Stochastic Dynamical System for SDE driven by Nonlinear Brownian Motion

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1 Stochastic Dynamical System for SDE driven by Nonlinear Brownian Motion Peng Shige Shandong University, China School of Stochastic Dynamical Systems and Ergodicity Loughborough University, Dec

2 Stochastic Dynamical System for SDE driven by Nonlinear Brownian Motion Peng Shige Shandong University, China School of Stochastic Dynamical Systems and Ergodicity Loughborough University, Dec

3 Optimal Unbiased Estimation for Maximal Distribution 1. Preliminary: Sublinear expectation space (Ω, H, Ê) Stochastic Dynamical System fo Let Ω be a given set and let H be a linear space of real valued functions defined on Ω. We assume c H, c R, X H, if X H

4 Sublinear expectation space (Ω, H, Ê) Ê[λX ] = λê[x ] for λ 0. Stochastic Dynamical System fo Definition A Sublinear expectation (i) Monotonicity: Ê is a functional E : H R satisfying Ê[X ] Ê[Y ] if X Y, (ii) Constant preserving: Ê[c] = c for c R, (iii) Sub-additivity: For each X, Y H, Ê[X + Y ] Ê[X ] + Ê[Y ], (iv) Positive homogeneity:

5 (Ω, H, Ê) is called a sublinear expectation space A sublinear expectation Ê[ ] defined on (Ω, H) is said to be regular if Ê[X i ] 0 for each sequence {X i } i=1 of random variables in H such that X i (ω) 0 for each ω Ω. If (i) and (ii) are satisfied, Ê is called a nonlinear expectation and the triple (Ω, H, Ê) is called a nonlinear expectation space

6 1. Sublinear Expectations Stochastic Dynamical System fo Ω: a given set H: a linear space of real valued functions defined on Ω, s.t. a) c H for each constant c, b) X H = X H

7 (Ω, H, E): a sublinear expectation space. Stochastic Dynamical System fo Proof. [Definition 1.1] A Sublinear expectation E is a functional E : H R satisfying Monotonicity: Cash translatability: E[X ] E[Y ] if X Y. E[X + c] = E[X ] + c for c R. Sub-additivity: For each X, Y H, Positive homogeneity: E[X + Y ] E[X ] + E[Y ]. E[λX ] = λe[x ] for λ 0.

8 Proof. [Definition 1.2] Let E 1 and E 2 be two nonlinear expectations defined on (Ω, H). E 1 is said to be dominated by E 2 if E 1 [X ] E 1 [Y ] E 2 [X Y ] for X, Y H. (1.1)

9 Proof. [Remark. 1.4] (iii)+(iv) is called sublinearity. This sublinearity implies (v) Convexity: E[αX + (1 α)y ] αe[x ] + (1 α)e[y ] for α [0, 1]. If a nonlinear expectation E satisfies convexity, we call it a convex expectation.

10 2. Representation of a Sublinear Expectation Stochastic Dynamical System fo Theorem (THM 2.1) Let E a sublinear functional defined on (Ω, H). Then a family of linear functionals {E θ : θ Θ} on (Ω, H) s. t. E[X ] = max θ Θ E θ[x ] for X H Furthermore, if E is a sublinear expectation, then each E θ is a linear expectation.

11 3. Distributions, Independence and Product Spaces Stochastic Dynamical System fo Definition Let X 1 and X 2 be two n dim. random vectors in a nonlinear expectation spaces (Ω, H, E). They are called identically distributed, denoted by X 1 d = X2, if E 1 [ϕ(x 1 )] = E 2 [ϕ(x 2 )] for ϕ C Lip (R n ).

12 Lemma Let (Ω, H, E) be a sublinear expectation space and let F X [ϕ] := E[ϕ(X )] be the sublinear distribution of X H d. Then there exists a family of probability measures {F θ } θ Θ defined on (R d, B(R d )) such that ˆ F X [ϕ] = sup ϕ(x)f θ (dx), ϕ C l,lip (R d ). (1) θ Θ R d

13 A simple and useful property Stochastic Dynamical System fo Proposition. Let (Ω, H, E) be a sublinear expectation space and X, Y be two random variables such that E[Y ] = E[ Y ], i.e., Y has no mean-uncertainty. Then we have E[X + αy ] = E[X ] + αe[y ] for α R. In particular, if E[Y ] = E[ Y ] = 0, then E[X + αy ] = E[X ].

14 Proof. We have E[αY ] = α + E[Y ] + α E[ Y ] = α + E[Y ] α E[Y ] = αe[y ] for α R Thus E[X + αy ] E[X ] + E[αY ] = E[X ] + αe[y ] E.Q. = E[X ] E[ αy ] E[X + αy ].

15 A more general form of the above proposition is: Proposition. We make the same assumptions as the previous proposition. Let Ẽ be a nonlinear expectation on (Ω, H) dominated by the sublinear expectation E. If E[Y ] = E[ Y ], then we have as well as In particular Ẽ[αY ] = αẽ[y ] = αe[y ], α R (2) Ẽ[X + αy ] = Ẽ[X ] + αẽ[y ], X H, α R. (3) Ẽ[X + c] = Ẽ[X ] + c, for c R. (4)

16 Definition A sequence of n-dimensional random vectors {η i } i=1 defined on a sublinear expectation space (Ω, H, E) is said to converge in distribution (or converge in law) under E if for each ϕ C b.lip (R n ), the sequence {E[ϕ(η i )]} i=1 converges.

17 The following notion of independence plays a key role in the nonlinear expectation theory. Definition In a nonlinear expectation space (Ω, H, E), a random vector Y H n is said to be independent from another random vector X H m under E[ ] if for each test function ϕ C Lip (R m+n ) we have E[ϕ(X, Y )] = E[E[ϕ(x, Y )] x=x ]. Remark. Y is independent from X does not implies X is independent from Y.

18 1. Maximal distribution and nonlinear normal Distribution Stochastic Dynamical System fo Definition A d-dimensional random vector η = (η 1,, η d ) on (Ω, H, E) is called maximal distributed if there exists a bounded, closed and convex subset Γ R d such that E[ϕ(η)] = max y Γ ϕ(y).

19 Remark. A maximal distributed random vector η satisfies aη + b η d = (a + b)η for a, b 0, where η is an independent copy of η.

20 Remark. When d = 1 we have Γ = [µ, µ], where µ = E[η] and µ = E[ η]. The distribution of η is F η [ϕ] = E[ϕ(η)] = sup ϕ(y) µ y µ for ϕ C Lip (R).

21 Law of Large Numbers g(p) := E[ p, Y 1 Stochastic ], p R d. Dynamical System fo Theorem (Law of large numbers) Let {Y i } i=1 be a sequence of real valued i.i.d. random sequince on a sublinear expectation space Ω, H, E) such that Ê[( Y 1 c) + ] c 0. Then { S n } n=1 defined by S n := 1 n n Y i i=1 converges in law to a maximal distribution: lim E[ϕ( S n )] = max ϕ(y), (5) n y [µ,µ] for all ϕ C (R), where µ = Ê[ Y 1 ], µ = Ê[Y 1 ].

22 Let X 1,, X n be n copies of the same maximal distribution X i d = M[µ,µ], i = 1,, n. with unknown parameters µ µ, and X i is independent of {X j } j=1,,i 1. In short, we say our X 1,, X n are i.i.d.. It is clear that maximum distribution is completely determined by these two parameters. It is then important to construct statistics to estimate parameters µ and µ properly.

23 It is natural to think of the following statistics: µ (X 1,, X n ) := max{x 1,, X n }, (6) µ (X 1,, X n ) := min{x 1,, X n }. (7)

24 Like in the classical statistical theory, we want our estimator to be unbiased in some sense, which should be redefined properly. Definition Let f n C (R n ), a statistic T n = f n (X 1,, X n ) is called an unbiased estimator of µ (resp. for µ) if Ê[f n (X 1,, X n )] = µ (resp. Ê[ f n (X 1,, X n )] = µ), for all < µ µ <.

25 We have the following lemma. Lemma If f n C Lip (R n ), the estimator T n = f n (X 1, X 2,, X n ) is unbiased for the upper mean µ, then for all µ µ, we have, max f n(x 1,, x n ) = µ, (8) (x 1,,x n ) [µ,µ] n (resp. min f n(x 1,, x n ) = µ) (9) (x 1,,x n ) [µ,µ] n Consequently, for all µ µ, and (x 1,, x n ) [µ, µ] n, f n (x 1,, x n ) µ, (10) (resp. f n (x 1,, x n ) µ). (11)

26 Proof. Since { Ê[f n (X 1,, X n )] = Ê = Ê [ =... = max Ê[ max µ x n µ f n(x 1,, x n )] } max f n(x 1,, X n 1, x n ) µ x n µ 1 i n ] max f n(x 1,, x n ). µ x i µ Denote f n 1 (x 1,, x n 1 ) := max µ xn µ f n (x 1,, x n 1, x n ), then f n 1 (a 1,, a n 1 ) f n 1 (b 1,, b n 1 ) max f n(a 1,, a n 1, x n ) f n (b 1,, b n 1, x n ) x n [µ, µ] max L (a 1,, a n 1 ) (b 1,, b n 1 ) x n [µ, µ] = L (a 1,, a n 1 ) (b 1,, b n 1 ), x 1 =X 1,,x n 1 =X n which means f n 1 C Lip (R n 1 ), hence we can continue our equality Stochastic Dynamical System fo

27 Ê[f n (X 1,, X n )] = Ê[f n 1 (X 1,, X n 1 )] = Ê[f 1 (X 1 )] = max x 1 [µ, µ] f 1(x 1 ) = max x 1 [µ, µ] max f 2(x 1, x 2 ) x 2 [µ, µ] = max (x 1,x 2 ) [µ, µ] 2 f 2(x 1, x 2 ) = max (x 1,,x n ) [µ, µ] n f n(x 1,, x n ). Then we have (8) and thus (10). The proof of (9) and (11) are similar. E.Q.

28 is the smallest unbiased estimator for the lower mean µ. Stochastic Dynamical System fo Now, we present our main result. Theorem Let X 1,, X n be i.i.d. sample of size n from the population of maximal distribution X i d = M[µ,µ], i = 1,, n, with unknown parameters µ µ. Then we have, quasi surely (i.e., P θ -almost surely for any θ Θ), µ min{x 1 (ω),, X n (ω)} max{x 1 (ω),, X n (ω)} µ. Moreover, µ n = max{x 1,, X n } is the largest unbiased estimator for the upper mean µ, µ n = min{x 1,, X n }

29 PROOF It is easy to check that µ = max{x 1,, X n } is an unbiased estimator for the unknown upper mean µ and µ = min{x 1,, X n } is an unbiased estimator for the unknown lower mean µ. Stochastic Dynamical System fo

30 Let T n = f n (Y 1,, Y n ) be a given unbiased estimator for the upper mean µ. For any y 1,, y n R, we set µ = max{y 1,, y n }, µ = min{y 1,, y n }, d and consider the case Y i = M[ µ, µ]. According to Lemma 9, the unbiased estimator T n = f n (Y 1,, Y n ), f n must satisfy (10), namely, f n (y 1,, y n ) µ = max{y 1,, y n }. Since y 1,, y n can be arbitrarily chosen, we then have f n (y 1,, y n ) max{y 1,, y n }, y 1,, y n R. Thus µ n is the largest estimator for the upper mean. We can prove that µ n is the smallest estimator for the lower mean. [ENDPF]

31 The next proposition tell us that the two estimators are both maximal distributed with the same lower mean µ and upper mean µ. Proposition. Let X 1,, X n be a maximal distributed i.i.d. sample with X 1 = M [µ,µ]. Then max{x 1 (ω),, X n (ω)} d = min{x 1 (ω),, X n (ω)} d = M [µ,µ].

32 Proof. It is clear that, for each ϕ C (R), we have Ê[ϕ(max{X 1,, X n })] = max (x 1,,x n ) [µ, µ] n ϕ(x 1 x 2 x n ) = max x [µ, µ] ϕ(x) = Ê[ϕ(X 1 )]. Thus max{x 1,, X n } d = M [µ,µ]. Similarly we can prove that min{x 1,, X n } d = M [µ,µ].

33 Remark. In fact, we can prove that, for continuous function f C (R n ), we have f (X 1,, X n ) d = M [µf,µ f ] (12) where µ f := max (x 1,,x n ) [µ, µ] n f (x 1,, x n ), µ f := min (x 1,,x n ) [µ, µ] n f (x 1,, x n ). Indeed, for each ϕ C (R), Ê[ϕ(f (X 1,, X n ))] = which implies (12). max ϕ(f (x 1,, x n )) = max ϕ(y), (x 1,,x n ) [µ, µ] n y [µ,µ f f ]