A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION TRISTRAM DE PIRO Abstract. We derive a nonstandard version of the Fokker-Planck equation. Definition 0.1. We let v N \ N, and η =. We let Ω η = {x R : 0 x < 1} and T = {t R : 0 x 1} We let C η consist of internal unions of the intervals [ i, i+1 ), for η η 0 i η 1, and let D consist of internal unions of [ i, i+1 ) and {1}, for 0 i 1. We define counting measures µ η and λ on C η and D respectively, by setting µ η ([ i, i+1)) = 1, λ η η η ([ i, i+1)) = 1, for 0 i η 1, 0 i 1 respectively, and λ ({1}) = 0. We let (Ω η, C η, µ η ) and (T, D, λ ) be the resulting -finite measure spaces, in the sense of []. Definition 0.. We let V (C η ) = {f : Ω η C, f(x) = f( [ηx] )} and η W (C η ) V (C η ) be the set of measurable functions f : Ω η C, with respect to C η, in the sense of []. Lemma 0.3. W (C η ) is a -finite vector space over C, of dimension η, ( 1 ). 1 By a -vector space, one means an internal set V, for which the operations + : V V V of addition and scalar multiplication. : C V V are internal. Such spaces have the property that -finite linear combinations Σ i I λ i.v i, ( ), for a -finite index set I, belong to V, by transfer of the corresponding standard result for vector spaces. We say that V is a -finite vector space, if there exists a -finite index set I and elements {v i : i I} such that every v V can be written as a combination ( ), and the elements {v i : i I} are independent, in the sense that if ( ) = 0, then each λ i = 0. It is clear, by transfer of the corresponding result for 1

TRISTRAM DE PIRO Proof. This follows straightforwardly, by transfer, from the corresponding result for C-valued functions on finite sets. Namely, any C-valued f : N [1, n] C can be written as n i=1 λ ie i, where e i (j) = δ ij, for 1 j n, and λ i = f(i), for 1 i n. Definition 0.4. Given n N >0, we let Ω n = {m N : 0 m < n }, and let C n be the set of sequences of length n, consisting of 1 s and 1 s. We let θ n : Ω n N n be the map which associates m Ω n with its binary representation, and let φ n : Ω n C n be the composition φ n = (γ θ n ), where, for m N n, γ( m) =. m 1. For N \ N, we let φ : Ω C be the map, obtained by transfer of φ n, which associates i N, 0 i <, with an internal sequence of length, consisting of 1 s and 1 s. Similarly, for η =, we let ψ η : Ω η C be defined by ψ η (x) = φ ([ηx]). For 1 j, we let ω j : C {1, 1} be the internal projection map onto the j th coordinate, and let ω j : Ω η {1, 1} also denote the composition (ω j ψ η ), so that ω j W (Ω η ). By convention, we set ω 0 = 1. For an internal sequence t C, we let ω t : Ω η {1, 1} be the internal function defined by; ω t = 1 j ω t(j)+1 j Again, it is clear that ω t W (Ω η ). Lemma 0.5. The functions {ω j : 1 j } are -independent in the sense of [1], (Definition 19), in particular they are orthogonal with respect to the measure µ η. Moreover, the functions {ω t : t C } form an orthogonal basis of V (Ω η ), and, if t 1, E η (ω t ) = 0, and V ar η (ω t ) = 1, where, E η and V ar η are the expectation and variance corresponding to the measure µ η. Proof. According to the definition, we need to verify that for an internal index set J = {j 1,..., j s } {1,..., }, and an internal tuple (α 1,..., α s ) C s, where s = J ; µ η (x : ω j1 (x) < α 1,..., ω jk (x) < α k,..., ω js (x) < α s ) = s k=1 µ η(x : ω jk (x) < α k ) ( ) finite dimensional vector space over C, that V has a well defined dimension given by Card(I), see [3], even though V may be infinite dimensional, considered as a standard vector space.

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 3 Without loss of generality, we can assume that each α jk > 1, as if some α jk 1, both sides of ( ) are equal to zero. Let J = {j J : 1 < α j 1} and J = {j J : 1 < α j }, so J = J J. Then; µ η (x : ω j1 (x) < α 1,..., ω js (x) < α s ) = 1 η Card(z C : z(j ) = 1 for j J, z(j ) { 1, 1} for j J ) = 1 Card(z C : z(j ) = 1 for j J ) = s = s where s = Card(J ). Moreover; s k=1 µ η(x : ω jk (x) < α k ) = j J µ η (x : ω j (x) = 1) = s as µ η (x : ω j (x) = 1) = 1, for 1 j. Hence, ( ) is shown. That -independence implies orthogonality follows easily by transfer, from the corresponding fact, for finite measure spaces, that E(X j1 X j ) = E(X j1 )E(X j ), for the standard expectation E and independent random variables {X j1, X j }, ( ). Hence, by ( ); E η (ω j1 ω j ) = E η (ω j1 )E η (ω j ) = 0, (j 1 j ) ( ) as clearly E η (ω j ) = 0, for 1 j. If t 1, let J = {j : 1 j, t(j ) = 1}, then; E η (ω t ) = E η ( 1 j ω t(j)+1 j ) = E η ( j J ω j ) = j J E η (ω j ) = 0 ( ) where, in ( ), we have used the facts that J and internal, and a simple generalisation of ( ), by transfer from the corresponding fact for finite measure spaces. Hence, 1 = ω 1 is orthogonal to ω t, for t 1. If t 1 t are both distinct from 1, then, if J 1 = {j : 1 j, t 1 (j) = 1} and J = {j : 1 j, t (j) = 1}, so J 1 J and J 1, J, we have; E η (ω t1 ω t ) = E η ( j J 1 ω j. j J ω j ) ( ) = E η ( j (J 1 \J ) ω j. j (J \J 1 ) ω j) ( )

4 TRISTRAM DE PIRO = E η ( j (J 1 \J ) ω j)e η ( j (J \J 1 ) ω j) = 0 ( ) In ( ), we have used the definition of J 1 and J, and in ( ), we have used the fact that (J 1 J ) = (J 1 J ) (J 1 \J ) (J \J 1 ), and ωj = 1, for 1 j. Finally, in ( ), we have used the facts that (J 1 \ J ) and (J \ J 1 ) are disjoint, and at least one of these sets is nonempty, the result of ( ) and a similar generalisation of ( ). This shows that the functions {ω t : t C } are orthogonal, ( ). That they form a basis for V (Ω η ) follows immediately, by transfer, from ( ) and the corresponding fact for finite dimensional vector spaces. The final calculation is left to the reader. We require the following; Definition 0.6. For 0 l, we define l, on C, to be the internal equivalence relation given by; t 1 l t iff t 1 (j) = t (j) ( j l) We extend this to an internal equivalence relation on Ω η, which we denote by l ; x 1 l x iff ψ η (x 1 ) l ψ η (x ) ( ) We let Cη l be the -finite algebra generated by the partition of Ω η into the l equivalence classes with respect to l, ( ). As is easily verifed, we have C l l η C l η, if l 1 l, Cη 0 = {, Ω η } and C η = Cη. For 0 l, we let W (Cη) l W (C η ) be the set of measurable functions f : Ω η C, with respect to Cη. l We will refer to the collection {Cη l : 0 l } of -finite algebras, as the nonstandard filtration associated to Ω η. We also require a slight modification of the construction of nonstandard Brownian motion in [1]. Namely, we define χ : T Ω η C by; χ(t, x) = 1 ( [t] i=1 ω i), [t] 1 χ(t, x) = 0, 0 [t] < 1 If λ : T Ω η C is measurable, we say that λ is progressively measurable if;

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 5 (i). For t T, Y [t] is measurable with respect to C [t] η. If λ is progressively measurable, we define the stochastic integral µ(t, x) = t 0 λ(t, x)dχ(t, x) by; [t] j=1 λ(, x)(χ( j µ(t, x) = 0, for 0 [t] < 1, x) χ(, x)), for [t] 1 Remarks 0.7. Observe that, for t T, E η (µ(t, x) = 0. This follows as; E η ( [t] j=1 λ(, x)(χ( j, x) χ(, x))) = [t] j=1 E η(λ(, x)(χ( j, x) χ(, x))) = [t] j=1 E η(λ(, x))e η(χ( j, x) χ(, x)) = 0 Lemma 0.8. For 0 l, a basis of the -finite vector space W (C l η) is given by D l = 0 m l B m, where, for 1 m, B m = {ω t : t(m) = 1, t(m ) = 1, m < m }, and B 0 = {ω 1 }. Proof. The case when l = 0 is clear as ω 1 = 1, and using the description of Cη 0 in Definition 0.6. Using the observation ( ) there, we have, for 1 l, that W (Cη) l is a -finite vector space of dimension l. Using Lemma 0.5, and the fact that Card(D l ) = l, it is sufficient to show each ω t D l is measurable with respect to Cη. l We have that, for 1 j l, ω j is measurable with respect to Cη j Cη. l Hence, the result follows easily, by transfer of the result for finite measure spaces, that the product X j1 X j, of two measurable random variables X j1 is measurable. and X j Definition 0.9. We define a nonstandard martingale to be a D C η - measurable function Y : T Ω η C, such that; (i). For t T, Y [t] (ii). E η (Y [t] C [s] η is measurable with respect to C [t] η. ) = Y [s], for (0 s t 1).

6 TRISTRAM DE PIRO Lemma 0.10. Let Y : T Ω η R be a nonstandard martingale, then; Y t (x) = [t] j=0 c j(t, x)ω j (x) where, if s = j, j 1, c j : [s, 1] Ω η C is D Cη measurable, c 0 : [0, 1] Ω η C is D Cη, 0 measurable, c j (s, x) = c j (t, x), for s t 1. Proof. Using (ii) of Definition 0.9, we have that E η (Y t ) = E η (Y t C 0 η) = Y 0. Replacing Y t by Y t Y 0, we can, without loss of generality, assume that E η (Y t ) = 0, for t [0, 1]. By (i) of Definition 0.9 and Lemma 0.8; Y t = [t] j=1 c j(t, x)ω j (x) where; c j (t, x) = a=1 i 1 <...<i a;1 p(i 1,...,i a) j (t)ω i1... ω ia (x) Using Lemma 0.8, we obtain the first claim for s = j. Again, using (ii) of Definition 0.9, and the fact that c [t ](t, x)ω t is orthogonal to c [s ](t, x)ω [s ], for [s ] [s] < [t ] [t], ( ), we have; [s] j=1 c j(t, x)ω j (x) = [s] j=1 c j(s, x)ω j (x) Equating coefficients, and using the fact that D [t] is a basis for W (C η [t] ), we obtain c j (s, x) = c j (t, x), for all 0 s t 1. Lemma 0.11. Martingale Representation Theorem Let Y : T Ω η R be a nonstandard martingale, with E η (Y t ) = 0, then there exists a progressively measurable Z : T Ω η R such that; Y t = t 0 Z(t, x)dχ(t, x) Proof. By Lemma 0.10, we can find c j : [ j, 1] C, j 1, such that; Y t (x) = [t] j=1 c j(t, x)ω j (x), for [t] 1

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 7 Set Z(t, x) = c [t]+1 (1, x), 0 [t] < 1, and Z(1, x) = 0,,then Z is progressively measurable and; Y t = t 0 Z(t, x)dχ(t, x) as required. Definition 0.1. If N \ N, we define R = [, ) R. We let F consist of internal unions of intervals of the form [ i, i+1 ), for i 1. We define a measure φ on F, by setting φ ([ i, i+1 )) = 1, for i 1. We let W (R ) consist of the set of measurable functions on R. If f W (R ), we define f by; f ( i ) = (f( i+1 ) f( i )), for i f ( 1 ) = 0 and R fdφ = 1 1 i= f( i ) If f C (R), we let f W (R ) be defined by; f( i ) = f( i ), for i 1 Remarks 0.13. Observe that if f W (R ); f ( i ) = (f( i+ ) f( i+1 f ( ) = (f( 1 ) f( )) f ( 1 ) = 0 ) + f( i )), for i 3 We now introduce distribution functions; Definition 0.14. Let Y : T Ω η R be measurable, and suppose that Y, with N \ N, and + 1. Let t T. Define R t : R [0, 1] and r t : R R by; R t (x) = µ η (Y [t] [xη] ), r η t(x) = R t(x)

8 TRISTRAM DE PIRO Lemma 0.15. We have that; R t ( 1 ) = r R t(x)dφ = 1 R t ( ) = 0 Proof. The facts about R t follow immediately from the definition and the property that Y 1. We have that; R r t(x)dφ = 1 1 i= R t( i ) = 1 i= η η(r t ( i+1 ) R t ( i )) = R t ( 1 ) R t ( η) = 1 Lemma 0.16. Let W C 3 (R), and let max(w, dw, d W, d3 W ) K, dx dx dx 3 then (W ) ( dw ) dx on R \ [η 1, ) and (W ) ( d W ) dx on R \ [, ) Proof. We have, by the mean value theorem, that, for x 0 R, n N ; W (x 0 + 1 n ) W (x 0) = dw dx x 0 +c 1 n, where c (0, 1 n ) Therefore; dw dx x 0 n(w (x 0 + 1 n ) W (x 0)) = dw dx x 0 dw dx x 0 +c = c d W dx x0 +d, where d (0, c) Therefore; dw dx x 0 n(w (x 0 + 1 n ) W (x 0)) K n We have that; n (W (x 0 + n ) W (x 0 + 1 n ) + W (x 0))

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 9 = n( dw dx x 0 +c+ 1 n dw dx x 0 +c) = d W dx x0 +c+e where e (0, 1 n ) Hence; d W dx x0 n (W (x 0 + n ) W (x 0 + 1 n ) + W (x 0)) = d W dx x0 d W dx x0 +c+e = (c + e) d3 W dx 3 x0 +f, where f (0, c + e) K n Then; R = ( x R)( n N )max( dw dx n(w (x + 1 n ) W (x)), d W dx n (W (x + n ) W (x + 1 n ) + W (x)) ) K n Hence, R satisfies the transferred statement. In particular, for any infinite N, as K 0, we obtain the result from Definition 0.1 and Remark 0.13. Lemma 0.17. Let hypotheses be as in the previous lemma. Then, there exist {f, g} W (R ), with f K, g K, and C R such that; ( dw dx ) = (W ) + f + δ 1 ( dw dx ) (η 1 ) ( d W ) dx = (W ) + g + δ (( d W ) dx ( ) + ( dw ) dx (η ) + C) + δ 1 (( d W ) dx ( 1 )) Proof. The proof just requires the observation that; W (η 1 ) = W (η 1 ) = 0 and W (η )

10 TRISTRAM DE PIRO = (W ( )) = (( dw ) dx (η ) + C ) = (( dw ) dx (η ) C where C R. Definition 0.18. If V C 3 (R), and Y W (Ω η ), we let (V Y ) η be defined by; (V Y ) η ( i ) = V (Y ( i )), for 0 i η 1 η η Suppose that Y, and = + 1, let V be defined by; V ( i ) = V ( i ), for i 1 so that (V Y ) η W (Ω η ) and V W (R ). Lemma 0.19. Let Y W (Ω η ), with Y, and + 1. Let V C 3 (R), with max(v, dv ) K. Then; dx E η ((V Y ) η ) R V r Y dφ Proof. We have that; γ 1 = R V r Y dφ = 1 i 1 V ( i )r Y ( i ) = i V ( i )(R Y ( i+1 ) R Y ( i )) = 1 η i j J i V ( i ) ( ) where J i = {j Z [0, η 1] : Y ( j η ) ( i, i+1 ]} γ = E η ((V Y ) η ) = 1 η 0 j η 1 V (Y ( j η ))

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 11 = 1 η i j J i V (Y ( j )) ( ) η γ 1 γ 1 η i j J i V ( i ) V (Y ( j )) η = 1 η i j J i V ( i ) V ( i+c(j) ), 0 c(j) < 1 1 η i K j J i = K 0 where, in the penultimate step, we have used the proof of Lemma 0.16. Definition 0.0. If t R and V C 3 ([0, t 0 ] R, and Y W (T Ω η ), we define (V (t, Y )),η by; (V (t, Y )),η (t, x) = V ( [t], Y (t, x)) for (t, x) (T [0, t 0 ]) Ω η so that (V (t, Y )),η W ((T [0, t 0 ]) Ω η ). Lemma 0.1. Let Y : T Ω η R be a nonstandard martingale with associated Z : T Ω η R, and let D = Z. For t 0 [0, 1], let V t0 and W be as given. Let W (t, x) = V t0 (t)w (x), and let S t0,η be defined by; S t0,,η(t, x) = W ( [t], Y (t, x)) ( Vt 0 (t, Y )),η (t, x) = Vt 0 ( [t] Then, for t µ(t 0 ) [0, t 0 ]; S t0,,η(t, x) ( Wt 0 0, Y (t, x)) (t, Y )),η +D( W t0 x (t, Y )),η dλ + Z( Wt 0 0 x (t, Y )),ηdχ Proof. We have that; S t0,,η(t, x)

1 TRISTRAM DE PIRO = W ( [t ], Y (t, x)) W (0, Y (0, x)) = [t ] j=1 ( W ( j, Y ( j, x)) W (, Y (, x))) = [t ] j=1 W ( + [t ] j=1 W + [t ] j=1 ( 1 ( x W + [t ] j=1 ( 1 W + [t ] j=1 ( W, Y (, x))) 1, Y (, x)))(y ( j (, Y (, x)) + ɛ tt) 1 ( x ( x W (t, x) = e x sin( πt t 0 ), x) Y (, x)), Y (, x)) + ɛ xx)(y ( j, Y (, x)) + ɛ tx) 1(Y ( j max(w xxx, W txx, W ttx, W ttt ) K W xx (t + h 1, x + h ) W xx (t, x), x) Y (, x)), x) Y (, x)) W xx (t + h 1, x + h ) W xx (t + h 1, x) + W xx (t + h 1, x) W xx (t, x) h K + h 1 K Similarly, for W tx, W tt. R = ( (t, x) (t 1, t ) R < 1 3 ) max(w xx (x+h) W xx (x), W tx (x+h) W tx (x), W tt (x+h) W tt (x)) K 1 3 = ɛ Assume that Z < min{ 1 6, ɛ 1 3 } = min( 1 6, (K ) 1 3 1 9 ) = (K ) 1 3 1 9 ) (Y ( j (Y ( j, x) Y (, x)) Z( j, x) e j < min{ 1 3, ɛ 1 3 1 }, x) Y (, x)) min{ 3, ɛ 3 1 } It follows that; [t ] j=1 ɛ xx(y ( j, x) Y (, x))

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 13 ɛ xx ɛ 3 1 ɛ xx ɛ 3 where ɛ xx = sup δ u s (W xx (s + δ) W xx (s)), for W (u) = ɛɛ 3 = ɛ 1 3 0, as ( j, Y ( j Similarly, [t ] j=1 ɛ tt 1 ɛ tt 1 = ɛ 0 Finally; )) ( [t ] j=1 ɛ tx 1(Y ( j, x) Y (, x)) ɛ tx 1ɛ 1 3 1 = ɛ tx ɛ 1 3 1 = ɛɛ 1 3 1 = ɛ 3 1 0 Then; (i). [t ] j=1 W ( (ii). [t ] j=1 W = [t ] j=1 W = Z( Wt 0 0 x (iii). [t ] ( x ( x 1 j=1, Y (, x))) 1 = ( Wt 0 0, Y (, x)))(y ( j, Y (, x)))z( j, x) e j (t, Y )),ηdχ W where K = max( W, W ). x, Y ( )) 1 3 (t, Y )),η dλ, x) Y (, x)) (, Y (, x)) 1 K 0

14 TRISTRAM DE PIRO (iv). [t ] = [t ] 1 W j=1 ( x 1 W j=1 ( x = D( W t0 0 x (v). [t ] j=1 W, Y (, x))(y ( j, Y (, x))z ( j, x) 1 (t, Y )),η dλ ( x, Y (, x)) 1(Y ( j, x) Y (, x)), x) Y (, x)) K 1 6 1 0 giving the result. Lemma 0.. With assumptions as in the previous lemma, we have that; E η ( ( Wt 0 0 R ( Wt 0 0 (t, Y )),η + D( W t0 x (t, Y )),η dλ ) (t, x)), + D( W t0 x (t, x)),p(t, x)dλ dφ 0 Proof. As V t0 (t ) 0, we have that S t0,,η(t, x) 0 on Ω η, hence E η (S t0,,η(t, x)) 0. We have E η ( Z( Wt 0 (t, Y )) 0 x,η dχ) = 0, by Remark 0.13. This gives the first part of the claim. For the second part, observe that, using Lemma 0.19 and the fact that fdλ 0 0, if f 0; E η ( ( Wt 0 0 0 0 Ω η ( Wt 0 ( Wt 0 R (t, Y )),η (t, Y )),η dµ η dλ (t, x)),p(t, x)dφ dλ = R 0 ( Wt 0 (t, x)),p(t, x)dλ dφ and a similar calculation for E η ( Lemma 0.3. We have that; R 0 D( W t0 x (t, Y )),η dλ ). 0 ( (Wt 0 ), + D (W t0 ), x )p(t, x))dλ dφ 0 Proof. By Lemma 0.16, we have that Wt 0 (t, x)), = (Wt 0 ), +g +ɛ, where g K and ɛ is an error term supported at the endpoint, and

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 15 W t0 x (t, x)), = (W t0 ), x + h + δ, where h K and δ is an error term supported at endpoints. Hence; R 0 ( Wt 0 (t, x)),p(x, t)dλ dφ R (g + ɛ)p(t, x)dλ R 0 dφ K t p(t, x) dλ 0 R dφ + R Kt 0 δ [t ] 1 + D = Kt + D 0 W t0 (t, x)),([t ] 1 )p(t, x) dλ dφ R p([t 0] 1, x) dφ 0 ( (Wt 0 ), p(t, x)dλ dφ where we have used the fact that p(t, x) is a probability distribution, and Wt 0 D. Similarly; R 0 D( W t0 x (t, x)),p(t, x)dλ dφ R D(h + δ)p(t, x)dλ R 0 dφ KC 9 1 p(t, x) dλ 0 R dφ + t (δ 0 R (( W t0 (t, x)) x,( ) + ( Wt 0 x ) + C) + δ 1 (( W t0 x Kt C 1 9 0 (t, x)),( 1 )))Dp(x, t) dφ dλ 0 D (W t0 ), x p(t, x))dλ dφ (t, x)),(η for sufficiently large, where we have used the fact that p(t, ) = p(t, 1 ) = 0, p(t, x) 0, and R p(t, x)dφ = 1, for t T Lemma 0.4. If f W (R ), then; R f dφ = f( 1 ) f( )

16 TRISTRAM DE PIRO R f gdφ = fg( 1 ) fg( ) R f sh g dφ where f sh ( i ) = f( i+1 ) for i, f sh ( 1 ) = 0. R f gdφ = (f g( 1 ) f g( )) (f sh g ( 1 ) f sh g ( )) (f( 1 )g ( )) + R f sh g dφ Proof. For the first part, we have that; R f dφ = 1 0 i f ( + i ) = 1 0 i η (f ( + i+1 ) f( + i ) = 0 i f ( + i+1 ) f( + i ) = f( 1 ) f( ) For the second part observe, that, for i ; (fg) ( i ) = (fg( i+1 ) fg( i )) = (fg( i+1 ) g( i )f( i+1 ) + g( i = g ( i )f(( i+1 )) + g( i )f ( i ) = (g f sh + gf )( i ) )f( i+1 ) fg( i )) and that (fg) ( 1 ) = (g f sh + gf )( 1 ) = 0 to obtain (fg) = (f g + f sh g ), ( ). Then, by the first part and ( ); fg( 1 ) fg( ) = R (fg) dφ = (f g + f sh g )dφ R as required. Using the second part, we have that; (f g)dφ R

A NONSTANDARD VERSION OF THE FOKKER-PLANCK EQUATION 17 = f g( 1 ) f g( ) R (f sh g )dφ, ( ) For i 3, we have that; f sh ( i ) = f ( i+1 ) = (f( i+ ) f( i+1 )) = (f sh ( i+1 ) f sh ( i )) = (f sh ) ( i ) f sh ( 1 ) = (f sh ) ( 1 ) = 0 f sh ( ) = f ( 1 ) = 0 (f sh ) ( ) = (f sh ( 1 ) f sh ( )) = f( 1 ) Therefore, f sh = (f sh ) + δ η η f( 1 ) We therefore have that; R (f sh g )dφ = R ((f sh ) g + δ η η f( 1 )g )dφ = f( 1 )g ( ) + R (f sh ) g dφ = f( 1 )g ( )+(f sh g ( 1 ) f sh g ( )) R (f sh )g dφ, ( ) Combining ( ) and ( ) gives the result. Lemma 0.5. We have that; R 0 (W t 0 ) sht, p + (W t 0 ) sh x, Dp x dλ dφ 0

18 TRISTRAM DE PIRO Lemma 0.6. If p R and (Dp) x 0 (W t 0 ),( p + Dp x )dλ dφ 0 are S-integrable, we have that; Proof. To see this, observe that, as (W t0 ), is S-continuous, (W t0 ) sht, = (W t0 ), + ɛ 1, (W t0 ) sh x,η = (W t0 ), + ɛ, where ɛ 1, ɛ are infinitesimal functions. We have that; (W R 0 t 0 ) sht p,η dλ dφ R ɛ p dλ R 0 dφ = Dɛ 0 Similarly; (W R 0 t 0 ) sh x Dp,η dλ x dφ R ɛ Dp dλ R 0 x dφ 0 (W t 0 ),( p )dλ dφ 0 (W t 0 ),( Dp x )dλ dφ = Cɛ 0 Lemma 0.7. If p and (Dp) x are Lebesgue-integrable, we have that; p + (Dp) 0 x References [1] A Non-Standard Representation for Brownian Motion and Ito Integration, R. Anderson, Israel Journal of Mathematics, Volume 5, (1976). [] Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory, Peter Loeb, Transactions of the American Mathematical Society, (1975). [3] Applications of Nonstandard Analysis to Probability Theory, Tristram de Piro, M.Sc Dissertation in Financial Mathematics, University of Exeter, (013). Mathematics Department, The University of Exeter, Exeter E-mail address: t.depiro@curvalinea.net