theorem 2.2.9(The general case) For the g-expexctation, then the following conclusions are equivalent:(i)ε_g[·] is law invariant;(ii) g(y,z,s)≡0 ,(?)(y,z,s)∈R×R×[0,T].But under the definition of the law invariance in the sense thatζ～_gη, we only get the sufficient condition of law invariance of g-expectation.Theorem 2.2.11 Suppose that there exist two continuous functions {α(t)}, {β(t)} such that g does not depend on y and with the formg(y,z,t) =α(t)|z|+β(t)z, then the g-expectationε_g[·] is law invariant in the sense thatζ～_gη.Secondly, we study the additivity of g-probability. Capacities are nature and widely used measures of imprecision (or ambiguity) and a generalization of probability measures. Usually capacities fail to have a additivity property except probability measures, so they are not easy to be measured in practice. However, in this section, we consider a natural set of probability measures arising from financial markets. We show that the upper prior probability measures are additive for a large collection of sets. This result is somewhat surprising and unusual.Theorem 2.3.1 For any a, b∈R, let A = {W_T≤α} and B = {W_τ≥b) for someτ≤T in BSDE , then(i) z_t^A≤0, and z_t^B≤0, a.e. t∈[0,T). (ii)z_t^A∩B≤0,a.e.t∈[0,T).(iii) If k > 0, then the solutions y_t^A∪B, y_t^A, y_t^B, y_t^A∩B at time t = 0 satisfyy₀^A∪B = y₀^A + y₀^B - y₀^ABthat is ,V(A∪B) + V(A∩B) = V(A) + V(B).Thirdly, we study the convexity of g-probability. Most work on capacities has focused on the 2-alternating and sub 2-alternating. It is well known that g-probability defined from peng's g-expectation is a capacity. We show that the g-probability fails to be 2-alternating but is sub 2-alternating when g satisfies the condition of sub-additivity. We have BSDE as the tool to study the properties of the g-probability all through. Thus it is more convenient to study than other capacities.Next is an important example, this shows although g is convex, the g-probability can not be always 2-alternating.Example 2.4.6 Let g = log(l+z⁺) in BSDE and A = {(W_T- W_Ï„)≤0}, B = {W_τ≥a}, (?)a∈R. ThenV(A∪B)> V(A) + V(B) - V(A∩B) where the definition of V(·) is defined as the above example.Before we give the main result, let's introduce the definition of sub 2-alternating.Definition 2.4.3 We say that conjugate capacities V and v are sub 2-alternating if (?)A, B∈F_T,v(A∪B)≤V(A) + V(B) - V(A∩B)andV(A∪B)≥v(A) + v(B) - v(A∩B). Theorem 2.4.7 Suppose that g in BSDE satisfies (H1-H4), then the g-probability defined from BSDE is sub 2-alternating.In Chapter three we consider the problems about optimal stopping. First we study the optimal stopping problem derived by a class of BSDEs. Consider the following problem: find u(x) andÏ„^*∈χsuch thatwhere g(·) is a bounded continuously differentiate function in Rⁿ.We study the above question and get the properties of the value function and the high contact principle as follows.Theorem 3.2.8 Suppose X_t, g andε(·) are described as be described as at the beginning of this section. If there exists an open set D (?) Rⁿ with C¹-boundary and a functionφ(x) on (?) such that: (i)φ(·) G C¹((?))∩C²(D),g(·)∈C²(G\(?))φ(x)≥g(x) x∈D Ag(x)≤0 x∈(G\(?));(ii) ("The second order condition ") (D, q) solves the following free boundaryproblemAφ(x) = 0 x∈Dφ(x)= g(x) x∈(?)Dâ–½_xφ(x) =â–½_xg(x) x∈(?)D∩G,Then, extendingφ(x) to all of G by puttingφ(x) = g(x) for x∈G\Dwhere D = {x:φ(x)>g(x)}Secondly, we consider the stopping time where a reward function g depends not only on X_t, but also on the initial point x. That is, g = g(X_Ï„^x, x). The qustion is as follows: find u(x) andÏ„^*∈χsuch sthatwhere g(·,x) is a bounded continuously differentiate function in R^k. For simplicity we also assume that the diffusion process X_t in equation (3.2) satisfiesΣa_i,jζ_iζ_j≥δΣζ_i²(δ> 0) with a = [a_i,j] = 1/2σσ^T.Next we examine a class of general optimal stopping problems in which reward functions depend on initial points. Two points of view on the initial point are introduced: one is to view it as a constant, the other is to view it as a constant process starting from the point. Based on the two different views, two versions of the generalized high contact principle are derived.Theorem 3.3.5 Let X_{t, g} be described as at the beginning of this section. For any given initial point x, if there exist an open set D(x) (?) R^k with C¹-boundary and a function q_x(y) on (?)(x) such that: (i) q_x (·)∈C¹((?)(x))∩C²(D(x)), g(·,x)∈C²(G\(?)(x)),q_x(y)≥g(y,x) y∈D(x)Ag(y,x)≤0 y∈(G\(?)(x));and (ii) ("The second order condition") (D(x),q) solves the following free boundary problemAq_x(y) = 0 for all y∈D(x)q_x(y) = g(y,x)for all y∈(?)D(x)â–½_yq_x(y)=â–½_yg(y,x) for all y∈(?)D(x)∩G,then, extending q_x(y) to all of G by putting q_x(y) = g(y, x) for y∈G\D(x), we havewhere q(x) = q_x(x),andD(x) = {y : q_x(y) >g(y,x)}.So far we have introduced our first point of view on the original initial point x. That is to view it as a parameter or a constant. Next, we will introduce our second point of view on it. That is, instead of viewing it as a constant, to view it as a process starting from x.To this end, we change the initial point of the Ito diffusion process X_t from x to ydX_t = b(X_t)dt +σ(X_t)dB_t, X₀ = y,and define a constant Ito diffusion process Z_t as followsdZ_t = 0, Z₀ = x. Put them together, we have a new Ito diffusion process Y_t = Y_t^(y,x) in R^2k bywhereη= (ζ,x)∈R^k×R^k. Thus, Y_t is an Ito diffusion process starting at (y,x). Let P^(y,x) denote the probability law of Y_t and E^(y,x) denote the expectation w.r.t. P^(y,x) In terms of Y_t, equation (3.25) can be rewritten as which is a special case of the problemwhere the optimal stopping time associated with u(y,x) is denoted byÏ„^*.From the above, we have the second version of the generalized high contact principle:Theorem 3.3.7 Let g be described as the above and X_t be described as in equation (3.37), and G (?) R^k be an open set and denote W = G×G. For any given initial point (y, z), if there exist an open set D (?) R^2k with C¹-boundary and a functionφ(y, x) on (?) such that: (i)φ(·,x)∈C¹((?))∩C²(D),φ(y,x)≥g(y,x) forall(y,x)∈D Ag(y,x)≤0 for all (y,x)∈(G\(?));and (ii) ("The second order condition") (D,φ) solves the following free boundary problemAφ(y,x)= 0 for all (y, x)∈Dφ(y, x) = g(y, x) for all (y, x)∈(?)Dâ–½_yφ(y,x) =â–½_yg(y,x) for all (y,x)∈(?)D∩W,then, extendingφ(y,x) to all of W by puttingφ(y,x) = g(y,x) for (y,x)∈W\D and letting y = x, we havewhereandD = {(y,x) :φ(y,x) > g(y,x)}.In Chapter 4 we explore the properties of Z in BSDEs. A control theorem with respect to z is obtained. As the application of the results, we get a constrained interval of the portfolio in the incomplete financial market. We first get the new representation theorem.Theorem 4.3.1 For FBSDE (4.3). suppose (H3) holds, g∈C_b^0,1([0, T]×R^d×R×R^d;R),Φis continuous on R^d. Denote A = (x∈R^d :Φ_x dones't exist}. Assume further that P{X_T∈A} = 0. then we havewhere (?),â–½X_s is the solution of the variational equationwhere I_n×n denotes the nx n identity matrix.In the following we get a comparison theorem basing on the representation theorem.Theorem 4.3.3 Suppose (X,Y,Z) is the solution of FBSDE (4.11). assumptions (H1 - H2) hold.Φand the set A are defined as in Theorem 4.3.1 and alsoσ(s, x) is increasing in x andσ(s, x)≥0, (?) (s, x)∈[t, T]×Rⁿ, then(i) If b(s,x) is decreasing in x, g(s,y,z) is decreasing in y,z respectively, and x(?)A,Φ_x(x)≤K(K≥0), thenZ_s≤K_σ(s,X_s), a.e, s∈[t,T](ii) If b(s,x) is increasing in x, g(s,y,z) is increasing in y,z respectively, and x(?)A,Φ_x(x)≥K(K≥0), thenZ_s>Kσ(s,X_s),a.e,s∈[t,T]...