Background of the problem BSDEs with Markov Chains BSDEs with Singularly Perturbed Markov Chains Application: Homogenization of Systems of PDEs BSDEs with Markov Chains and The Application: Homogenization of Systems of PDEs Zhen WU School of Mathematics, Shandong University, Jinan 250100, China. Email: wuzhen@sdu.edu.cn joint with Dr. Huaibin TANG INRIA-IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France Brest, March, 2010 1 / 52
Background of the problem BSDEs with Markov Chains BSDEs with Singularly Perturbed Markov Chains Application: Homogenization of Systems of PDEs Main results in this talk : we consider one kind of backward stochastic differential equations (BSDEs in short) where the coefficient f is affected by a Markovian switching. 1. Theoretical result: 1) We obtain the existence and uniqueness results for the solution to this kind of BSDEs. 2) We get the weak convergence result of BSDEs with a singular-perturbed Markov chain which is involved in a large state space. 2. Application: homogenization of one system of PDE. 2 / 52
Background of the problem BSDEs with Markov Chains BSDEs with Singularly Perturbed Markov Chains Application: Homogenization of Systems of PDEs Contents 1 Background of the problem 2 BSDEs with Markov Chains Motivation BSDEs with Markov Chains 3 BSDEs with Singularly Perturbed Markov Chains Preliminary BSDEs with singularly perturbed Markov chains 4 Application: Homogenization of Systems of PDEs 3 / 52
Background of the problem BSDEs with Markov Chains BSDEs with Singularly Perturbed Markov Chains Application: Homogenization of Systems of PDEs Background 1. BSDEs: Bismut (1973) : stemmed BSDEs from stochastic control problem. Pardoux & Peng (1990, 1991) : general BSDEs driven by Brownian motion and probabilistic representation of PDE. Tang & Li (1994), Barles, Buchdahn & Pardoux (1997) : BSDEs with respect to both a Brownian motion and a Poisson random measure. Nualart & Schoutens (2001), Bahlali, Eddahbi & Essaky (2003) : BSDEs driven by a L´ evy process. 4 / 52
Background of the problem BSDEs with Markov Chains BSDEs with Singularly Perturbed Markov Chains Application: Homogenization of Systems of PDEs 2. Singularly perturbed Markov chains: Zhang and Yin (1998) : consider the Markov chain involved in a large state space, they introduced a small parameter ( ε > 0 ) to highlight the contrast between the fast and slow transitions among different Markovian states and lead the Markov chain to a singularly perturbed one with two time-scale: the actual time t and the stretched time t ε . As ε → 0 , the asymptotic probability distribution of such Markov chain can be studied. Zhang and Yin (1999, 2004) : Applications in optimal control problem and mathematical finance: finding the near-optimal control of random-switching LQ optimal control problem, and nearly optimal asset allocation in hybrid stock investment models. 5 / 52
Background of the problem BSDEs with Markov Chains BSDEs with Singularly Perturbed Markov Chains Application: Homogenization of Systems of PDEs Our question : BSDEs with singular-perturbed Markov chains and their application. 6 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs BSDEs with Markov Chains (Ω , F , P ) , T > 0 , {H t , 0 ≤ t ≤ T } . H t (0 , T ; R n ) : ϕ = { ϕ t ; t ∈ [0 , T ] } satisfying M 2 � T 0 | ϕ t | 2 dt < ∞ ; E H t (0 , T ; R n ) : ϕ = { ϕ t ; t ∈ [0 , T ] } satisfying S 2 E (sup 0 ≤ t ≤ T | ϕ t | 2 ) < ∞ . B : B 0 = 0 , d -dimensional H t -Brownian motion. α : continuous-time Markov chain independent of B with the state space M = { 1 , 2 , . . . , m } . Let N denote the class of all P -null sets of F . Denote F t = F B t ∨ F α t,T ∨ N . Denote M 2 (0 , T ; R n ) = M 2 F t (0 , T ; R n ) and S 2 (0 , T ; R n ) = S 2 F t (0 , T ; R n ) . Remark : {F t ; 0 ≤ t ≤ T } is neither increasing nor decreasing, and it does not constitute a filtration . 7 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs Suppose the generator of the Markov chain Q = ( q ij ) m × m is given by � q ij △ + o ( △ ) , if i � = j, P { α ( t + △ ) = j | α ( t ) = i } = 1 + q ij △ + o ( △ ) , if i = j, where △ > 0 . Here q ij ≥ 0 is the transition rate from i to j if i � = j , while q ii = − � m j =1 ,i � = j q ij . The generator Q is called weakly irreducible if the system of equations νQ = 0 and � m i =1 ν i = 1 has a unique nonnegative solution. This nonnegative solution ν = ( ν 1 , · · · , ν m ) is called the quasi-stationary distribution of Q . 8 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs Contents 1 Background of the problem 2 BSDEs with Markov Chains Motivation BSDEs with Markov Chains 3 BSDEs with Singularly Perturbed Markov Chains Preliminary BSDEs with singularly perturbed Markov chains 4 Application: Homogenization of Systems of PDEs 9 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs Motivation Some references can be seen Zhang and Yin(1999) , Li and Zhou(2002) and Zhou and Yin(2003) Consider the following stochastic LQ control problem with Markov jumps �� T min J ( v ( · )) = 1 � [ x ′ t R ( t, α t ) x t + v ′ t N ( t, α t ) v t ] dt + x ′ 2 E T Q ( α T ) x T 0 � dx t = [ A ( t, α t ) x t + B ( t, α t ) v t ] dt + [ C ( t, α t ) x t + D ( t, α t ) v t ] dB t , s.t. x 0 = x ∈ R n , H t (0 , T ; R n u × d ) . Admissible controls set: U ad ≡ M 2 Our aim is to find an admissible control u ( · ) such that J ( u ( · )) = inf v ∈U ad J ( v ( · )) . 10 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs Here, we use FBSDE to derive its optimal control: Theorem 2.1 For any admissible control v ( · ) , if the following FBSDE admits a unique solution ( x v t , y v t , z v t ) dx v t = [ A ( t, α t ) x v t + B ( t, α t ) v t ] dt + [ C ( t, α t ) x v t + D ( t, α t ) v t ] dB t , t = [ A ′ ( t, α t ) y v t + C ′ ( t, α t ) z v − dy v t + R ( t, α t ) x v t ] dt − z v t dB t , x v y v T = Q ( α T ) x v 0 = x, T Then there exists a unique optimal control for the above LQ problem u ( t ) = − N − 1 ( t, α t )[ B ′ (( t, α t )) y t + D ′ ( t, α t ) z t ] . Here ( x t , y t , z t ) is the solution of FBSDE with respect to the control u ( · ) . 11 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs Contents 1 Background of the problem 2 BSDEs with Markov Chains Motivation BSDEs with Markov Chains 3 BSDEs with Singularly Perturbed Markov Chains Preliminary BSDEs with singularly perturbed Markov chains 4 Application: Homogenization of Systems of PDEs 12 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs BSDEs with Markov Chains Firstly, we will study the following BSDEs with Markov chains � T � T Y t = ξ + f ( s, Y s , Z s , α s ) ds − Z s dB s . (1) t t Assumption 2.1 (i) ξ ∈ L 2 ( F T ; R k ) ; (ii) f : Ω × [0 , T ] × R k × R k × d × M → R k satisfies that ∀ ( y, z ) ∈ R k × R k × d , ∀ i ∈ M , f ( · , y, z, i ) ∈ M 2 t (0 , T ; R k ) , and F B f ( t, y, z, i ) is uniformly Lipschitz continuous with respect to y and z , i.e., ∃ µ > 0 , such that ∀ ( ω, t ) ∈ Ω × [0 , T ] , ( y 1 , z 1 ) , ( y 2 , z 2 ) ∈ R k × R k × d , | f ( t, y 1 , z 1 , i ) − f ( t, y 2 , z 2 , i ) | ≤ µ ( | y 1 − y 2 | + | z 1 − z 2 | ) . 13 / 52
Background of the problem BSDEs with Markov Chains Motivation BSDEs with Singularly Perturbed Markov Chains BSDEs with Markov Chains Application: Homogenization of Systems of PDEs Theorem 2.2 Under Assumption 2.1, there is a unique solution pair ( Y, Z ) ∈ S 2 (0 , T ; R k ) × M 2 (0 , T ; R k × d ) for BSDE (1) . The following extension of Itˆ o’s martingale representation theorem and its corollary play key role during the proof of this theorem. Define a filtration ( G t ) 0 ≤ t ≤ T by G t = F B t ∨ F α Proposition 2.1 T where α and B are independent with each other. For M ∈ L 2 ( G T ; R k ) , there exists a unique random variable M 0 ∈ L 2 ( F α T ; R k ) and a unique stochastic process G t (0 , T ; R k × d ) such that Z = { Z t ; 0 ≤ t ≤ T } ∈ M 2 � T M = M 0 + 0 ≤ t ≤ T. Z t dB t , 0 Actually, M 0 = E [ M |F α T ] . 14 / 52
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