Large deviations for multidimensional SDEs with reflection.pdf

a r X i v : 0 7 0 5 .0 4 0 5 v 1 [ m a t h .P R ] 3 M a y 2 0 0 7 Large deviations for multidimensional SDEs with reflection ? Zongxia Liang ? Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China Abstract The large deviations principles are established for a class of multidi- mensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and anticipated. MSC(2000): Primary 60F10, 60H10 ; Secondary 60J50, 60J60. Keywords: Large deviation principle; Stochastic differential equa- tions with reflecting boundary; Generalized contraction principle; An- ticipated stochastic integrals and initial conditions. 1 Introduction and main results Let O be a smooth bounded open set in ?d. n(x) denotes the cone of unit outward normal vectors to ?O at x, that is, (i) ? C0 ≥ 0,?x ∈ ?O,?x′ ∈ O?,? k ∈ n(x) =? (x? x′, k) + C0|x? x′|2 ≥ 0, (1.1) (ii) ?x ∈ ?O, if ?C ≥ 0,? k ∈ ?d,?x′ ∈ O?, (x? x′, k) +C|x? x′|2 ≥ 0,=? k = θn(x) (1.2) for some θ ≥ 0. Moreover, we assume that (iii) ?n ≥ 1,?α 0,?R 0;?a1, · · · , an ∈ ?d, |ai| = 1,?i, ?x1, · · · , xn ∈ ?O : ?O ? ∪ni=1B(xi, R), ?i,?x ∈ ?O ∩B(xi, 2R),?ξ ∈ n(x), (ξ, ai) ≥ α 0, (1.3) where ?O denotes the boundary of O, O? denotes the closure of O, B(x, r) denotes the ball of radius r at x. We assume that Bt is an ?d- valued Ft- Brownian motion on a stochastic basis (?,F , {Ft}t∈[0,1],P) satisfying the ?This work is supported by NSFC and SRF for ROCS, SEM ?email: zliang@math.tsinghua.edu.cn 1 usual assumptions. For ε 0 we consider the following perturbed stochastic differential equations on domain O with reflecting boundary conditions,????? Xεt (x) = x+ ∫ t 0 b(X ε s (x))ds + √ ε ∫ t 0 σ(X ε s (x))dBs ? Lεt (x), Lεt (x) = ∫ t 0 ξ(X ε s (x))d|Lε(x)|s, |Lε(x)|t = ∫ t 0 I{s:Xεs (x)∈?O}d|Lε(x)|s (1.4) for ? t ∈ [0, 1] and x ∈ O?, where b : ?d 7→ ?d and σ : ?d 7→ ?d × ?d are continuous functions,