Diffusion Processes via Parabolic Equations: an Infinitesimal Approach to Lindeberg's Limit Theorem

We approach infinitesimal diffusion processes via a linkage to the diffusion equation. By this we obtain Lindeberg's limit theorem and a Lindeberg type limit theorem for diffusion processes by an application of the underspill principle.


Introduction
The evolution of diffusions is in standard mathematics described either by ordinary stochastic differential equations, or by partial differential equations, called diffusion equations.Both descriptions are usually connected via the Ito formula.(See [20] Section 5.2, [31] Sections 5.1-5.3,[28] Section 7.3, [18] Theorem 4.8.6., [32] Section 6, [14] Section 1.8 and [16] Section 5.1.)Nonstandard diffusion theory is usually approached by linking the nonstandard stochastic process defined on a near interval by the concept of Loeb measure [25] to a corresponding standard stochastic process, as described in [8] or [35].This approach to diffusion theory started with a nonstandard construction of the Brownian motion and the Ito integral in [1] and was further extended to stochastic integration in a broader context in [22] and applied to the analysis of the Ornstein-Uhlenbeck Process [23] and the Malliavin Calculus [9] (see also [24]).Some further important developments concerning Loeb measures on Hyperfinite spaces can be found in [12].Their abstraction leads to the concept of neocompactness that can be used to prove the existence of solutions of stochastic differential equations with special properties [17].
Another way of linking nonstandard stochastic processes to the standard mathematical world -by Nelson's reduction algorithm -is described in [27].A third possibility [3] is to use hyper-finite combinatorics and the concept of equivalent processes ( [27] Chapter 17) together with path-wise versions of Ito's formula and Girsanov's theorem.
We approach diffusions via infinitesimal stochastic difference equations and call the solutions of these equations infinitesimal diffusion processes.To connect the infinitesimal diffusion processes to the world of standard mathematics we employ Kolmogorov's backward equation.
By this connection we further prove that the standard parts of the expectations of the infinitesimal diffusion processes are independent of the choice of the underlying infinitesimal model of white noise.This fact can be interpreted as an infinitesimal version of a Lindeberg type limit theorem.By the underspill-principle of nonstandard analysis this enables us to prove a Lindeberg type limit theorem for diffusions in standard mathematical terms.This limit theorem is contrasted by various approximation theorems provided in [18].
Note that the relation between nonstandard diffusion processes and the diffusion equation has -in a different way -already been investigated in [4].Another, similar connection between a stochastic differential equation and a partial differential equation is the connection between the Navier-Stokes equation and the Foias ¸equation.Investigation on this connection by nonstandard methods was done in [6] and [7].
Diffusion processes Y : Ω × [t 0 , T] → R with initial state x 0 are in the standard mathematical literature described as solutions of stochastic differential equations ( 1) (with B t denoting Brownian motion).Note that the existence of the solution of a stochastic differential equation presupposes in the Ito interpretation ([28] Section 5.1) the notion of stochastic integration.Equation ( 1) is in the Ito interpretation just an abbreviation of (See also [31] Section 3.4 or [16] Sections 5.2 and 5.3.) In our approach we investigate the solution Here (δW t ) t∈[ t 0 ...T ) denotes an arbitrary infinitesimal model of white noise (see Definition 3.11).Note that our approach does not rely on the concept of Ito integration.We also do not use martingale arguments, Ito's formula, Girsanov's theorem, Nelson's reduction algorithm, Loeb measures, combinatorics or the Fourier-Laplace transform.
The existence of solutions u constitutes the starting point of our investigation.Sufficient regularity conditions on the functions a, σ and f that grant the existence of a solution 4 of Kolmogorov's backward equation are provided in [34] Theorem 3.2.1 and [18] Theorem 4.8.6.An analytic existence proof can be found in [13] Section 6.
Given a random variable Z we denote its expectation by E[Z].The solution u(t, x) of Kolmogorov's backward equation is (following [31] Section 5.1) linked to (1) via ( 4) We prove (see Theorem 4.13), that the solution X of (2) fulfills The closeness of E[f • X T ] and u(x 0 , t 0 ) expressed by ( 5) is the starting point for all further investigations.It enables us to prove a Lindeberg type limit theorem for discrete time processes that approximate diffusion processes.It further shows together with ( 4) that and the solution u of (3) exist). 4If u(t, x) is a solution of the terminal value problem (3), then v(t, x) We use the notation provided by internal theories to formulate our mathematical results.The notation can be obtained form the st-∈-languages IST or BST (see [26], [10], [15]).We suppose that the reader is familiar with the notion of standard, internal and external formula and with the principles of transfer, idealization and standardization and some elementary consequences of these notions and principles (see [26], [10]).
We denote by st or st(.) the unary predicate standard.We denote by N and R the set of all natural numbers {0, 1, 2, . . .} and the set of all real numbers, respectively, i.e., the sets N and R contain standard as well as nonstandard elements.We use the term "set" synonymously with the term "internal set".We say that a set S is countable if S is either finite or possesses the cardinality of N.
Notation 2.1 Let (X, .) be a normed space.We say that x ∈ (X, .) is limited and write x << +∞ if ∃ st n ∈ N such that x < n; otherwise we say that x is unlimited.In the case that (X, .) = (R, |.|) we also write −∞ << x << +∞ instead of x << +∞.For positive unlimited r ∈ (R, |.|) we write r ≈ ∞.We say that x ∈ (X, .) is infinitely small or infinitesimal if ∀ st ε > 0 x < ε.If x − x is infinitely small we write x ≈ x .Thus if x is infinitely small we write x ≈ 0. Note that all the concepts introduced above are external.
Notation 2.2 We further introduce a symbol .It is used as a replacement character for a non explicitly stated infinitesimal quantity.Let F(ξ) and G(ξ) denote functions of a variable ξ that are possibly constant in ξ .We define The symbol is used in the same manner if the character ≤ in ( 6) is replaced by the character =.For example let F(ξ) be defined by Further our definition implies that since this is just an abbreviation of (∀ st ε)(∀o ≈ 0) (o ≤ ε).We also note that For the use of the symbol as an external number (differing from our use) see [19].
The following proposition is an elementary consequence of idealization.We state it, since we are going to use it explicitly in the proofs of Lemma 4.7, Theorem 5.2 and Lemma A.2 (and thus implicitly in the proofs of Theorem 4.13, Theorem 5.4 and Example 6.6).
Proposition 2.3 Let φ be an internal formula.Then For convenience of the reader we state the definitions of the important concepts of S-convergence and uniform S-continuity.
Definition 2.4 A sequence is a function defined on N. Let (Y, .) be a normed space.We say that the sequence Let Z be a subset of a normed space (X, .).We say that a function f : (We use the term "uniformly S-continuous" since "S-continuity" has been used in different ways, i.e., compare with [29] We further use the following notations for near intervals: [ t 0 . . .

Some probabilistic concepts
For the purpose of simplicity we suppose that the random variables under consideration are defined on a countable set Ω, endowed with a probability P determined point wise by its values P({ω}), i.e., P is defined on the power set P(Ω) of Ω, P is countably additive and thus determined by its values P({ω}) on {{ω} | ω ∈ Ω}.
Let (X, A) be an arbitrary measurable space.We note that any internal function f : (Ω, P(Ω)) → (X, A) is measurable.Thus our restriction to countable Ω relieves us from measurability arguments.However there is a close relation between internal and Loeb-measurable mappings for measure spaces [8], [21] and stochastic processes [12]).

Distribution and Convergence of Random Variables
Definition 3.1 We denote by C 0 the space of all continuous functions f : R → R with compact support.For n ∈ N ∪ {∞} we denote by C n the space of all n-times continuously differentiable functions f : R → R. We further let C n 0 := C n ∩ C 0 .Note that f ∈ C n 0 implies that all derivatives of f up to order n are uniformly continuous.
where f (y) dQ(y) denotes the Riemann-Stieltjes integral of f with respect to Q and Proof: All we have to show is, that (8) implies (7).By the Stone-Weierstrass-Theorem and transfer any standard f ∈ C 0 can be uniformly approximated up to ε 3 for an arbitrary standard ε > 0 by a standard function g ∈ C ∞ 0 .Thus (8) implies ( 9) Since ε > 0 was assumed to be standard but otherwise arbitrary and f ∈ C 0 was an arbitrary standard function, we obtain that ( 9) implies (7). 2 Proof: The proof of Proposition 3.6 is analogous to the proof of Proposition 3.4.2 Proposition 3.7 Let a sequence (Y i ) i∈N of almost limited random variables Y i and a standard sequence if and only if Proof: (10) holds by transfer if and only if thus further by Remark 2.5 if and only if and further by Definition 3.3 if and only if we obtain that X and Y are nearly equivalent if and only if This relates the concept of near equivalence to the concept of approximate S-convergence and further shows that for almost limited X the random variables X and Y are nearly equivalent if and only if they possess a common distribution Q.

Stochastic processes
Notation 3.10 For convenience of notation we switch freely between the following formulations of the concept of a stochastic process X with time [ t 0 . . .T ] and state space M : Let x 0 ∈ R be fixed.Let a stochastic process X : Ω × [ t 0 . . .T ] → R be recursively defined by X t 0 (ω) := x 0 and X t+δt (ω) := [K(t, X t (ω))](ω).
We let δX t := X t+δt − X t and say that the process X is the solution of the infinitesimal stochastic difference equation with initial condition X t 0 = x 0 .

Diffusion Equations and Processes
Definition 4.1 Given a function u if u is a bounded uniformly continuous function such that all its derivatives up to order m with respect to its first variable and all its derivatives up to order n with respect to its second variable are bounded and uniformly continuous (as functions of both variables).
Provided that the respective derivatives exist, we introduce the following notations: Notation 4.2 Let ψ : R → R. We denote by ψ (x) the first order derivative of ψ at x and by ψ (x) the second order derivative of ψ at x.Given a function u : [t 0 , T] × R → R, we denote the partial derivative of u with respect to its first variable at (t, x) by ∂u(s, x) ∂s s=t resp.for simplicity of notation by ∂u(t, x) ∂t .
We denote the first and second order derivatives of u with respect to its second variable by ∂u(t, y) ∂y y=x and ∂ 2 u(t, y) ∂ 2 y y=x .
resp. for simplicity of notation by Proposition 4.3 Let x ∈ R and let v : R → R be a function such that v(x) = 0 and v (x) = 0. Suppose further that x → v (x) is S-continuous and limited.Then and call R x v the remainder of v at x. Definition 4.5 We denote by (L t : C 2 → C 0 ) t∈[t 0 ,T] a standard indexed family of differential operators of the form There exists an infinitesimal η > 0 such that (iii) Then by ( 13) and ( 15) Lemma 4.7 Let (L t ) t∈[t 0 ,T] be a standard indexed family of differential operators that fulfills the hypotheses of Definition 4.5.Let f : R → R be a standard bounded C 2 function.Suppose that there exists a C 1,2 b solution u : [t 0 , T] × R → R of the terminal value problem (18) [L t u(t, .)](x) Let Remark 4.8 Note that by standardness of (L t : C 2 → C 0 ) t∈[t 0 ,T] it follows that b(., .)and a(., .)are standard functions and that [t 0 , T] is a standard interval.
Proof: By standardness of (L t ) t∈[t 0 ,T] and f , and by the existence of a continuous solution u of (18), we obtain by transfer that there exists a continuous standard solution u of (18).For a given family (Z Part I: Note that by Definition 4.4 We further remark that By the hypothesis that u ∈ C 1,2 b and the fact that u(., .) is standard by Corollary 4.9, there exists a standard ρ ∈ R such that ( 24) and for any t ∈ [t 0 , T] we have that (25) x → u t (x) is uniformly S-continuous.
Note that by ( 22), ( 23) and ( 24) we obtain Part II: We show that for x ∈ R and t ∈ [ t 0 . . .T ) To do this we have to show that for x ∈ R and t ∈ [ t 0 . . .T ) Since the proofs of ( 30) and ( 31) are completely analogous, we just prove (30) and regard (31) and thus further (28) as being proved.

Heinz Weisshaupt
Let now x ∈ R and t ∈ [ t 0 . . .T ) be arbitrarily given.Then The equality (a) follows from an application of (21), while (b) is obtained by splitting E[[R x u t+δt ](x + Z t,x )] into two integrals and by an application of Definition 4.5 (i).The inequality (c) follows from an application of (26), and by ( 22), ( 23), Proposition 4.3 and ( 25).The equality (d) follows by an application of ( 13) and ( 15) while equality (e) is obtained by (16).We further obtain (f) by the use of ( 17) and (g) by application of ( 16) and (24).The relation (h) is obtained by calculation and rearrangement of terms, while (i) follows from the fact that c ≈ ∞ and the definition of L.
Part III: We show that Since we assumed that u ∈ C 1,2 b we have that t → u t (x), t → u t (x) are uniformly S-continuous mappings and thus we obtain that 28) and ( 33) we obtain for x ∈ R and t ∈ [ t 0 . . .T ) that From (18) and From ( 27), (34) and (35) we obtain for x ∈ R and t ∈ [ t 0 . . .T ) that i.e., (32) has been proved.

Part IV: We prove now by backward induction on
Since (36) is external we can not apply (internal) induction directly to (36).We therefore replace (36) by an external family of formulas as follows: For any standard ε > 0 we consider
To complete the proof of (ε) by backward induction we just have to show that if (ε) holds true for s = t + δt, then (ε) holds true for s = t.Thus suppose that (ε) holds for s = t + δt.From ( 19), (32) and the validity of (ε) in the case s = t + δt we obtain that for x ∈ R i.e., holds.Thus application of (internal) induction to (37) shows that (ε) holds for any s ∈ [ t 0 . . .T ].Since the ε in our consideration was assumed to be standard but otherwise arbitrary, we obtain from Proposition 2.3 that (36) holds for all s ∈ [ t 0 . . .T ]. (36) implies by ( 14) and ( 29) that (20) holds. 2 Theorem 4.13 Let (L t ) t∈[t 0 ,T] be a standard indexed family of differential operators that fulfills the hypotheses of Definition 4.5.Let f : R → R be a standard bounded C 2 function.Suppose that there exists a C 1,2 b solution u : [t 0 , T] × R → R of the terminal value problem Let [ t 0 . . .T ] be a near interval and let (δW t ) t∈[ t 0 ...T ) be a model of white noise.Let σ : [t 0 , T) × R → R be such that σ 2 (., .)≈ b(., .) and let ã ≈ a.Let X be the solution of the infinitesimal stochastic difference equation with initial condition X t 0 = x 0 .Then From the definition of û by (19) we obtain that Induction over (39) and u(T, x) = f (x) gives Formula (40) gives together with the conclusion of Lemma 4.7 the approximate identity (38). 2 Remark 4.14 Note that boundedness (and standardness) have been the only hypotheses imposed on the functions a and b.Our proofs worked out mainly because of the strong hypotheses that the solution u of ( 18) is a (standard) element of C 1,2 b .
Remark 4.15 Since a, b, t 0 and T are (by boundedness and standardness) limited, we obtain that the random variables X t -that constitute the stochastic processes under consideration -fulfill −∞ << E[X t ], Var(X t ) << ∞.From this we obtain further by [5] Theorem 2.11 that the random variables X T are almost limited.
Heinz Weisshaupt 5 Lindeberg's Theorem and related results We consider now equation (18) in the case that b(t, x) = 1, a(t, x) = 0 and f ∈ C 2 0 , i.e., we consider the following special case of the terminal value problem ( 18): A solution of the terminal value problem (41) on [t 0 , T] × R is provided by the function u given by ( 42) If φ denotes the density function of the N(0, 1) distribution, then also Definition 5.1 Let Ξ = (ξ i,j ) (i,j)∈N×N be an array of random variables ξ i,j ∈ R.
Suppose that there exists a function i → J i from N to N such that lim i→∞ J i = ∞ and that Var(ξ i,j ) = 0 ≡ ξ i,j if and only if j ≥ J i .Suppose further that for any fixed i the vector (ξ i,j )) j∈N consists of independent random variables.We call Ξ a triangular array of independent random variables with sum-variables ζ i := 0≤j≤Ji−1 ξ i,j = j∈N ξ i,j .
The following theorem is known as Lindeberg's limit theorem.(Compare with [11] Section 9.6.) Theorem 5.2 Let Ξ be a triangular array of independent random variables with sumvariables ζ i .Suppose that E[ξ i,j ] = 0 for all i, j ∈ N. Suppose further that and that Then the sequence of distributions of (ζ i ) i∈N converges weakly to N(0, σ 2 ).
Proof: The theorem stated above is a theorem in the terms of standard mathematics.We can therefore by transfer suppose that the theorem is stated with the addition that any object named in the theorem is standard.Especially we can replace (44) by From (46) we obtain by Proposition 2.3 that From (43) we obtain by standardness of (ζ i ) i∈N and Remark 2.5 that Choose an arbitrary nonstandard i ∈ N. Let for j ∈ {0, . . ., J i − 1} increments δt j be given, such that and define points t j by (48) t 0 := 0, t j+1 := t j + δt j , T : on the interval [t 0 , T] is for t ∈ [t 0 , T) provided by the function Proof: Straightforward calculation shows that (59) is a solution of (58).Thus there exists a one-one correspondence between the solutions of (58) and (60).
Proof: This follows with y =  We show that the infinitesimal Lindeberg condition displayed by equation ( 12) of this article is equivalent with the near Lindeberg condition given by equation (14.1) of [27].

Definition 3 . 2 (
Compare with [27] Chapter 7.) We say that a random variable Y is almost limited if for any standard ε > 0 there exists a standard n ∈ N such that P(Y ∈ [−n, n]) > 1 − ε.Definition 3.3 Let Y : Ω → R be an almost limited random variable and let Q : R → [0, 1] be a monotone increasing function such that 0
[2]]finally by Proposition 3.4 if and only if (11) holds. 2 Remark 3.8 For the definition of weak convergence see[30]definition B.80.For results on weak convergence in the setting of Loeb measure spaces consult[2].
[27]rk 3.9 We may define (analogous to[27]Chapter 17) that two random variables X, Y : Ω → R are nearly equivalent if for all limited, S-continuous h : R → R we are given thatE[h • X] ≈ E[h • Y].Suppose that X : Ω → R is an almost limited random variable and let Y be a second random variable.By the fact that for any limited, S-continuous h : R → R, any standard ε > 0 and any standard n ∈ N there exists a standard [27]nition 3.11Let [ t 0 ...T ] be a near interval and let (δW t ) t∈[ t 0 ...T ) denote an indexed family of independent random variables δW t : Ω → R with laws ν t such thatE[δW t ] = • δt , Var(δW t ) = (1 + ) • δtWe say that (δW t ) t∈[ t 0 ...T ) is a model of white noise and that W is a model of Brownian motion.Remark 3.12 Note that (12) is an infinitesimal formulation of the Lindeberg condition.It is equivalent with the near Lindeberg condition (14.1) in[27].A precise statement of the equivalence together with a proof is provided in the appendix of this article.
[33]rk 4.10 Note that uniqueness results for solutions of diffusion equations (compare with[33]sections 11.3 and 11.5) are non trivial.
t 0 ).Proof: Let us denote the conditional expectation of a random variable Y with respect to a second random variable Z by E[Y | Z].We apply Lemma 4.7 with Z t,x := ã(t, x) δt + σ(t, x) δW t .
T) is a model of white noise.Let W be the model of Brownian motion associated with (δW t ) t∈[ t 0 ...T ) .Then t with a(t, x) = 0 and b(t, x) = 1.Thus by Theorem 4.13 we have that for any standard f ∈ C 2 0 Corollary 6.5 There exists a unique C 1,2 b solutions of the terminal value problem (60).
(12)ly of random variables (δW t ) t∈[ t 0 ...T ) fulfills the near Lindeberg condition if Suppose that we are given a family of real valued random variables (δW t ) t∈[ t 0 ...T ) with distributions ν t .Then the infinitesimal Lindeberg condition(12)and the near Lindeberg condition (61) are equivalent.