Numerical Methods For Solving Forward-backward Stochastic Differential Equations With Their Applications In Finance And The Cauchy Problem For Hyperbolic Equations

The study of backward stochastic differential equations (BSDEs) is a relatively new field and has been developed quite rapidly comparing with that of stochastic differential equations (SDEs) which lasts for more than half a century. This is because, besides hav-ing many interesting mathematical properties, BSDEs show a broad application prospect in more and more fields. The linear BSDEs which studied by Bismut [16] in1973can be regarded as the extension of the well-known Girsanov theorem while the basic framework of nonlinear BSDEs was introduced by Pardoux and Peng [95] in1990. In [98], Peng first got the relation between backward stochastic differential equations and partial differen-tial equations (PDEs), and then in [97], he studied the stochastic maximum principle for optimal control problems based on BSDEs. From then on, significant progresses have been made in the study of the theory and applications of BSDEs, including Forward-Backward Stochastic Differential Equations (FBSDEs), Reflected BSDEs, Constrained BSDEs, Backward Doubly Stochastic Differential Equations, Application in Stochastic Control, Application in Mathematical Finance, Nonlinear Expectations and Nonlinear Martingale Theory, Recursive Utility, Partial Differential Equations and Differential Ge-ometry, etc.Along with the fast development in the theory and applications of BSDEs and FB-SDEs, the research of their numerical methods has attracted more and more attention. By using the relation between FBSDEs and PDEs, a four step scheme for solving FB-SDEs was proposed by Ma and Yong [87]. Based on the four step scheme, a numerical method was developed for solving FBSDEs in [37], in which the Euler scheme was used for the forward SDEs and the characteristic finite difference method was used for the PDEs. Based on similar ideas,[89,92,93,128] proposed some numerical methods for solving FBSDEs. In [6], Bally introduced a random time discrete scheme and used the jump time of the Poisson process to discrete BSDEs. Later, for pricing American options Bally and Pages [7] studied an optimal quantization method that can also be used to solving a certain kind of reflected BSDEs whose generators do not depend on z. Zhang [125,126] studied the properties of the solutions of BSDEs, proposed the Euler scheme and obtained its half order convergence rate. Gobet and Labart [51] generalized Zhang’s result [126], gave error expansion for the Euler scheme and proved the accuracy of half order convergence for the Euler scheme. In [52], the authors proposed a numerical regression-based Monte Carlo method which can be used to solve the high-dimensional problems. Through the regression operator deduced by the kernel estimation or Malli-avin calculus to calculate the conditional mathematical expectation on each time level, Bouchard and Touzi [18] developed a backward discrete scheme. Bender and Denk [12] proposed a forward scheme for solving BSDEs. Delarue and Menozzi introduced a time-space discrete scheme for solving quasi-linear parabolic PDEs in [34] and then improved this scheme through a special interpolation procedure in [35]. Cvitanic and Zhang [32] transformed the FBSDEs to a control problem and presented a steepest descent method to solve it. In2006, Zhao, Chen and Peng [129] proposed a θ-scheme for BSDEs, and used Monte Carlo simulation and spatial interpolation method to approximate the conditional mathematical expectation on each time level. In2009, Zhao, Wang and Peng [131] gave the L1-error estimates for the θ-scheme when used to solve BSDEs whose generators do not depend on z. In2010, using the variational equations of the BSDEs whose generators do not depend on z, Wang, Luo and Zhao [119] proposed the Crank-Nicolson scheme which has the second-order convergence accuracy for solving z.This doctoral thesis consists of three parts. The main part is focused on the numer-ical methods for solving forward-backward stochastic differential equations and theirs applications in finance and the Cauchy problem for hyperbolic partial differential equa-tions. The second part concerns the risk measure induced by gΓ-solution of a constrained backward stochastic differential equation (CBSDE) and uses the inf-convolution of the convex risk measures to deal with some optimization problems involving the transforma-tion of the initial risk measures. Finally, the profit and risk of ruin have been studied for some gambling strategies, such as the doubling bets strategy, Fibonacci betting strategy and the exposure controlling strategy. Moreover, the applications of these strategies in the money management systems are also analyzed and compared. The thesis is organized by seven chapters. In the following, we list the main results of this thesis. Chapter1:We will first give a brief introduction to SDEs,BSDEs and FBSDEs, and then review some numerical methods for solving BSDEs,FBSDEs and hyperbolic PDEs.Finally,we will recall some fundamental results on the homotopy analysis method and the vanishing viscosity method that play an important role in the nonlinear problems or the Cauchy problem for hyperbolic PDEs.Chapter2:In this chapter,we will mainly consider the following BSDEs where Wr=(Wr1,...,Erd)T is a standard d-dimensional Brownian motion,Wst,x x+Ws-Wt(t≤s≤T).In Section2.1,we will first recall the θ-scheme proposed by[129]for BSDEs(1), and then in Lemma2.1we will give the estimates for the truncation error terms Rynand Rzn by using Ito formula.Scheme2.1Let(yn,zn)(n=N-1,N-2,...,0)be the numerical solution of BSDEs(1)at the time tn.Given yN=yT.For n=N-1,N-2,…,1,0,(yn,zn)can be solved by the following equations-{θ3△tnzn+(1-θ3)△tnEtnx[zn+1]}, where θ1,θ2∈[0,1],θ3∈(0,1].Lemma2.1(1).If f∈Cb1,3and φ∈Cb3+α,then the following estimates hold for θ1,θ2∈[0,1],θ3∈(0,1].(2).If f∈Cb2,5and φ∈Cb5+φ,then the following estimates hold for θi=1/2(i=1,2,3).Here,C is a constant depending only on T,upper bounds of derivatives of φ, f and u,where u is the classical solution of PDE(2.3).Based on[119,129],in Section2.2we will propose the following semi-discrete scheme for BSDEs(1).Scheme2.3Let(yk,n,zk,j,n)(n=N-1,N-2,…,0,1≤k≤m,1≤j≤d)be the approximations of(ysk,zsk,j)tn≤s≤T at the time s=tn.Given yk,N=yTk and zk,j,N=zTk,j for1≤k≤m and1≤j≤d.Then for n=N-1,N-2,…,1,0,(yk,n,zk,j,n,zk,j,n)can be solved by the following equations where θ1,θ2∈[0,1],yn=(y1,n,…,ym,n)T,zn=(zk,j,n)m×d.For the error estimates of the above scheme,we have the following theorem.Theorem2.3Let(yt,zt)be the solution of BSDEs(1)and(yn,zn)(n=N,N-1,...,0) be the numerical solution of Scheme2.3.(1).For1≤k≤m,if fk∈Cb1,3and φk∈Cb3+α, then for sufficiently small time step△t and any θ1∈[0,1]we have (2).For1≤k≤m,if fk∈Cb2,5and φk∈Cb5+α,then for sufficiently small time step△t and θ1=1/2we haveHere C is a constant depending only on c0,m,d,T,upper bounds of derivatives of φ,f and u,where u is the classical solution of PDE(2.3).Theorem2.4Let(yt,zt)be the solution of BSDEs(1)and(yn,zn)(n=N,N-1,...,0) be the numerical solution of Scheme2.3.(1).For1≤k≤m,if fk∈Cb1,3and φk∈Cb3+α, then for sufficiently small time step△t and any θ2∈[0,1]we have (2).For1≤k≤m,if fk∈Cb2,5,φk∈Cb5+α,then for sufficiently small time step△t and θ1=θ2=1/2we haveHere C is a constant depending only on c0,m,d,T,upper bounds of derivatives of φ,f and u,where u is the classical solution of PDE(2.3). Chapter3:In this chapter,we will consider the following decoupled FBSDEs and propose θ-scheme3.1for solving decoupled FBSDEs(8).Scheme3.1Let(Yn,Zn)(n=N-1,N-2,…,0)be the approximations of {(Ystn,xn,Zstn,xn), tn≤s≤T}at time s=tn. Given YN=YT. For n=N-1,N-2,...,1,0,(Xn+1,yn,Zn)can be solved by the following equations where θ1,θ2∈[0,1],θ3∈(0,1].In the two cases of f=f(t,x,y)and f=f(t,x,y,z),we give the error estimates for θ-scheme3.1in Theorem3.2and Theorem3.4,respectively.Theorem3.2Let YN=φ(XN).Suppose that(Xttn,Xn,Yttn,Xn,Zttn,Xn)and(Xn,Yn,Zn)(n=N,N-1,...,0)are the solutions of FBSDEs(8)(f=f(t,x,y))and θ-scheme3.1, respectively θ3=1we have where C is a constant that depends only on c0,T,upper bounds of derivatives of b,σ, f,φ and u,where u is the classical solution of PDE(3.3).Theorem3.4Suppose that(Xttn,Xn,Yttn,Zttn,Xn)and(Xn,Yn,Zn)are the solu-tions of FBSDEs(8)(f=f(t,x,y,z))and θ-scheme3.1(θ1,θ2∈[0,1],θ3∈(1/2,1]), respectively where C is a constant that depends only on c0, T, upper bounds of derivatives of b, a, f, φ and u, where u is the classical solution of PDE (3.3).Chapter4:In this chapter we consider the following hyperbolic partial differential equations (PDEs) Based on the vanishing viscosity method and the Feynman-Kac formula, we will solve hyperbolic PDEs (9) numerically through a new numerical scheme proposed for the following weakly coupled FBSDEs Here, for (t, x, c)∈[0, T]×Rn×R, a and g are defined by a(t, x, c):=-A(T-t, x, c) and g(t, x, c):=G(T-t, x, c), respectively. We use the solution YT-tT-t,x,ε of FBSDEs (10) to approximate the solution U(t, x) of hyperbolic PDEs (9) when ε is small enough.For weakly coupled FBSDEs (10), we propose the following semi-discrete and fully discrete schemes. To demonstrate the effectiveness and accuracy of our new numerical method, we use the fully discrete scheme4.2to solve various initial-value problems for hyperbolic PDEs, such as linear advection equations, Burgers’ equation and the Buckley-Leverett equation. The numerical results show that Scheme4.2is a high-order scheme and is not subject to the CFL condition, that is to say, it can be used to solve hyperbolic PDEs (9) when CFL>1.Scheme4.1Let yn (n=N-1, N-2,...,0) be the approximations of{Ytntn,x,ε, tn≤s≤T} at time s=tn. Given yN=U0(x). For n=N-1, N-2,...,1,0,(yn, xn+1) can be solved by the following equationsScheme4.1is just a semi-discrete scheme for weak coupled FBSDEs (10). To solve (yn,xn+1) numerically, two more numerical approximations must be done, the approx-imations of the conditional mathematical expectation Etn [·] and the values of function yn+1at non-grid points. In order to reduce the time spent in computing Etn[·], we shall use Gauss-Hermite quadrature formula to approximate Etn[·] and propose a fully discrete numerical scheme for one dimensional FBSDEs (10).Scheme4.2Given yiN=U0(xi), i∈Z. For n=N-1,N-2,...,1,0,yin(i∈Z) can be solved by the following equations where Etn[yn+1] and Etn[gn+1] are defined by Here ηj,ωj(j=1,2,..., K) are defined by (2.39) and (2.40), Ihyn+1(xjn+1) is the in-terpolation of grid function{yin+1,i∈Z} at the spaced point xjn+1by using a finite number of spaced grid points near the spaced point xjn+1.The unknowns in (11) are xjn+1(j=1,...,K) and yin.Chapter5:As the homotopy analysis method (HAM) is a general analytic ap-proach to get series solutions for nonlinear problems and has a convenient way of con-trolling the convergence region of series solutions, we will apply HAM to nonlinear BSDEs and obtain approximate analytical solutions for nonlinear BSDEsFor nonlinear BSDEs (12), we first introduce embedded variable p∈[0,1] and auxiliary parameter h≠0, and then construct a family of backward stochastic differential equations It is obvious that, when p=0,(13) becomes a linear BSDE and its solution yt0can be obtained by Feynman-Kac formula; when p=1,(13) becomes a nonlinear BSDEBy the existence and uniqueness of solutions for BSDEs,(yt1, zt1) is just the solution of nonlinear BSDE (12).Supposed that the solutions (ytp,ztp) of BSDEs (13) can be written as Taylor series and the above series converge at p=1. If f=f(t, y) has continuous derivatives up to order l with respect to the variable y, then we have where k=1μ=yt0+θΣ∞(kl)/(ytk)pk,∪∈(0,1).Substituting these series of ytp, ztp, f(t,ytp) into BSDEs (13), we combine all like terms of p and let the coefficients be zero, then we will get a series of linear BSDEs whose solutions can be used to construct an approximate analytical solutions for nonlinear BSDE (12).Chapter6:In this chapter we consider the risk measure induced by gr-solution of a constrained backward stochastic differential equation (CBSDE) and uses the inf-convolution of the convex risk measures to deal with some optimization problems involv-ing the transformation of the initial risk measures. The idea comes from hedging and super-pricing contingent claims in incomplete markets. In general case, when there are constraints and ambiguity in the market, wealth processes obtained by solving BSDEs are g-super-martingales, thus in the constrained case, we only consider g-supersolutions of BSDEs. Definition6.1(g-supersolution, cf.[100]) We say a triple (y, z, C)∈DFt2(0,T;R)×LFt2(0, T; Rd)×CFt2(0,T; R) is a g-supersolution of CBSDE with terminal value ζ if hold.For any given ζ∈LFT2(R), we denote HΦ(ζ) as the set of all triples (y, z, C) satisfying Constrained condition (C) and BSDEs (14). Note that for a given ζ∈LFT2(R), HΦ(ζ) maybe empty or contain more than one elements. If it is not empty, for the purpose of analysis, we are mainly interested in the smallest one which is also called as gr-solution. Definition6.2(gr-solution, cf.[100,101]) A g-supersolution (yt, zt, Ct) is called as gΓ-solution with Constrained condition (C) and terminal value ζ, if for any g-supersolution (yt’, zt’, Ct’) with Constrained condition (C) and terminal value ζ, yt≤yt’, a.e., a.s. holds.gΓ-solution with Constrained condition (C) and terminal value ζ is denoted by εt,Tg,Φ(ζ) and simply denoted by εtg,Φ(ζ) when g(t, y,0)=0and Φ(t, y,0)=0a.s. hold for any t∈[0,T].As the origin idea of CBSDEs comes from hedging and pricing contingent claims in incomplete market or constrained market generally, we will study the risk measure in-duced by gΓ-solution of a CBSDE. As an application, we investigate optimal risk transfer problem by the inf-convolution of such risk measures. The optimal risk transfer problem was considered in Barrieu and El Karoui [11] where constraints were characterized by closed convex sets. Instead of risk measures generated by closed convex sets, we consider risk measures induced by CBSDEs directly.For two agents with their own risk measure pj(ζ)=ε0gi,Φi(-ζ) generated by CBSDE with gi,Φi,i=1,2respectively, we consider the following optimization problemTo make the problem above be well defined, we should assure that gΓ-solutions of CBSDEs (14) exist for all ζ∈L∞(FT). This can be done by a mild assumption that h=g,Φ, h(t, y,0)=0,(?)t∈[0, T], y∈R hold for h=g,Φ. The optimization problem (15) is a model for risk transfer between two agents, for more arguments see Barrieu and El Karoui [11]. Based on the analysis of the structure of CBSDEs, we got the following results under some suitable assumptions. Theorem6.1If both g and Φ satisfy assumptions(Ai),i=1,2,3and hold for h=g,Φ,then ζ=0is a optimal value for problem(15)when gi=g,Φi=Φ,i=1,2.The result above tells us that if the two agents have same risk measure induced by CBSDEs with same coefficients,then there is no need for them to do any trading activities to transfer risk.Our next results need a new notation for convex functions.For any λ＞0,we define the dilatation of a convex function f(.)as fλ(·)=λf(·/λ).Theorem6.2Suppose g and Φ satisfy the assumptions(Ai),i=1,2,3,Φ(λz)=λΦ(z) holds for any λ＞0.Let ρ(ζ)=ε0g,Φ(-ζ),9λ(z)=λ9(z/λ),then we haveThe above result says that,under some mild assumptions,the dilatation of gΓ-solution with the parameter g coincides with gΓ-solution with the dilatation of g as the parameter.Theorem6.3Suppose g and Φ satisfy the assumptions(Ai),i=1,2,3,two agents have risk measures with different risk tolerance coefficient gλ and gγ respectively, then one optimal value of problem(15)isAt last,we state a dynamic version of inf-convolution of gΓ-solutions.Theorem6.5Suppose gi,i=1,2,Φare convex functions satisfying the assumptions (Ai),i=1,2,3,Φ(t,z1+z2)≤Φ(t,z1)+Φ(t,Z1),(?)z1,z2and there exist some a,b∈R such that gi(t,z)≥az+b,i=1,2.The inf-convolution of g1and g2is given by Let(εtg3,Φ(η),z3(t),C3(t))be the gΓ-solution with terminal value ζ∈L∞(FT)satisfying constraint(C)and z be a measurable process such that z=arg miny{91(t,z3(t)-y)+g2(t,y)}dt×dP-a.s.,then the following results hold:(1)For any t∈[0,T] and any ζ∈L∞(FT), (2) If Φ(t, z(t))=0,Φ(t, z3(t)-z(t))=0and then ζ*is an optimal value for problem (15), furthermore, we haveChapter7:The problem of ruin probability of gamblers is a classic ruin problem, and this probability is directly related with the vital interests of the gamblers. In this chapter we first studied the profit and risk of ruin for some gambling strategies, such as the doubling bets strategy, Fibonacci betting strategy and the exposure controlling strategy. Then the applications of these strategies in the money management systems are also analyzed and compared.Assumes that the player has1million dollars, and the minimum bet is100dollars. The probability of winning in each gamble is p. If there is no limit for the maximum bet and the player is allowed to be on credit in the game until he ruin. Then how much the risk of ruin is when the player gamble N times using the doubling bets strategy? Here we think that the player ruins when his loss is not less than1million dollars.Due to213=8192,214=16384, when N≤13, the player will not ruin at all; When14≤N≤6397, the player will ruin only if he loses14consecutive times. As a result, We can divide ruin events into independent small events and calculate each probability. For large enough N, if random variable X denotes the gamble times of the player have attend until game over, then we haveWhen N≤73, we can give easily the formula of the ruin probability when the player gamble N times using the doubling bets strategy by summing the probability of those independent small events. But as the number of gambling N increases, the calculation of ruin probability becomes increasingly complex and it is difficult to give the explicit expression for the risk of ruin.