Solution Of Reflected Backward Stochastic Differential Equation And Related Partial Differential Equation

An equation in the form ofis so-called Backward Stochastic Differential Equation (BSDE for short).Linear BSDE was first introduced by Bismut [7] in 1973 when he study the maximum principle in stochastic optimal control. General nonlinear BSDE was first introduced by Pardoux and Peng [56] in 1990. They showed that there is a unique adapted solution when the coefficient was Lipshctz continuous. Independently, Duffie and Epstein [22] introduced stochastic differential utilities in economics, as solutions to a certain type of BSDEs. After that time, this kind of equations has received considerable research attention due to their nice structure and wide applicability in numbers of different areas, such as mathematical finance, stochastic control, economical management and etc..In the year 1997, El Karoui, Kapoudjian, Pardoux, Peng and Quenez [24] generalized the equations to reflected case, that is, the solution is forced to stay above a given stochastic process, which is called the obstacle. They introduced an increasing process to push the solution upward. Furthermore, the push is minimal. The equation is in the form ofBesides the existence and uniqueness of the solution, we can also learn from the paper that, formulated with a Markovian framework, the solution of reflected BSDE can be represented as a uniqueness viscosity solution of an obstacle problem for a partial differential equation. In 1997 Matoussi [51] studied the case when f is of linear growth in y and z. When the terminal valueξis square integrable he proved the existence of a maximal and a minimal solution. Later reflected BSDE's, whose coefficients are quadratic growth in z, have been studied by Kobylanski, Lepeltier, Quenez, Torres in [42], but they required the terminal valueξis bounded.As we have known, BSDE can be widely used in finance. In a complete market, it is possible to construct a portfolio to copy an expected profit which gainsξat time T. The value of the portfolio is given by a BSDE, and Z corresponds to the strategy. Sometimes, we have to restrict our wealth in a given field, which can be described by a reflected BSDE. One of another application is to give the probabilistic representation of PDEs and generalized the classical Feynman-Kac formula to nonlinear case.This thesis is focus on the reflected BSDE whose coefficient is of quadratic growth in z and of linear growth in y. Moreover, the terminal value is unbounded. We also give an existence result on fully coupled forward-backward SDEs. In the following, we list the main results of this thesis.Chapter 1: We introduce problems studied from Chapter 2 to Chapter 5.Chapter 2: We study the properties of reflected BSDE whose terminal value is unbounded and the coefficient is of quadratic growth in z and of linear growth in y. Theorem 2.1.2. (Existence) Under the assumptions 2.1.1-2.1.3, the reflected BSDE associated to (ξ, f, L) admits at least a solution, i.e. there exists a triplet (Y_t, Z_t, K_t)_{0≤t≤T}, with Y∈S_n²(0, T), and K∈S_ci²(0, T), such thatMoreover if assumption 2.1.4 holds, then Z∈H_n²(0, T).Furthermore, if f is convex in z, the solution is unique.Theorem 2.1.3. (Uniqueness) Let the assumption 2.1.1-2.1.5 hold, then reflected BSDE(ξ, f, L) has a unique solution (Y, Z, K)∈S_n²(0, T)×H_n²(0, T)×S_ci²(0, T).In the second part of this chapter, we give the stability property of the solution. Theorem 2.6.3. (stability) Ifξⁿ→ξP-a.s. and for each (y, z)∈R×R^d, fⁿ(t, y, z)→f(t, y, z), then, for each p≥1, we haveChapter 3: To prove by contradiction, we first show in this chapter that the function defined by a reflected BSDE is a viscosity solution of an obstacle problem for a partial differential equation.Theorem 3.2.1. (Existence) Define u(t,x) := Y_t^t,x, then it is a viscosity solution of the following PDEUnder the same assumptions as that in Kobylanski [41], the viscosity is unique.Theorem 3.3.1. (Comparison Theorem) Under assumptions 3.3.1, there is a comparison result for the viscosity solutions of (34). More precisely, if u is a bounder continuous viscosity subsolution of (3.4) and v a bounded continuous viscosity supersolution of (3.4), such thatu(T,x)≤v(T,x) in Rⁿ,thenu≤v on [0, T)×Rⁿ.Theorem 3.3.4. (Uniqueness) Under the assumption 3.1.1-3.1.5 and 3.3.1, the PDE (3.4) has at most one viscosity solution in the class of continuous bounded functions .Chapter 4: We study the following fully coupled Forward-Backward Stochastic Differential Equation systemOne distinct character is that the forward coefficient b contains the backward solution variable Y. Constructing a sequence of solutions we can prove thatTheorem 4.1.1. Assume that the coefficients in the equation satisfy: (a) b is increasing in y and f is increasing in x;(b) there exists a constant M > 0, such that|b(s,x,y)|≤M(1 + |x| + |y|), |f(t,x,y,z)|≤M(1 + |y| + |z|);(c) |σ(s,x)|≤M(1 + |x|), |σ(s,x)-σ(s,x')|≤M|x - x'|for all s∈[0, T],w∈Ωand x, x', y,z∈R.Then there exists at least one solution (X, Y, Z, K)∈S² (?) S² (?) H² (?) S_ci² of the abovesystem.Chapter 5: In this chapter, we study one kind of corporate international optimal portfolio and consumption choice problem for the investor who can invest his wealth either in a domestic bond (bank account) or in a oversea real project. The bank pays a lower interest rate for deposit and takes a higher rate for any loan. On the other hand, the investor can invest his money to a real project with production in a foreign country. Let W(t) denote the total wealth at time t, n(t) the amount invest in the foreign market, then the total wealth W satisfies:dW(t) = [r(t)W(t) - C(t) + (g(t) - r(t))Ï€(t) - (r'(t) - r(t))(W(t) -Ï€(t))^-]dt +σ(t)Ï€(t)(1 +β)dB_t +Ï€(t)σ_e(t)d(?)-t.Suppose the investor want to maximize the expected utilitywhere l, h are strictly increasing, concave and differentiable with respect to C, W.We show that the celebrated dynamic programming principle still holds for this kind of optimal control problem.Theorem 5.2.1. For any (W,s)∈R×[0,T),And then using this famous principle method we provide the general solution of the optimal proportion for the investor and give the economic analysis. For the specific Hyperbolic Absolute Risk Aversion (HARA) case, here L and K are constants, R∈(0,1)andγ> 0 is the discount factor. We get the explicit optimal investment and consumption solution using a conjecturing method which has been used to solved the classical linear quadratic (LQ) optimal control problem.Proposition 5.3.1. Denote thatâ–³(s) = (1 +β)²Ïƒ²(s) +σ_e²(s) + 2ρ(1 +β)σ(s)σ_e(s). Under all the above assumptions, the optimal strategies to the optimal portfolio and consumption choice problem (5.9)-(5.10) for the specific HARA case is given byandFurther more, the value function is given bywhereΦ₁(s),Φ₂(s),Φ₃(s) satisfy (5.18) (5.20) (5.21) respectively.At the end of this chapter, we give some simulation results to illustrate the influence of the volatility parameters on the optimal investment strategy.