Numerical approximations of one-point large deviations rate functions of stochastic differential equations with small noise
aa r X i v : . [ m a t h . P R ] F e b NUMERICAL APPROXIMATIONS OF ONE-POINT LARGE DEVIATIONSRATE FUNCTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITHSMALL NOISE
JIALIN HONG, DIANCONG JIN, AND DERUI SHENG
Abstract.
In this paper, we study the numerical approximation of the one-point large deviationsrate functions of nonlinear stochastic differential equations (SDEs) with small noise. We showthat the stochastic θ -method satisfies the one-point large deviations principle with a discrete ratefunction for sufficiently small step-size, and present a uniform error estimate between the discreterate function and the continuous one on bounded sets in terms of step-size. It is proved that theconvergence orders in the cases of multiplicative noises and additive noises are 1 / AMS subject classifications : 60F10, 60H35, 65C301.
Introduction
The large deviations principle (LDP) provides exponential-order estimates for probabilities ofrare events occurring in the stochastic systems, where the decay speed of probabilities is character-ized by the large deviations rate functions. Since the probabilities of rare events that observablesdeviate from their typical values are exponentially small in large deviations estimates, the directsimulation based on Monte–Carlo, which requires tremendous samples, fails to provide an effectiveapproximation of rate functions in practice ([3]). To solve this problem, the popular algorithmsare based on two basic methods used in the simulation of rare events: splitting and importancesampling (see, e.g., [1, 4, 5, 6, 11, 15] and references therein). These methods usually deal with theadditive path functionals of stochastic processes in the long-time limit, for example the observable T R T f ( M t )d t provided that { M t } t> is a regular stochastic process and f is a smooth function([14]). Most algorithms in the literature mentioned above are devoted to simulating the long-timelarge deviations rate functions of observables. The numerical investigation of the one-point largedeviations rate functions of SDEs with small noise is few.In this work, we focus on the numerical approximations of the one-point large deviations ratefunctions of general SDEs with small noise. To be more precise, we consider the following Itˆo SDE ( d X ǫ ( t ) = b ( X ǫ ( t ))d t + √ ǫσ ( X ǫ ( t ))d W ( t ) , t ∈ (0 , T ] ,X ǫ (0) = x ∈ R d , (1.1)where the initial value x is non-random, and the positive constant ǫ denotes the intensity of thenoise and is sufficiently small. { W ( t ) , t ∈ [0 , T ] } is a standard m -dimensional Brownian motiondefined on a complete filtered probability space (Ω , F , { F t } t ∈ [0 ,T ] , P ), with { F t } t ∈ [0 ,T ] satisfying Key words and phrases. large deviations principle, rate function, convergence analysis, stochastic numericalmethods.This work is supported by National Natural Science Foundation of China (Nos. 11971470, 11871068, 12031020,12022118, 12026428 and 11926417). the usual conditions. b : R d → R d and σ : R d → R d × m are such that (1.1) admits a unique strongsolution. It can be shown that { X ǫ ( T ) } ǫ> satisfies an LDP with a rate function I as ǫ → I is implicitly determined by the minimum of a functional.Hence, one usually needs to resort to the numerical simulation in order to approximately estimate I . In order to do this, our idea is based on the direct time discretization for (1.1). In detail,for a numerical method { X ǫn } Nn =0 with X ǫn being the approximation of X ǫ ( t n ), where t n = nh , n = 0 , , . . . , N and h = TN is the step-size, we use the rate function I h of { X ǫN } ǫ> to approximate I . In this process, two main problems are:(P1) Is there any numerical method { X ǫn } Nn =0 such that { X ǫN } ǫ> satisfies the small noise LDPand its rate function I h can approximate well I ?(P2) If so, how to give the rigorous error estimate between I h and I in terms of h ?Concerning (P1), we show in [8] that stochastic symplectic methods can asymptotically preservethe LDPs of observables of the test equation, while stochastic nonsymplectic ones can not. In[16], we find that common numerical methods asymptotically preserve the LDP of the invariantmeasure of the linear Langevin equation in the small noise limit, while only specific method canasymptotically preserve the LDP of the invariant measure of the original equation in the strongdissipation limit. These results indicate that a numerical method may not provide an effectiveapproximation for the rate function of the underlying SDE, despite the fact that this numericalsolution converges to the exact solution in the strong or weak sense. As for (P2), in reference to thenumerical study for one-point large deviations rate functions of SDEs with small noise, we are onlyaware of the work [7], where the author gives an error estimate between the discrete rate functionbased on the midpoint scheme and the original rate function of linear stochastic Maxwell equationswith additive noises. However, for general nonlinear SDEs with small noise in the form of (1.1), asfar as we know, there is no any result about the convergence rate of discrete rate functions basedon the time discretizations.Our contributions lie in providing a numerical approximation of the rate function I of { X ǫ ( T ) } ǫ> and giving the convergence order of the discrete rate function I h . First, we apply the stochastic θ -method to discrete (1.1) and obtain the corresponding numerical solution { X ǫN } Nn =0 (see (2.7)).Then for sufficiently small step-size, we prove that the LDP holds for { X ǫN } ǫ> with the good ratefunction I h , by means of the sample path large deviations of Brownian motion and the contractionprinciple. Concerning the convergence analysis, the main difficulty lies in the error estimate betweenthe minimums of two functionals occurring in the expressions of I and I h . Based on the uniform errorbound between the two functionals on any given bounded set, we finally prove that the uniformconvergence orders of I h in the cases of multiplicative noises and additive noises are 1 / { X ǫ ( T ) } ǫ> forthe nonlinear SDE (1.1) with small noise.The rest of this paper is organized as follows. Section 2 gives the LDPs of both { X ǫ ( T ) } ǫ> of theSDE (1.1) and { X ǫN } ǫ> based on stochastic θ -method. Section 3 proves several useful lemmas forestimating the error between I h and I . We prove in Section 4 that the uniform convergence ordersof I h on a given bounded set are 1 / DP 3 One-point large deviations principles
In this section, we give the one-point LDP for both the exact solution of (1.1) and numer-ical solution based on the stochastic θ -method. We begin with some notations. Throughoutthis paper, let a ∧ b denote the minimum of a and b for any a, b ∈ R . Let N + be the set ofall positive integers. Denote by | · | the 2-norm of a vector or matrix. For T ∈ (0 , + ∞ ) and d ∈ N + , denote by C (cid:0) [0 , T ] , R d (cid:1) the space of all continuous functions f : [0 , T ] → R d , equippedwith the supremum norm k f k = sup t ∈ [0 ,T ] | f ( t ) | . And for given x ∈ R d , denote C x (cid:0) [0 , T ] , R d (cid:1) := (cid:8) f ∈ C (cid:0) [0 , T ] , R d (cid:1) : f (0) = x (cid:9) . Let L (0 , T ; R d ) stand for the space of all square integrable func-tions with the inner product h f, g i L = R T h f ( t ) , g ( t ) i d t and the norm k f k L := (cid:16)R T | f ( t ) | d t (cid:17) / for any f, g ∈ L (0 , T ; R d ), where h· , ·i denotes the inner product on R d . Denote H (0 , T ; R d ) := (cid:8) f : [0 , T ] → R d (cid:12)(cid:12) f is absolutely continuous and f ′ ∈ L (0 , T ; R d ) (cid:9) and endow this space with thesemi-norm k f k H := k f ′ k L . Also for given x ∈ R d , denote H x (0 , T ; R d ) := (cid:8) f ∈ H (0 , T ; R d ) : f (0) = x (cid:9) .For a metric space ( E, ρ ), x ∈ E and r >
0, denote by ¯ B ( x, r ) := { y ∈ E : ρ ( x, y ) ≤ r } the closedball centered at x and with the radius r . The infimum of a function over an empty set is interpretedas + ∞ .In this section let X be a Polish space , i.e., complete and separable metric space. The followingare the concepts of the rate function and LDP (see, e.g., [9, 12]).
Definition 2.1.
A real-valued function I : X → [0 , ∞ ] is called a rate function if it is lowersemicontinuous, i.e., for each a ∈ [0 , ∞ ) , the level set I − ([0 , a ]) is a closed subset of X . If all levelsets I − ([0 , a ]) , a ∈ [0 , ∞ ) , are compact, then I is called a good rate function. Definition 2.2.
Let I be a rate function and { µ ǫ } ǫ> be a family of probability measures on X . Wesay that { µ ǫ } ǫ> satisfies an LDP on X with the rate function I if (LDP1) lim inf ǫ → ǫ ln( µ ǫ ( U )) ≥ − inf I ( U ) for every open U ⊆ X , (LDP2) lim sup ǫ → ǫ ln( µ ǫ ( C )) ≤ − inf I ( C ) for every closed C ⊆ X . Analogously, a family of random variables { Z ǫ } ǫ> valued on X is said to satisfy an LDP with therate function I , if its distribution satisfies (LDP1) and (LDP2) in Definition 2.2. The followingresult, called the contraction principle , is useful to derive a new LDP, through a continuous map,based on the known one. Proposition 2.3. [12, Theorem 4.2.1]
Let Y be another Polish space, f : X → Y be a continuousfunction, and I : X → [0 , ∞ ] be a good rate function. (a) For each y ∈ Y , define ˜ I ( y ) , inf { I ( x ) : x ∈ X , y = f ( x ) } . Then ˜ I ( y ) is a good rate function on Y . (b) If I controls the LDP associated with a family of probability measures { µ ǫ } ǫ> on X , then ˜ I ( y ) controls the LDP associated with the family of probability measures (cid:8) µ ǫ ◦ f − (cid:9) ǫ> on Y . Next we introduce the sample path large deviations for Brownian motion W . Denote W ǫ ( t ) = √ ǫW t , t ∈ [0 , T ]. Then denote by ν ǫ the law of W ǫ on C ([0 , T ] , R m ). The following propositionindicates that { ν ǫ } ǫ> obeys an LDP in C ([0 , T ] , R m ) (see e.g., [12, Theorem 5.2.3]). JIALIN HONG, DIANCONG JIN, AND DERUI SHENG
Proposition 2.4. { ν ǫ } ǫ> satisfies, in C ([0 , T ] , R m ) , an LDP with the good rate function I w ( φ ) = ( R T | φ ′ ( t ) | d t, φ ∈ H (0 , T ; R m ) , + ∞ , otherwise . In this paper, we are interested in the LDP of { X ǫ ( T ) } ǫ> and its discrete versions. For this end,we first give the following assumption. Assumption (A1). b and σ are globally Lipschitz continuous, i.e., there is some constant L > such that | b ( x ) − b ( y ) | + | σ ( x ) − σ ( y ) | ≤ L | x − y | ∀ x, y ∈ R d . (2.1)It follows from (2.1) that both b and σ grow at most linearly. For convenience, we assume that | b (0) | + | σ (0) | ≤ L so that | b ( x ) | + | σ ( x ) | ≤ L (1 + | x | ) , x ∈ R d . (2.2)Define the map Γ : L (0 , T ; R m ) → C x ([0 , T ] , R d ), which takes f ∈ L (0 , T ; R m ) to the solutionof the following equation: ϕ ( t ) = x + Z t b ( ϕ ( s ))d s + Z t σ ( ϕ ( s )) f ( s )d s, t ∈ [0 , T ] . Then under Assumption (A1), { X ǫ } ǫ> satisfies an LDP on C x ([0 , T ] , R d ) with the good ratefunction J given by J ( ϕ ) := inf { f ∈ L (0 ,T ; R m ):Γ( f )= ϕ } Z T | f ( t ) | d t ∀ ϕ ∈ C x ([0 , T ] , R d ) . (2.3)Readers can refer to [2] for this result. Define the coordinate map ξ T : C x ([0 , T ] , R d ) → R d by ξ T ( f ) = f ( T ), for each f ∈ C x ([0 , T ] , R d ). Then we have X ǫ ( T ) = ξ T ( X ǫ ). Hence the continuityof the map ξ T and Proposition 2.3 give the following result. Theorem 2.5.
Let Assumption (A1) hold. Then { X ǫ ( T ) } ǫ> satisfies an LDP on R d with the goodrate function I given by I ( x ) = inf { ϕ ∈ C x ([0 ,T ] , R d ): ϕ ( T )= x } J ( ϕ ) ∀ x ∈ R d . Let X be the solution of the following ordinary differential equation X ( t ) = x + Z t b ( X ( s ))d s, t ∈ [0 , T ] . (2.4)It can be verified that X ǫ ( T ) converges to X ( T ) in probability as ǫ →
0, i.e., for any δ > ǫ → P ( | X ǫ ( T ) − X ( T ) | ≥ δ ) = 0. Applying the LDP of { X ǫ ( T ) } ǫ> , we can characterize the decayspeed of the probability P ( | X ǫ ( T ) − X ( T ) | ≥ δ ). More precisely, we will show that P ( | X ǫ ( T ) − X ( T ) | ≥ δ ) decays exponentially as ǫ → Corollary 2.6.
Under Assumptions (A1), we have the following. (1) I ( x ) = 0 if and only if x = X ( T ) . (2) Let δ > be fixed and define C ( δ ) := inf { x ∈ R d : | x − X ( T ) |≥ δ } I ( x ) . Then C ( δ ) > and for any η > , there exists some constant ǫ ( η ) > such that for any ǫ < ǫ ( η ) , P ( | X ǫ ( T ) − X ( T ) | ≥ δ ) < e − ǫ ( C ( δ ) − η ) . (2.5) DP 5
Proof. (1) On one hand, it follows from the definition of I that 0 ≤ I ( X ( T )) ≤ J ( X ). Notingthat Γ( ) = X , by the definition of J , one has J ( X ) ≤
0, which leads to I ( X ( T )) = 0.On the other hand, let x be such that I ( x ) = 0. It follows from the definitions of I and J thatthere exist two sequences of functions { f n } n ≥ ⊆ L (0 , T ; R m ) and { ϕ n } n ≥ ⊆ C x ([0 , T ] , R d ) suchthat for any n = 1 , , . . . , ϕ n (0) = x , ϕ n ( T ) = x , J ( ϕ n ) < n , R T | f n ( t ) | d t < J ( ϕ n ) + n and ϕ n ( t ) = x + Z t b ( ϕ n ( s ))d s + Z t σ ( ϕ n ( s )) f n ( s )d s, t ∈ [0 , T ] . (2.6)Thus, k f n k L < J ( ϕ n ) + n < n , which gives lim n →∞ k f n k L = 0. It follows from (2.2), (2.6) and theH¨older inequality that | ϕ n ( t ) | ≤ | x | + 3 t Z t | b ( ϕ n ( s )) | d s + 3 Z t | σ ( ϕ n ( s )) | d s Z t | f n ( s ) | d s ≤ | x | + 6 L T Z t (1 + | ϕ n ( s ) | )d s + 6 L k f n k L Z t (cid:0) | ϕ n ( s ) | (cid:1) d s ≤ | x | + 6 L T (cid:0) T + k f n k L (cid:1) + 6 L (cid:0) T + k f n k L (cid:1) Z t | ϕ n ( s ) | d s ∀ t ∈ [0 , T ] . By the Gronwall inequality, k ϕ n k ≤ (cid:2) | x | + 6 L T (cid:0) T + k f n k L (cid:1)(cid:3) exp (cid:8) L T (cid:0) T + k f n k L (cid:1)(cid:9) . Sincelim n →∞ k f n k L = 0, there is some constant K > n ≥ k ϕ n k ≤ K . As a consequence, (2.1),(2.2), (2.4) and (2.6) yield | ϕ n ( t ) − X ( t ) | ≤ Z t (cid:12)(cid:12) b ( ϕ n ( s )) − b ( X ( s )) (cid:12)(cid:12) d s + Z t | σ ( ϕ n ( s )) || f n ( s ) | d s ≤ L Z t | ϕ n ( s ) − X ( s ) | d s + Z t L (1 + sup n ≥ k ϕ n k ) | f n ( s ) | d s ≤ L (1 + K ) √ T k f n k L + L Z t | ϕ n ( s ) − X ( s ) | d s ∀ t ∈ [0 , T ] . Based on the above formula and the Gronwall inequality, | ϕ n ( t ) − X ( t ) | ≤ e Lt L (1 + K ) √ T k f n k L for any t ∈ [0 , T ], which along with lim n →∞ k f n k L = 0 leads to lim n →∞ k ϕ n − X k = 0. In this way, weobtain x = lim n →∞ ϕ n ( T ) = X ( T ). Hence, we prove the first conclusion.(2) In fact, there exists some point x ∗ ∈ { x ∈ R d : | x − X ( T ) | ≥ δ } such that I ( x ∗ ) =inf { x ∈ R d : | x − X ( T ) |≥ δ } I ( x ). Here we use the fact that a good rate function can achieve its infimum on anon-empty closed set (see, e.g., [12, Chapter 1.2]). Further, it follows from the first conclusion that I ( x ∗ ) >
0. Hence, C ( δ ) = I ( x ∗ ) >
0. By Definition 2.2, the LDP of { X ǫ ( T ) } ǫ> implies thatlim sup ǫ → ǫ ln P ( | X ǫ ( T ) − X ( T ) | ≥ δ ) ≤ − inf { x ∈ R d : | x − X ( T ) |≥ δ } I ( x ) = − C ( δ ) < . This deduces that (2.5) holds. (cid:3)
Remark 2.7.
The LDP of { X ǫ ( T ) } ǫ> shows that it formally satisfies P ( X ǫ ( T ) ∈ [ a, a + da ]) ≈ e − ǫ I ( a ) d a , i.e., the hitting probability decays exponentially. Corollary 2.6(1) indicates that X ( T ) isthe unique minimizer of I . Hence, one can conclude that X ǫ ( T ) most likely visits X ( T ) as ǫ → .In addition, by Corollary 2.6(2), the event that X ǫ ( T ) visits other points except X ( T ) is rare andits probability decays exponentially. These are consistent to the fact that X ǫ is the small randomperturbation of X . JIALIN HONG, DIANCONG JIN, AND DERUI SHENG
For N ∈ N + , let { t < t < · · · < t N − < t N } be a uniform partition of [0 , T ] with t n = nh , n = 0 , , . . . , N , where h = TN is the step-size. The stochastic θ -method ( θ ∈ [0 , X ǫn +1 = X ǫn + b (cid:0) (1 − θ ) X ǫn + θX ǫn +1 (cid:1) h + √ ǫσ ( X ǫn )∆ W n , n = 0 , , . . . , N − , (2.7)where ∆ W n = W ( t n +1 ) − W ( t n ) is the increment of Brownian motion. In addition, denoteˆ s := max ( { t , t , . . . , t N } ∩ [0 , s ]) and ˇ s := min ( { t , t , . . . , t N } ∩ [ s, T ]) for each s ∈ [0 , T ]. Forsufficiently small h , define the map Γ h : L (0 , T ; R m ) → C x ([0 , T ] , R d ) by ϕ = Γ h ( f ), where ϕ isthe unique continuous solution of ϕ ( t ) = x + Z t b ((1 − θ ) ϕ (ˆ s ) + θϕ (ˇ s )) d s + Z t σ ( ϕ (ˆ s )) f ( s )d s ∀ t ∈ [0 , T ] . Then we are able to give the one-point LDP of the numerical solution based on the stochastic θ -method. Theorem 2.8.
Let Assumption (A1) hold. Then for any h ∈ (0 , L ] , { X ǫN } ǫ> satisfies the LDPon R d with the good rate function I h ( x ) given by I h ( x ) := inf { ϕ ∈ C x ([0 ,T ] , R d ) ,ϕ ( T )= x } J h ( ϕ ) ∀ x ∈ R d . Here J h : C x ([0 , T ] , R d ) → R is defined as J h ( ϕ ) := inf { f ∈ L (0 ,T ; R m ):Γ h ( f )= ϕ } Z T | f ( t ) | d t ∀ ϕ ∈ C x ([0 , T ] , R d ) . Proof.
In this proof, we use K ( g , h, x ) to denote a generic constant depending on g , h and x but independent of g , which may vary from one place to another, where g and g will be specifiedbelow. First we introduce the continuous version { ¯ X ǫ ( t ) , t ∈ [0 , T ] } of the stochastic θ -method(2.7): ¯ X ǫ ( t ) = x + Z t b (cid:0) (1 − θ ) ¯ X ǫ (ˆ s ) + θ ¯ X ǫ (ˇ s ) (cid:1) d s + √ ǫ Z t σ ( ¯ X ǫ (ˆ s ))d W ( s ) ∀ t ∈ [0 , T ] . Then it suffices to show that { ¯ X ǫ ( T ) } ǫ> satisfies the LDP with the good rate function I h , due to¯ X ǫ ( T ) = X ǫN .For any fixed h ∈ (0 , L ], define the map F h : C ([0 , T ] , R m ) → C x ([0 , T ] , R d ) by f = F h ( g ),where f is the unique continuous solution of f ( t ) = x + Z t b ((1 − θ ) f (ˆ s ) + θf (ˇ s )) d s + Z t σ ( f (ˆ s ))d g ( s ) ∀ t ∈ [0 , T ] . Next we prove that F h is continuous. Let g ∈ C ([0 , T ] , R m ) be fixed and denote f = F h ( g ). Bythe definition of F h , f (0) = x and for any t ∈ [ t n , t n +1 ], n = 0 , , . . . , N − f ( t ) = f ( t n ) + b ((1 − θ ) f ( t n ) + θf ( t n +1 )) ( t − t n ) + σ ( f ( t n ))( g ( t ) − g ( t n )) . (2.8)It follows from the above formula and (2.2) that | f ( t n +1 ) | ≤ | f ( t n ) | + hL (1 + | f ( t n ) | + | f ( t n +1 ) | ) + 2 L (1 + | f ( t n ) | ) k g k , n = 0 , , . . . , N − . Noting that hL ≤ for h ∈ (0 , L ], we have | f ( t n +1 ) | ≤ (cid:18)
32 + 2 L k g k (cid:19) | f ( t n ) | + 12 + 2 L k g k + 12 | f ( t n +1 ) | , n = 0 , , . . . , N − , which yields | f ( t n +1 ) | ≤ (3 + 4 L k g k ) | f ( t n ) | + 1 + 4 L k g k ≤ C ( g ) (1 + | f ( t n ) | ) , n = 0 , , . . . , N − DP 7 with C ( g ) := 3 + 4 L k g k . Hence, it holds that | f ( t n ) | ≤ C ( g ) + C ( g ) | f ( t n − ) |≤ C ( g ) + C ( g ) + C ( g ) | f ( t n − ) |· · ·≤ C ( g ) + C ( g ) + · · · C n ( g ) + C n ( g ) | f (0) | , n = 0 , , . . . , N. Accordingly, one immediately hassup n =0 , ,...,N | f ( t n ) | ≤ N X i =1 C i ( g ) + C N ( g ) | x | = C T/h +1 ( g ) − C ( g ) C ( g ) − C T/h ( g ) | x | . This is to say, sup n =0 , ,...,N | f ( t n ) | ≤ K ( g , h, x ), which along with (2.8) gives that for any t ∈ [ t n , t n +1 ], n = 0 , , . . . , N − | f ( t ) | ≤ sup n =0 , ,...,N | f ( t n ) | + hL − θ ) sup n =0 , ,...,N | f ( t n ) | + θ sup n =0 , ,...,N | f ( t n ) | ! + 2 L n =0 , ,...,N | f ( t n ) | ! k g k ≤ K ( g , h, x ) . In this way, we have k f k ≤ K ( g , h, x ).Take g ∈ ¯ B ( g ,
1) and set f = F h ( g ). Then f (0) = x and for any t ∈ [ t n , t n +1 ], n =0 , , . . . , N − f ( t ) = f ( t n ) + b ((1 − θ ) f ( t n ) + θf ( t n +1 )) ( t − t n ) + σ ( f ( t n ))( g ( t ) − g ( t n )) . (2.9)Denote e ( t ) := f ( t ) − f ( t ) for any t ∈ [0 , T ]. It follows from (2.8) and (2.9) that for any t ∈ [ t n , t n +1 ], n = 0 , , . . . , N − e ( t ) = e ( t n ) + [ b ((1 − θ ) f ( t n ) + θf ( t n +1 )) − b ((1 − θ ) f ( t n ) + θf ( t n +1 ))] ( t − t n )+ σ ( f ( t n )) ( g ( t ) − g ( t n ) − ( g ( t ) − g ( t n ))) + ( σ ( f ( t n )) − σ ( f ( t n ))) ( g ( t ) − g ( t n )) . Applying the estimate k f k ≤ K ( g , h, x ), (2.1) and (2.2), we have | e ( t ) | ≤| e ( t n ) | + hL ( | e ( t n ) | + | e ( t n +1 ) | ) + 2 L (1 + k f k ) k g − g k + 2 L | e ( t n ) | (1 + k g k ) ≤ K ( g , h, x ) ( | e ( t n ) | + k g − g k ) + hL | e ( t n +1 ) | ∀ t ∈ [ t n , t n +1 ] , n = 0 , , . . . , N − , (2.10)where we have used the fact k g k ≤ k g k + 1 for any g ∈ ¯ B ( g , h ≤ L , | e ( t n +1 ) | ≤ K ( g , h, x ) ( | e ( t n ) | + k g − g k ) , n = 0 , , . . . , N − . Using the iteration argument, one has | e ( t n ) | ≤ K n ( g , h, x ) | e (0) | + n X i =1 K i ( g , h, x ) k g − g k , n = 1 , , . . . , N. From the above formula and e (0) = 0, it follows thatsup n =0 , ,...,N | e ( t n ) | ≤ K ( g , h, x ) k g − g k . (2.11) JIALIN HONG, DIANCONG JIN, AND DERUI SHENG
Substituting (2.11) into (2.10) yields k e k ≤ K ( g , h, x ) k g − g k , which immediately leads tolim g → g k F h ( g ) − F h ( g ) k = lim g → g k e k = 0. This shows that for given h ≤ L , F h is continuous.Noting ¯ X ǫ = F h ( √ ǫW ), we use Proposition 2.3 and the continuity of F h to conclude that (cid:8) ¯ X ǫ (cid:9) ǫ> satisfies the LDP on C x (cid:0) [0 , T ] , R d (cid:1) with the good rate function ¯ J h given by¯ J h ( ϕ ) = inf { g ∈ C ([0 ,T ] , R m ): F h ( g )= ϕ } I w ( g )= inf { g ∈ H (0 ,T ; R m ): F h ( g )= ϕ } Z T | g ′ ( t ) | d t = inf { g ∈ H (0 ,T ; R m ): ϕ ( t )= x + R t b ((1 − θ ) ϕ (ˆ s )+ θϕ (ˇ s ))d s + R t σ ( ϕ (ˆ s )) g ′ ( s )d s,t ∈ [0 ,T ] } Z T | g ′ ( t ) | d t for any ϕ ∈ C x ([0 , T ] , R d ). Since H (0 , T ; R m ) is isomorphic to L (0 , T ; R m ), we have ¯ J h = J h .According to the definition of the coordinate map ξ T , ¯ X ǫ ( T ) = ξ T ( ¯ X ǫ ). Again by Proposition 2.3and the continuity of ξ T , we finally complete the proof. (cid:3) We have shown that both { X ǫ ( T ) } ǫ> and its numerical approximation { X ǫN } ǫ> based on thestochastic θ -method satisfy the LDPs. In next sections, we will study whether the rate function I h can approximate well the rate function I .3. Some properties of rate functions
In order to estimate the error between the rate functions I h and I in terms of the step-size h ,we give some properties of them in this section, including their simplified presentations and thecontinuity of I . These results depend on the following assumption. Assumption (A2). m = d and σ ( x ) is invertible for each x ∈ R d . Lemma 3.1.
Under Assumptions (A1) and (A2), the rate functions I and I h have the followingrepresentations: I ( x ) = inf ϕ ∈ A x A ( ϕ ) and I h ( x ) = inf ϕ ∈ A x B h ( ϕ ) ∀ x ∈ R d , h ∈ (0 , L ] , where A x := (cid:8) ϕ ∈ H x (0 , T ; R d ) : ϕ ( T ) = x (cid:9) , A ( ϕ ) := R T (cid:12)(cid:12) σ − ( ϕ ( t )) ( ϕ ′ ( t ) − b ( ϕ ( t ))) (cid:12)(cid:12) d t and B h ( ϕ ) := R T (cid:12)(cid:12) σ − ( ϕ (ˆ t )) (cid:0) ϕ ′ ( t ) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1)(cid:12)(cid:12) d t .Proof. Denote D J h := (cid:8) ϕ ∈ C x (cid:0) [0 , T ] , R d (cid:1) : J h ( ϕ ) < ∞ (cid:9) . We first show that D J h = H x (cid:0) , T ; R d (cid:1) .In fact, for any ϕ ∈ D J h , one has J h ( ϕ ) < ∞ . It follows from the definition of J h that there is somefunction f ∈ L (0 , T ; R d ) such that ϕ ( t ) = x + Z t b ((1 − θ ) ϕ (ˆ s ) + θϕ (ˇ s )) d s + Z t σ ( ϕ (ˆ s )) f ( s )d s ∀ t ∈ [0 , T ] . Clearly, ϕ is absolutely continuous and ϕ ′ ( t ) = b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1) + σ ( ϕ (ˆ t )) f ( t ) for a.e. t ∈ [0 , T ]. It follows from the boundedness of b ((1 − θ ) ϕ (ˆ · ) + θϕ (ˇ · )), σ ( ϕ (ˆ · )) and f ∈ L (0 , T ; R d )that ϕ ′ ∈ L (0 , T ; R d ), i.e., ϕ ∈ H x (0 , T ; R d ). On the other hand, if ϕ ∈ H x (0 , T ; R d ), weset g ( t ) = σ − ( ϕ (ˆ t )) (cid:0) ϕ ′ ( t ) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1) for a.e. t ∈ [0 , T ]. It can be verified that g ∈ L (0 , T ; R d ) by means of the continuity of σ − , b and the boundedness of ϕ . By the definition DP 9 of J h , J h ( ϕ ) ≤ R T | g ( t ) | d t < + ∞ , i.e., ϕ ∈ D J h . Hence, we prove D J h = H x (cid:0) , T ; R d (cid:1) . Basedon this result, we obtain J h ( ϕ ) = ( R T (cid:12)(cid:12) σ − ( ϕ (ˆ t )) (cid:0) ϕ ′ ( t ) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1)(cid:12)(cid:12) d t, if ϕ ∈ H x (cid:0) , T ; R d (cid:1) , + ∞ , otherwise.Thus, it holds that I h ( ϕ ) = inf { ϕ ∈ C x ([0 ,T ] , R d ): ϕ ( T )= x } J h ( ϕ )= inf { ϕ ∈ H x (0 ,T ; R d ): ϕ ( T )= x } Z T (cid:12)(cid:12) σ − ( ϕ (ˆ t )) (cid:0) ϕ ′ ( t ) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1)(cid:12)(cid:12) d t = inf ϕ ∈ A x B h ( ϕ ) . Similarly, one can verify that I ( ϕ ) = inf ϕ ∈ A x A ( ϕ ), which completes the proof. (cid:3) Lemma 3.2.
The following properties hold. (1)
Under Assumptions (A1) and (A2), σ − is locally Lipschitz continuous, i.e., for each R > ,there exists some constant L R > such that | σ − ( x ) − σ − ( y ) | ≤ L R | x − y | ∀ x, y ∈ ¯ B (0 , R ) . (2) For any ϕ ∈ H x (0 , T ; R d ) , | ϕ ( t ) − ϕ ( s ) | ≤ ( t − s ) / (cid:18)Z ts | ϕ ′ ( r ) | d r (cid:19) / ∀ ≤ s ≤ t ≤ T, k ϕ k ≤ | x | + √ T k ϕ k H . (3) Let Assumptions (A1) and (A2) hold. Then there exists some constant C > independentof h such that for any ϕ ∈ H x (0 , T ; R d ) and h ∈ (0 , L ] , k ϕ k H ≤ C e C A ( ϕ ) , k ϕ k H ≤ C e C B h ( ϕ ) . Proof. (1) Let σ ∗ and det( σ ) denote the adjoint matrix and determinant of σ , respectively. Since σ is globally Lipschitz continuous, σ ∗ and det( σ ) are locally Lipschitz continuous. Notice that σ isinvertible everywhere and continuous. Hence for any R >
0, there is some constant K R > | x |≤ R | det( σ ( x )) | ≥ K R . Further, for any R > x, y ∈ ¯ B (0 , R ), | σ − ( x ) − σ − ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) σ ∗ ( x )det( σ ( x )) − σ ∗ ( y )det( σ ( y )) (cid:12)(cid:12)(cid:12)(cid:12) = 1 | det( σ ( x )) det( σ ( y )) | | ( σ ∗ ( x ) − σ ∗ ( y )) det( σ ( y )) + σ ∗ ( y ) (det( σ ( y )) − det( σ ( x ))) |≤ K R | σ ∗ ( x ) − σ ∗ ( y ) | + 1 K R sup | y |≤ R | σ ∗ ( y ) | | det( σ ( x )) − det( σ ( y )) |≤ L R | x − y | , where L R > R and σ . In this way, we conclude the first conclusion.(2) By the H¨older inequality, for any ϕ ∈ H x (0 , T ; R d ) and 0 ≤ s ≤ t ≤ T , | ϕ ( t ) − ϕ ( s ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ts ϕ ′ ( r )d r (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( t − s ) / (cid:18)Z ts | ϕ ′ ( r ) | d r (cid:19) / . Based on the above formula, for any t ∈ [0 , T ], | ϕ ( t ) − x | ≤ t / (cid:16)R t | ϕ ′ ( r ) | d r (cid:17) / ≤ √ T k ϕ k H ,which implies the second conclusion.(3) Next we use K ( x , T, L ) to denote some constant depending on x , T and L , but independent ofthe step-size h , which may vary from one place to somewhere. Denote f ( t ) = σ − ( ϕ ( t )) ( ϕ ′ ( t ) − b ( ϕ ( t ))),for a.e. t ∈ [0 , T ]. Then k f k L = 2 A ( ϕ ) and ϕ ( t ) = x + Z t b ( ϕ ( s ))d s + Z t σ ( ϕ ( s )) f ( s )d s. By the H¨older inequality and (2.2), for each t ∈ [0 , T ], | ϕ ( t ) | ≤ | x | + 3 t Z t | b ( ϕ ( s )) | d s + 3 Z t | σ ( ϕ ( s )) | d s Z t | f ( s ) | d s ≤ | x | + 6 T L Z t (cid:0) | ϕ ( s ) | (cid:1) d s + 12 L A ( ϕ ) Z t (cid:0) | ϕ ( s ) | (cid:1) d s ≤ K ( x , T, L ) (1 + A ( ϕ )) + K ( x , T, L ) (1 + A ( ϕ )) Z t | ϕ ( s ) | d s. According to the Gronwall inequality, | ϕ ( t ) | ≤ K ( x , T, L ) (1 + A ( ϕ )) e K ( x ,T,L )(1+ A ( ϕ )) t ≤ K ( x , T, L ) e K ( x ,T,L ) A ( ϕ ) ∀ t ∈ [0 , T ] , where we have used the fact 1+ x ≤ e x for any x ∈ R . Further, we obtain k ϕ k ≤ K ( x , T, L ) e K ( x ,T,L ) A ( ϕ ) .Hence, k b ( ϕ ) k + | σ ( ϕ ) k ≤ L (1 + k ϕ k ) ≤ K ( x , T, L ) e K ( x ,T,L ) A ( ϕ ) . Noting that ϕ ′ = b ( ϕ ) + σ ( ϕ ) f and p A ( ϕ ) ≤ p A ( ϕ ) ≤ e A ( ϕ ) , we have k ϕ k H ≤k b ( ϕ ) k L + k σ ( ϕ ) k k f k L ≤√ T k b ( ϕ ) k + k σ ( ϕ ) k p A ( ϕ ) ≤ K ( x , T, L ) e K ( x ,T,L ) A ( ϕ ) . Next we prove that k ϕ k H ≤ C e C B h ( ϕ ) . Denote g ( t ) = σ − ( ϕ (ˆ t )) (cid:0) ϕ ′ ( t ) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1) ,for a.e. t ∈ [0 , T ]. Then k g k L = 2 B h ( ϕ ) and for any t ∈ [0 , T ], ϕ ( t ) = x + Z t b ((1 − θ ) ϕ (ˆ s ) + θϕ (ˇ s )) d s + Z t σ ( ϕ (ˆ s )) g ( s )d s. (3.1)Hence for any n = 1 , , . . . , N , ϕ ( t n ) = ϕ ( t n − ) + hb ((1 − θ ) ϕ ( t n − ) + θϕ ( t n )) + σ ( ϕ ( t n − )) Z t n t n − g ( t )d t. It follows from (2.2) that | ϕ ( t n ) | ≤| ϕ ( t n − ) | + hL (1 + | ϕ ( t n − ) | + | ϕ ( t n ) | ) + L (1 + | ϕ ( t n − ) | ) Z t n t n − | g ( t ) | d t = hL + L Z t n t n − | g ( t ) | d t ! | ϕ ( t n − ) | + Lh | ϕ ( t n ) | + Lh + L Z t n t n − | g ( t ) | d t. DP 11
Thus for any h ∈ (0 , L ] and n = 1 , , . . . , N , | ϕ ( t n ) | ≤ (cid:16) hL + L R t n t n − | g ( t ) | d t (cid:17) | ϕ ( t n − ) | − Lh + Lh − Lh + L − Lh Z t n t n − | g ( t ) | d t = | ϕ ( t n − ) | + 2 Lh + L R t n t n − | g ( t ) | d t − Lh | ϕ ( t n − ) | + Lh − Lh + L − Lh Z t n t n − | g ( t ) | d t ≤| ϕ ( t n − ) | + Lh + 2 L Z t n t n − | g ( t ) | d t ! | ϕ ( t n − ) | + 2 Lh + 2 L Z t n t n − | g ( t ) | d t. Set k n = 4 Lh + 2 L R t n +1 t n | g ( t ) | d t , n = 0 , , . . . , N − | ϕ ( t n ) | ≤| ϕ ( t n − ) | + k n − | ϕ ( t n − ) | + k n − ≤| ϕ ( t n − ) | + k n − | ϕ ( t n − ) | + k n − | ϕ ( t n − ) | + k n − + k n − · · ·≤| x | + n − X j =0 k j | ϕ ( t j ) | + n − X j =0 k j ∀ n = 1 , , . . . , N. It follows from [19, Lemma 1.4.2] that for each n = 1 , , . . . , N , | ϕ ( t n ) | ≤ (cid:16) | x | + P n − j =0 k j (cid:17) e P n − j =0 k j ,which means sup n =0 , ,...,N | ϕ ( t n ) | ≤ (cid:16) | x | + P N − j =0 k j (cid:17) e P N − j =0 k j . Since N − X j =0 k j =4 LT + 2 L Z T | g ( t ) | d t ≤ LT + L + T k g k L =4 LT + L + 2 T B h ( ϕ ) ≤ K ( T, L )(1 + B h ( ϕ )) , we obtainsup n =0 , ,...,N | ϕ ( t n ) | ≤ ( | x | + K ( T, L )(1 + B h ( ϕ ))) e K ( T,L )(1+ B h ( ϕ )) ≤ K ( x , T, L ) e K ( x ,T,L ) B h ( ϕ ) . As a consequence,sup t ∈ [0 ,T ] (cid:12)(cid:12) b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:12)(cid:12) ≤ L − θ ) sup n =0 , ,...,N | ϕ ( t n ) | + θ sup n =0 , ,...,N | ϕ ( t n ) | ! ≤ K ( x , T, L ) e K ( x ,T,L ) B h ( ϕ ) , sup t ∈ [0 ,T ] (cid:12)(cid:12) σ ( ϕ (ˆ t )) (cid:12)(cid:12) ≤ L n =0 , ,...,N | ϕ ( t n ) | ! ≤ K ( x , T, L ) e K ( x ,T,L ) B h ( ϕ ) . By (3.1), ϕ ′ ( t ) = b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1) + σ ( ϕ (ˆ t )) g ( t ) for a.e. t ∈ [0 , T ]. Combining the aboveformulas, we finally obtain k ϕ ′ k L ≤ K ( x , T, L ) e K ( x ,T,L ) B h ( ϕ ) √ T + K ( x , T, L ) e K ( x ,T,L ) B h ( ϕ ) k g k L ≤ K ( x , T, L ) e K ( x ,T,L ) B h ( ϕ ) , where we have used the fact k g k L = p B h ( ϕ ) ≤ e B h ( ϕ ) . This completes the proof. (cid:3) Lemma 3.2(3) is enough to get the pointwise convergence of I h . However, in order to derive theuniform error between I h and I on a given bounded set, we need to resort to the continuity of I . Lemma 3.3.
Under Assumptions (A1) and (A2), D I = R d and I is continuous on R d , where D I := (cid:8) x ∈ R d : I ( x ) < + ∞ (cid:9) .Proof. For any x ∈ R d , set ϕ x ( t ) = x + x − x T t , t ∈ [0 , T ]. Then ϕ x ∈ A x and hence I ( x ) =inf ϕ ∈ A x A ( ϕ ) ≤ A ( ϕ x ) < + ∞ . In order to prove the continuity of I , it suffices to show that for anyfixed x ∈ R d , lim x → x I ( x ) = I ( x ). We divide the proof into two steps. Step 1: We prove lim sup x → x I ( x ) ≤ I ( x ) . In this proof, we denote by K ( x ) a constant depending on x but independent of x , which mayvary from one place to another. For arbitrary ε ∈ (0 , I that thereexists some function ϕ ε,x ∈ A x such that A ( ϕ ε,x ) < I ( x ) + ε ≤ I ( x ) + 1 ≤ K ( x ) . (3.2)By the above formula and Lemma 3.2, k ϕ ε,x k H ≤ C e C A ( ϕ ε,x ) ≤ K ( x ) , (3.3) k ϕ ε,x k ≤ | x | + √ T k ϕ ε,x k H ≤ K ( x ) . (3.4)Define e ϕ ε,x by e ϕ ε,x ( t ) = ϕ ε,x ( t ) + x − x T t , t ∈ [0 , T ]. It can be verified that e ϕ ε,x ∈ A x . It followsfrom the fact e ϕ ′ ε,x = ϕ ′ ε,x + x − x T and (3.3) that for any x ∈ ¯ B ( x , k e ϕ ε,x k H = k e ϕ ′ ε,x k L ≤ k ϕ ′ ε,x k L + | x − x | T √ T ≤ k ϕ ε,x k H + 1 √ T ≤ K ( x ) . (3.5)By Lemma 3.2(2) and (3.5), k e ϕ ε,x k ≤ | x | + √ T k e ϕ ε,x k H ≤ K ( x ) ∀ x ∈ ¯ B ( x , . (3.6)Define f ε,x ( t ) = σ − ( ϕ ε,x ( t )) (cid:0) ϕ ′ ε,x ( t ) − b ( ϕ ε,x ( t )) (cid:1) and g ε,x ( t ) = σ − ( e ϕ ε,x ( t )) (cid:0) e ϕ ′ ε,x ( t ) − b ( e ϕ ε,x ( t )) (cid:1) for a.e. t ∈ [0 , T ]. Then one has A ( ϕ ε,x ) − A ( e ϕ ε,x ) = 12 h f ε,x , f ε,x i L − h g ε,x , g ε,x i L = 12 h f ε,x + g ε,x , f ε,x − g ε,x i L . This immediately leads to | A ( ϕ ε,x ) − A ( e ϕ ε,x ) | ≤ k f ε,x + g ε,x k L k f ε,x − g ε,x k L ≤ (cid:0) k f ε,x k L + k g ε,x k L (cid:1) k f ε,x − g ε,x k L . Noting that A ( ϕ ε,x ) = k f ε,x k L , using (3.2) yields k f ε,x k L = p A ( ϕ ε,x ) ≤ K ( x ). It followsfrom (3.6) and the continuity of b , σ − that for any x ∈ ¯ B ( x , t ∈ [0 ,T ] (cid:0) | σ − ( e ϕ ε,x ( t )) | + | b ( e ϕ ε,x ( t )) | (cid:1) ≤ sup | x |≤k e ϕ ε,x k (cid:0) | σ − ( x ) | + | b ( x ) | (cid:1) ≤ K ( x ) . (3.7)Using (3.5) and (3.7) gives k g ε,x k L ≤ k σ − ( e ϕ ε,x ) k (cid:16) k e ϕ ε,x k H + √ T k b ( e ϕ ε,x ) k (cid:17) ≤ K ( x ). Con-sequently, for any x ∈ ¯ B ( x , | A ( ϕ ε,x ) − A ( e ϕ ε,x ) | ≤ K ( x ) k f ε,x − g ε,x k L . Further, for a.e. t ∈ [0 , T ], f ε,x ( t ) − g ε,x ( t )= σ − ( ϕ ε,x ( t )) ϕ ′ ε,x ( t ) − σ − ( e ϕ ε,x ( t )) e ϕ ′ ε,x ( t ) − (cid:2) σ − ( ϕ ε,x ( t )) b ( ϕ ε,x ( t )) − σ − ( e ϕ ε,x ( t )) b ( e ϕ ε,x ( t )) (cid:3) DP 13 =: M x ,x ( t ) − N x ,x ( t ) . Then we have | A ( ϕ ε,x ) − A ( e ϕ ε,x ) | ≤ K ( x ) ( k M x ,x k L + k N x ,x k L ) ∀ x ∈ ¯ B ( x , . (3.8)Next we show that lim x → x k M x ,x k L = lim x → x k N x ,x k L = 0. In fact, M x ,x ( t ) = σ − ( ϕ ε,x ( t )) ϕ ′ ε,x ( t ) − σ − ( e ϕ ε,x ( t )) (cid:18) ϕ ′ ε,x ( t ) + x − x T (cid:19) = (cid:0) σ − ( ϕ ε,x ( t )) − σ − ( e ϕ ε,x ( t )) (cid:1) ϕ ′ ε,x ( t ) − σ − ( e ϕ ε,x ( t )) T ( x − x ) . By Lemma 3.2(1), σ − is locally Lipschitz continuous, which together with (3.4) and (3.6) yieldsthat there is a constant L K ( x ) > (cid:12)(cid:12) σ − ( ϕ ε,x ( t )) − σ − ( e ϕ ε,x ( t )) (cid:12)(cid:12) ≤ L K ( x ) | ϕ ε,x ( t ) − e ϕ ε,x ( t ) |≤ L K ( x ) k ϕ ε,x − e ϕ ε,x k ≤ K ( x ) | x − x | ∀ t ∈ [0 , T ] , x ∈ ¯ B ( x , . (3.9)Combining the above formula and (3.7) gives k M x ,x k L ≤ K ( x ) | x − x |k ϕ ε,x k H + K ( x ) T | x − x |√ T ≤ K ( x ) | x − x | provided that x ∈ ¯ B ( x , x → x k M x ,x k L = 0. As for N x ,x ,we have N x ,x ( t ) = (cid:2) σ − ( ϕ ε,x ( t )) − σ − ( e ϕ ε,x ( t )) (cid:3) b ( ϕ ε,x ( t )) + σ − ( e ϕ ε,x ( t )) [ b ( ϕ ε,x ( t )) − b ( e ϕ ε,x ( t ))] . Based on (2.1), (3.9) and the fact k b ( ϕ ε,x ) k + k σ − ( e ϕ ε,x ) k ≤ K ( x ) for any x ∈ ¯ B ( x , k N x ,x k L ≤ K ( x ) | x − x | provided that x ∈ ¯ B ( x , x → x k N x ,x k L = 0.Then from (3.8) it follows thatlim x → x A ( e ϕ ε,x ) = A ( ϕ ε,x ) ∀ ε ∈ (0 , . Notice that for any ε ∈ (0 , I ( x ) = inf ϕ ∈ A x A ( ϕ ) ≤ A ( e ϕ ε,x ). The above formulas and (3.2) producelim sup x → x I ( x ) ≤ lim sup x → x A ( e ϕ ε,x ) = A ( ϕ ε,x ) < I ( x ) + ε. Since ε is arbitrary, lim sup ǫ → I ( x ) ≤ I ( x ). Step 2: We prove lim inf x → x I ( x ) ≥ I ( x ) . Fix ε ∈ (0 , x → x I ( x ) ≤ I ( x ) that there is some constant δ ( ε ) ∈ (0 ,
1] such that I ( x ) < I ( x ) + ε for any x ∈ ¯ B ( x , δ ( ε )). By the definition of I ( x ), thereexists some function e ψ ε,x ∈ A x such that A ( e ψ ε,x ) < I ( x ) + ε . Hence, A ( e ψ ε,x ) < I ( x ) + ε ∀ x ∈ ¯ B ( x , δ ( ε )) . By Lemma 3.2(3), k e ψ ε,x k H ≤ C e C A ( e ψ ε,x ) ≤ K ( x ). Define ψ ε,x by ψ ε,x ( t ) = e ψ ε,x ( t ) + x − x T t , t ∈ [0 , T ]. Then ψ ε,x ∈ A x and k ψ ε,x k H ≤ k e ψ ε,x k H + | x − x | T √ T ≤ K ( x ) provided that x ∈ ¯ B ( x , δ ( ε )). Similar to the proof in Step 1, one has lim x → x (cid:12)(cid:12)(cid:12) A ( e ψ ε,x ) − A ( ψ ε,x ) (cid:12)(cid:12)(cid:12) = 0. Thus,there is some constant δ ( ε ) ≤ δ ( ε ) such that A ( ψ ε,x ) < A ( e ψ ε,x ) + ε for any x ∈ ¯ B ( x , δ ( ε )).In this way, we obtain I ( x ) = inf ϕ ∈ A x A ( ϕ ) ≤ A ( ψ ε,x ) < A ( e ψ ε,x ) + ε < I ( x ) + ε ∀ x ∈ ¯ B ( x , δ ( ε )) , which yields lim inf x → x I ( x ) ≥ I ( x ) − ε . Letting ε → x → x I ( x ) ≥ I ( x ). Finally, combiningthe above conclusions, we finish the proof. (cid:3) Convergence of the discrete rate functions I h Throughout this section, let K ( R ) denote a generic constant depending on the parameter R but independent of the step-size h , which may vary from one place to another. Denote B R := (cid:8) ϕ ∈ H x (0 , T ; R d ) : k ϕ k H ≤ R (cid:9) and let I d be the ( d × d )-dimensional identity matrix. In whatfollows, we give the uniform error estimate between A and B h on any given bounded set of A x . Lemma 4.1.
Let Assumptions (A1) and (A2) hold. Then for any
R > , there exists some constant K ( R ) > such that for any h ∈ (0 , , sup ϕ ∈ B R | A ( ϕ ) − B h ( ϕ ) | ≤ K ( R ) h / . (4.1) In particular, if σ = I d , we have that for any R > , there is some constant K ( R ) > such thatfor any h ∈ (0 , , sup ϕ ∈ B R | A ( ϕ ) − B h ( ϕ ) | ≤ K ( R ) h. (4.2) Proof.
Denote f ( t ) = σ − ( ϕ ( t )) ( ϕ ′ ( t ) − b ( ϕ ( t ))) and g ( t ) = σ − ( ϕ (ˆ t )) (cid:0) ϕ ′ ( t ) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1) for a.e. t ∈ [0 , T ]. Then A ( ϕ ) = k f k L and B h ( ϕ ) = k g k L . Hence, | A ( ϕ ) − B h ( ϕ ) | = 12 |h f, f i L − h g, g i L | = 12 |h f + g, f − g i L |≤ k f + g k L k f − g k L ≤
12 ( k f k L + k g k L ) k f − g k L . By means of Lemma 3.2(2), for any ϕ ∈ B R , k ϕ k ≤ | x | + √ T k ϕ k H ≤ | x | + √ T R . Since σ − iscontinuous, there is some constant K ( R ) > k σ − ( ϕ ) k ≤ sup x ∈ ¯ B (0 , | x | + √ T R ) | σ − ( x ) | ≤ K ( R ) ∀ ϕ ∈ B R . (4.3)Therefore, k f k L ≤ K ( R ) (cid:0) k ϕ ′ k L + k b ( ϕ ) k L (cid:1) ≤ K ( R ) (cid:16) k ϕ k H + √ T k b ( ϕ ) k (cid:17) ≤ K ( R ) (cid:16) k ϕ k H + √ T L (1 + k ϕ k ) (cid:17) ≤ K ( R ) ∀ ϕ ∈ B R . Noting that sup t ∈ [0 ,T ] (cid:12)(cid:12) σ − ( ϕ (ˆ t )) (cid:12)(cid:12) ≤ k σ − ( ϕ ) k ≤ K ( R ) for every ϕ ∈ B R , we have k g k L ≤ K ( R ) k ϕ k H + √ T sup t ∈ [0 ,T ] (cid:12)(cid:12) b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:12)(cid:12)! ≤ K ( R ) (cid:16) k ϕ k H + L √ T (1 + k ϕ k ) (cid:17) ≤ K ( R ) ∀ ϕ ∈ B R . Accordingly, it holds that | A ( ϕ ) − B h ( ϕ ) | ≤ K ( R ) k f − g k L ∀ ϕ ∈ B R . (4.4) DP 15
Next, we decompose f − g into f ( t ) − g ( t ) = (cid:0) σ − ( ϕ ( t )) − σ − ( ϕ (ˆ t )) (cid:1) (cid:0) ϕ ′ ( t ) − b ( ϕ ( t )) (cid:1) − σ − ( ϕ (ˆ t )) (cid:0) b ( ϕ ( t )) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:1) . (4.5)By Lemma 3.2(2), for any t ∈ [0 , T ] and ϕ ∈ B R , (cid:12)(cid:12) ϕ ( t ) − ϕ (ˆ t ) (cid:12)(cid:12) ≤ ( t − ˆ t ) / (cid:18)Z t ˆ t | ϕ ′ ( s ) | d s (cid:19) / ≤ k ϕ k H h / ≤ Rh / and (cid:12)(cid:12) ϕ ( t ) − (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:12)(cid:12) ≤ (1 − θ ) | ϕ ( t ) − ϕ (ˆ t ) | + θ | ϕ ( t ) − ϕ (ˇ t ) |≤ (1 − θ )( t − ˆ t ) / (cid:18)Z t ˆ t | ϕ ′ ( s ) | d s (cid:19) / + θ (ˇ t − t ) / Z ˇ tt | ϕ ′ ( s ) | d s ! / ≤ (1 − θ ) k ϕ k H h / + θ k ϕ k H h / ≤ Rh / . Notice that σ − is locally Lipschitz continuous due to Lemma 3.2(1) and k ϕ k ≤ | x | + √ T R provided that ϕ ∈ B R . There exists a constant L R > ϕ ∈ B R ,sup t ∈ [0 ,T ] (cid:12)(cid:12) σ − ( ϕ ( t )) − σ − ( ϕ (ˆ t )) (cid:12)(cid:12) ≤ L R sup t ∈ [0 ,T ] | ϕ ( t ) − ϕ (ˆ t ) | ≤ K ( R ) h / . (4.6)Since b is globally Lipschitz continuous,sup t ∈ [0 ,T ] (cid:12)(cid:12) b ( ϕ ( t )) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:12)(cid:12) ≤ L sup t ∈ [0 ,T ] (cid:12)(cid:12) ϕ ( t ) − (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:12)(cid:12) ≤ K ( R ) h / . (4.7)Combining (4.3) and (4.5)-(4.7) leads to k f − g k L ≤ sup t ∈ [0 ,T ] (cid:12)(cid:12) σ − ( ϕ ( t )) − σ − ( ϕ (ˆ t )) (cid:12)(cid:12) (cid:0) k ϕ ′ k L + k b ( ϕ ) k L (cid:1) + √ T k σ − ( ϕ ) k sup t ∈ [0 ,T ] (cid:12)(cid:12) b ( ϕ ( t )) − b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1)(cid:12)(cid:12) ≤ K ( R ) h / (cid:16) k ϕ k H + √ T L(1 + k ϕ k ) (cid:17) + K ( R ) h / ≤ K ( R ) h / ∀ ϕ ∈ B R , (4.8)where we have used the inequality k ϕ k ≤ | x | + √ T R provided that ϕ ∈ B R . Substituting (4.8)into (4.4) yields sup ϕ ∈ B R | A ( ϕ ) − B h ( ϕ ) | ≤ K ( R ) h / . This proves (4.1).For the case σ = I d , we still have | A ( ϕ ) − B h ( ϕ ) | ≤ K ( R ) k f − g k L ∀ ϕ ∈ B R . In order to prove (4.2), it suffices to show that k f − g k L ≤ K ( R ) h . Notice that in this case, f ( t ) − g ( t ) = b (cid:0) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) (cid:1) − b ( ϕ ( t )) for a.e. t ∈ [0 , T ]. Hence, | f ( t ) − g ( t ) | ≤ L (cid:12)(cid:12) (1 − θ ) ϕ (ˆ t ) + θϕ (ˇ t ) − ϕ ( t ) (cid:12)(cid:12) ≤ L (cid:0) (1 − θ ) | ϕ ( t ) − ϕ (ˆ t ) | + θ | ϕ (ˇ t ) − ϕ ( t ) | (cid:1) ∀ t ∈ [0 , T ] . Further, we obtain k f − g k L ≤ L (1 − θ ) (cid:18)Z T | ϕ ( t ) − ϕ (ˆ t ) | d t (cid:19) / + Lθ (cid:18)Z T | ϕ (ˇ t ) − ϕ ( t ) | d t (cid:19) / . By Lemma 3.2(2), Z T | ϕ ( t ) − ϕ (ˆ t ) | d t = N − X n =0 Z t n +1 t n | ϕ ( t ) − ϕ ( t n ) | d t ≤ N − X n =0 Z t n +1 t n ( t − t n ) Z tt n | ϕ ′ ( s ) | d s d t ≤ h N − X n =0 Z t n +1 t n Z t n +1 t n | ϕ ′ ( s ) | d s d t = h N − X n =0 Z t n +1 t n | ϕ ′ ( s ) | d s = h k ϕ k H . This indicates that (cid:16)R T | ϕ ( t ) − ϕ (ˆ t ) | d t (cid:17) / ≤ k ϕ k H h ≤ Rh , for any ϕ ∈ B R . Similarly, one hasthat (cid:16)R T | ϕ (ˇ t ) − ϕ ( t ) | d t (cid:17) / ≤ Rh for any ϕ ∈ B R . As a consequence, we obtain that k f − g k L ≤ L (1 − θ ) Rh + LθRh = LRh , which completes the proof. (cid:3)
Remark 4.2.
The key to getting (4.2) , in the additive noises case, is the fact (cid:16)R T | ϕ ( t ) − ϕ (ˆ t ) | d t (cid:17) / ≤k ϕ k H h for any ϕ ∈ H x (0 , T ; R d ) . This is not applicable to the multiplicative noises case, dueto the presence of the term (cid:0) σ − ( ϕ ( · )) − σ − ( ϕ (ˆ · )) (cid:1) ϕ ′ ( · ) . In order to improve the estimate of R T (cid:12)(cid:12)(cid:0) σ − ( ϕ ( t )) − σ − ( ϕ (ˆ t )) (cid:1) ϕ ′ ( t ) (cid:12)(cid:12) d t , one needs ϕ ′ to be L p -integrable ( p > ). However, we cannot expect this to happen provided that ϕ ∈ H x (0 , T ; R d ) . Theorem 4.3.
Under Assumptions (A1) and (A2), we have the following. (1)
For any
R > , there is some constant K ( R ) > such that for any h ∈ (0 , L ∧ , sup | x |≤ R | I h ( x ) − I ( x ) | ≤ K ( R ) h / . (2) In particular, if σ = I d , we have that for any R > , there is some constant K ( R ) > suchthat for any h ∈ (0 , L ∧ , sup | x |≤ R | I h ( x ) − I ( x ) | ≤ K ( R ) h. Proof. (1) According to the definition of I , for any x ∈ ¯ B (0 , R ) and h ∈ (0 , L ∧ ϕ h,x ∈ A x such that A ( ϕ h,x ) < I ( x ) + h ≤ I ( x ) + 1 . (4.9) DP 17
By Lemma 3.3, I is continuous on ¯ B (0 , R ), which yields that there exists some constant K ( R ) > | x |≤ R I ( x ) ≤ K ( R ) . Hence, A ( ϕ h,x ) ≤ K ( R ) + 1 for any x ∈ ¯ B (0 , R ) and h ∈ (0 , L ∧ k ϕ h,x k H ≤ C e C A ( ϕ h,x ) ≤ K ( R ) ∀ x ∈ ¯ B (0 , R ) , h ∈ (0 , L ∧ . This implies ϕ h,x ∈ B K ( R ) = (cid:8) ϕ ∈ H x (0 , T ; R d ) : k ϕ k H ≤ K ( R ) (cid:9) for any x ∈ ¯ B (0 , R ) and h ∈ (0 , L ∧ K ( R ) > x ∈ ¯ B (0 ,R ) | A ( ϕ h,x ) − B h ( ϕ h,x ) | ≤ sup ϕ ∈ B K ( R ) | A ( ϕ ) − B h ( ϕ ) | ≤ K ( R ) h / ∀ h ∈ (0 , L ∧ . (4.10)Accordingly, for any x ∈ ¯ B (0 , R ) and h ∈ (0 , L ∧ I h ( x ) = inf ϕ ∈ A x B h ( ϕ ) ≤ B h ( ϕ h,x ) ≤ A ( ϕ h,x ) + K ( R ) h / . By the above formula and (4.9), for any x ∈ ¯ B (0 , R ) and h ∈ (0 , L ∧ I h ( x ) < I ( x ) + h + K ( R ) h / ≤ I ( x ) + ( K ( R ) + 1) h / . (4.11)On the other hand, due to the definition of I h , for any x ∈ ¯ B (0 , R ) and h ∈ (0 , L ∧ ψ h,x ∈ A x such that B h ( ψ h,x ) < I h ( x ) + h . Using (4.11) gives B h ( ψ h,x ) | x |≤ R | A ( ϕ h,x ) − B h ( ϕ h,x ) | ≤ K ( R ) h ∀ h ∈ (0 , L ∧ , sup | x |≤ R | A ( ψ h,x ) − B h ( ψ h,x ) | ≤ K ( R ) h ∀ h ∈ (0 , L ∧ . Analogous to the proof of (4.13), we obtainsup | x |≤ R | I h ( x ) − I ( x ) | ≤ K ( R ) h ∀ h ∈ (0 , L ∧ . Therefore the proof is completed. (cid:3)
Remark 4.4.
From the proof of Theorem 4.3, one can observe that the continuity of I is the keyto getting the uniform error between I h and I on a given bounded set. Otherwise, one can onlyget the pointwise convergence, i.e., for any x ∈ R d , there is some constant K ( x ) > such that forsufficiently small h > , | I h ( x ) − I ( x ) | ≤ K ( x ) h / (multiplicative noises) or | I h ( x ) − I ( x ) | ≤ K ( x ) h (additive noises). Application to the small time LDP of SDEs
The small time asymptotics of stochastic systems has received extensive attention in recent years(see e.g., [13, 17, 20] and references therein). This property characterizes the asymptotical behaviorof the underlying stochastic systems with the time tending to zero. In this section, we apply theresults, given in the previous sections, to numerically approximate the small time LDP rate functionsassociated with general Itˆo SDEs. Consider the following Itˆo SDE: ( d Y ( t ) = ˜ b ( Y ( t ))d t + ˜ σ ( Y ( t ))d W ( t ) , t ∈ (0 , ,Y (0) = y ∈ R d , (5.1)where W is the Brownian motion given in the SDE (1.1). Throughout this section, we assume thatfunctions ˜ b : R d → R d and ˜ σ : R d → R d × m are globally Lipschitz continuous. Next we use the smallnoise LDP of SDEs to give the LDP of { Y ( t ) } t> as t tends to 0, through a transform. (One canrefer to [18] for this approach.) Theorem 5.1. { Y ( t ) } t> satisfies the LDP on R d , as t → , with the good rate function I Y givenby I Y ( y ) = inf { ϕ ∈ C y ([0 , , R d ): ϕ (1)= y } J Y ( ϕ ) ∀ y ∈ R d , where J Y : C y ([0 , , R d ) → R is defined by J Y ( ϕ ) = inf { f ∈ L (0 , R m ): ϕ ( t )= y + R t ˜ σ ( ϕ ( s )) f ( s )d s,t ∈ [0 , } Z | f ( t ) | d t ∀ ϕ ∈ C y ([0 , , R d ) . Proof.
Let ˜ X ǫ ( t ) = Y ( ǫt ), t ∈ (0 , ǫ ∈ (0 , X ǫ ( t ) = Y ( ǫt ) = y + Z ǫt ˜ b ( Y ( s ))d s + Z ǫt ˜ σ ( Y ( s ))d W ( s )= y + ǫ Z t ˜ b ( Y ( ǫs ))d s + √ ǫ Z t ˜ σ ( Y ( ǫs ))d f W ( s ) , DP 19 where f W ( s ) = W ( ǫs ) √ ǫ . Noting that Y ( ǫs ) = ˜ X ǫ ( s ), we have˜ X ǫ ( t ) = y + ǫ Z t ˜ b ( ˜ X ǫ ( s ))d s + √ ǫ Z t ˜ σ ( ˜ X ǫ ( s ))d f W ( s ) , t ∈ [0 , . Since f W is still a standard m -dimensional Brownian motion, it follows from [10] that ˜ X ǫ satisfiesthe LDP on C y ([0 , , R d ) with the good rate function e J ( ϕ ) = inf { f ∈ L (0 , R m ): ϕ ( t )= y + R t ˜ σ ( ϕ ( s )) f ( s )d s,t ∈ [0 , } Z | f ( t ) | d t ∀ ϕ ∈ C y ([0 , , R d ) . By Proposition 2.3 and the continuity of the coordinate map ξ , ˜ X ǫ (1) satisfies the LDP on R d , as ǫ →
0, with a good rate function which is nothing but I Y . Due to ˜ X ǫ (1) = Y ( ǫ ), the asymptoticalbehavior of ˜ X ǫ (1) as ǫ → Y ( t ) as t →
0. Hence, we deduce theconclusion. (cid:3)
According to Theorem 2.5, the large deviations rate function I Y of Y ( t ), as t →
0, is that of Z ǫ (1), as ǫ →
0, where Z ǫ is the solution of ( d Z ǫ ( t ) = √ ǫ ˜ σ ( Z ǫ ( t ))d W ( t ) , t ∈ (0 , ,Z ǫ (0) = y ∈ R d . (5.2)Hence, we can apply the conclusions in Sections 2-4 to numerically approximate the rate function I Y . In detail, let { Z ǫn } Nn =0 be the solution of Euler–Maruyama method of (5.2), with h = N , N ∈ N + being the step-size. Then we immediately obtain the following conclusions from the results of theprevious sections. Theorem 5.2.
Assume that d = m and ˜ σ ( x ) is invertible for any x ∈ R d . Then we have thefollowing. (1) For sufficiently small h > , { Z ǫN } ǫ> satisfies the LDP on R d with the good rate function e I h ( y ) = inf { ϕ ∈ H y (0 ,T ; R d ): ϕ (1)= y } Z (cid:12)(cid:12) σ − ( ϕ (ˆ t )) ϕ ′ ( t ) (cid:12)(cid:12) d t. (2) For any
R > , there is some constant K ( R ) > such that for sufficiently small h > , sup | y |≤ R (cid:12)(cid:12)(cid:12) e I h ( y ) − I Y ( y ) (cid:12)(cid:12)(cid:12) ≤ K ( R ) h / . (3) In particular, if σ = I d , then for sufficiently small h > , e I h ( y ) = I Y ( y ) = 12 | y − y | . Proof.
The first conclusion and the second one are the direct corollaries of Theorems 2.8 and 4.3,along with Lemma 3.1. Next, we prove the third conclusion. In fact, since σ = I d , e I h ( y ) = I Y ( y ) =inf { ϕ ∈ H y (0 , R d ): ϕ (1)= y } R | ϕ ′ ( t ) | d t . On one hand, it follows from the Jensen inequality that for any ϕ ∈ H y (0 , R d ) with ϕ (1) = y , Z (cid:12)(cid:12) ϕ ′ ( t ) (cid:12)(cid:12) d t ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ′ ( t )d t (cid:12)(cid:12)(cid:12)(cid:12) = | y − y | . This indicates that I Y ( y ) ≥ | y − y | . On the other hand, define the function ψ by ψ ( t ) = y + ( y − y ) t , t ∈ [0 , ψ ∈ H y (0 , R d ), and ψ (1) = y . Hence we obtain I Y ( y ) ≤ R | ψ ′ ( t ) | d t = | y − y | . This completes the proof. (cid:3) As shown in the above theorem, one can use the Euler–Maruyama method of (5.2) to get thediscrete rate function e I h in the small noise limit, which can approximate well the small time largedeviations rate function of { Y ( t ) } t> . This provides an approach to numerically approximating thesmall time LDP rate functions associated with general nonlinear SDEs.6. Conclusions and future work
In this paper, we propose numerical methods to approximate the large deviations rate function I for the exact solution { X ǫ ( T ) } ǫ> of the nonlinear SDE (1.1) with small noise. We prove that { X ǫN } ǫ> based on the stochastic θ -method satisfies the LDP with the rate function I h . Therigorous convergence analysis of I h is given. Based on these results, we provide an effective methodto numerically approximate the small time large deviations rate functions of nonlinear SDEs. Inpractice, one needs to simulate I h indirectly. Although without being proved, the rate function I h is probably the Legendre transform of the so called logarithmic moment generating function Λ h ( λ ) = lim ǫ → ǫ ln E exp (cid:8) ǫ h X ǫN , λ i (cid:9) , λ ∈ R d , i.e., I h ( x ) = sup λ ∈ R d (cid:8) h x, λ i − Λ h ( λ ) (cid:9) . This happens if { X ǫN } ǫ> satisfies the conditions of G¨artner–Ellis theorem (see e.g., [12, Theorem 2.3.6]). Thus inpractice, one can simulate Λ h to approximate well I h . For the detailed procedure of the simulationof Λ h , we refer readers to [8, 21]. It is interesting to study the statistical errors of calculating I h bysimulating Λ h in the future. Concerning the convergence analysis of I h , we require σ is an invertiblesquare matrix everywhere. We expect to remove this assumption in our future work, e.g., maybeby means of the approximation argument. In addition, we will study the higher-order discrete ratefunctions to approximate I . References [1] G. Biondini.
An introduction to rare event simulation and importance sampling , volume 33 of
Handbook of Statist.
Elsevier/North-Holland, Amsterdam, 2015.[2] M. Bou´e and P. Dupuis. A variational representation for certain functionals of Brownian motion.
Ann. Probab. ,26(4):1641–1659, 1998.[3] C.-E. Br´ehier. Large deviations principle for the adaptive multilevel splitting algorithm in an idealized setting.
ALEA Lat. Am. J. Probab. Math. Stat. , 12(2):717–742, 2015.[4] C.-E. Br´ehier and T. Leli`evre. On a new class of score functions to estimate tail probabilities of some stochasticprocesses with adaptive multilevel splitting.
Chaos , 29(3):033126, 13, 2019.[5] F. C´erou, B. Delyon, A. Guyader, and M. Rousset. On the asymptotic normality of adaptive multilevel splitting.
SIAM/ASA J. Uncertain. Quantif. , 7(1):1–30, 2019.[6] F. C´erou and A. Guyader. Adaptive multilevel splitting for rare event analysis.
Stoch. Anal. Appl. , 25(2):417–443,2007.[7] C. Chen. A symplectic discontinuous galerkin full discretization for stochastic maxwell equations. arXiv:2009.09880 .[8] C. Chen, J. Hong, D. Jin, and L. Sun. Asymptotically-Preserving Large Deviations Principles by StochasticSymplectic Methods for a Linear Stochastic Oscillator.
SIAM J. Numer. Anal. , 59(1):32–59, 2021.[9] X. Chen.
Random walk intersections. Large deviations and related topics , volume 157 of
Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 2010.[10] A. Chiarini and M. Fischer. On large deviations for small noise Itˆo processes.
Adv. in Appl. Probab. , 46(4):1126–1147, 2014.[11] T. Dean and P. Dupuis. Splitting for rare event simulation: a large deviation approach to design and analysis.
Stochastic Process. Appl. , 119(2):562–587, 2009.
DP 21 [12] A. Dembo and O. Zeitouni.
Large deviations techniques and applications , volume 38 of
Stochastic Modelling andApplied Probability . Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.[13] Z. Dong and R. Zhang. On the small-time asymptotics of 3D stochastic primitive equations.
Math. Methods Appl.Sci. , 41(16):6336–6357, 2018.[14] G. Ferr´e and H. Touchette. Adaptive sampling of large deviations.
J. Stat. Phys. , 172(6):1525–1544, 2018.[15] A. Guyader and H. Touchette. Efficient large deviation estimation based on importance sampling.
J. Stat. Phys. ,181(2):551–586, 2020.[16] J. Hong, D. Jin, D. Sheng, and L. Sun. Numerically asymptotical preservation of the large deviations principlesfor invariant measures of langevin equations. arXiv:2009.13336 .[17] S. Li, W. Liu, and Y. Xie. Small time asymptotics for SPDEs with locally monotone coefficients.
Discrete Contin.Dyn. Syst. Ser. B , 25(12):4801–4822, 2020.[18] Y. Lin and L.-C. Tsai. Short time large deviations of the kpz equation. arXiv:2009.10787 .[19] A. Quarteroni and A. Valli.
Numerical approximation of partial differential equations , volume 23 of
SpringerSeries in Computational Mathematics . Springer-Verlag, Berlin, 1994.[20] M. R¨ockner and T. Zhang. Stochastic 3D tamed Navier-Stokes equations: existence, uniqueness and small timelarge deviation principles.
J. Differential Equations , 252(1):716–744, 2012.[21] C. M. Rohwer, F. Angeletti, and H. Touchette. Convergence of large-deviation estimators.
Phys. Rev. E ,92:052104, 2015.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email address : [email protected] Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email address : [email protected] (Corresponding author) Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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