Pathwise Itô integration

In the previous section we saw that our pathwise integral I(f,dg) is of Stratonovich type, i.e. it satisfies the usual integration by parts rule. But in applications it may be interesting to have an Itô integral. Here we show that a slight modification of I(f,dg) allows us to treat non-anticipating Itô-type integrals.

In our dyadic context, a natural approximation of a non-anticipating integral is given fork∈Nby

I_k^Itˆo(f,dg)(t) :=

2^k



ℓ=0

f(t⁰_kℓ)(g(t²_kℓ∧t)−g(t⁰_kℓ∧t))

=

2^k



ℓ=0



p,q



m,n

fpmgqnϕpm(t⁰_kℓ)(ϕqn(t²_kℓ∧t)−ϕqn(t⁰_kℓ∧t)).

Let us assume for the moment that t = m2^−k for some 0 ≤ m ≤ 2^k. In that case we obtain forp≥korq ≥kthatϕpm(t⁰_kℓ)(ϕqn(t²_kℓ∧t)−ϕqn(t⁰_kℓ∧t)) = 0. Forp, q < k, both ϕ_pmand ϕ_qn are affine functions on [t⁰_kℓ∧t, t²_kℓ∧t]. And for affinev and w and s < tit is not hard to see that

v(s)(w(t)−w(s)) =

 t s

v(r)dw(r)− 1

2[v(t)−v(s)][w(t)−w(s)].

Hence, we conclude that fort=m2^−k we have

I_k^Itˆo(f,dg)(t) =I(Sk−1f,dSk−1g)(t)−1

2[f, g]k(t), (4.27) where [f, g]_k is the k–th dyadic approximation of the quadratic covariation [f, g], i.e.

[f, g]_k(t) :=

2^k



ℓ=0

[f(t²_kℓ∧t)−f(t⁰_kℓ∧t)][g(t²_kℓ∧t)−g(t⁰_kℓ∧t)].

For the moment let us continue by studying the right-hand side of (4.27). Later we will show how to return from there toI_k^Itˆo(f,dg)(t) for generalt, not necessarily of the form t=m2^−k.

We write [w, w] := ([wⁱ, w^j])_1≤i,j≤d and L(w, w) := (L(wⁱ, w^j))_1≤i,j≤d, and similarly for all expressions of the same type.

Theorem 4.5.1. Letα ∈(1/3,1/2)and letw∈ C^α(R^d) andf, g∈ D_w^α(R). Assume that (L(S_kw, S_kw)) converges uniformly, with uniformly bounded C^2α norm. Also assume that ([w, w]_k) converges uniformly. Then I(Sk−1f,dSk−1g)(t)−1/2[f, g]_k(t) converges uniformly to a limitI^Itˆo(f,dg) that satisfies

∥I^Itˆo(f,dg)∥_∞.∥f∥_w,α∥g∥_w,α(1 +∥w∥²_α+∥L(w, w)∥_2α+∥[w, w]∥_∞).

The quadratic variation of I^Itˆo(f,dg) is given by [f, g] =

d



i,j=1

 · 0

f^w,i(s)g^w,j(s)d[wⁱ, w^j](s). (4.28) Moreover, for ε∈(0,3α−1)the speed of convergence can be estimated by









I^Itˆo(f,dg)−



I(Sk−1f,dSk−1g)−1 2[f, g]_k







_∞

.ε2^{−k(3α−1−ε)}∥f∥_w,α∥g∥_w,α^1 +∥w∥_α+∥w∥²_α^ +∥f∥_α∥g∥_α∥L(S_k−1w, Sk−1y)−L(w, y)∥_2α +∥f^w∥_∞∥g^w∥_∞∥[w, w]_k−[w, w]∥_∞.

Proof. Let us first treat the quadratic variation. Recall from Lemma 4.4.5 that f ∈ D_w^α(R) if and only if there exists R^f : [0,1]² →R, such that|R^f_s,t|.|t−s|^2α, and such

that for all 0≤s < t≤1 we havefs,t=f^w(s)ws,t+R_s,t^f . An analogous statement holds

It is easy to see that there exists C > 0, such that the first two terms on the right hand side are uniformly bounded by C2^{−k(3α−1)}∥f∥_w,α∥g∥_w,α. For the third term, let us fix i and j. Then this is just the integral of f^w,ig^w,j with respect to the measure µ^k_t =^_ℓδ_t⁰

where we write [u] := [u, u] and similarly for [u]k. By assumption, the right hand side converges to zero, from where we get the uniform convergence of [f, g]_kto [f, g]. Moreover, we have the explicit representation Theo-rem 4.4.15, to obtain convergence to a limit I(f,dg) that satisfies

∥I(f,dg)∥_∞.∥f∥_w,α∥g∥_w,α(1 +∥w∥²_α+∥L(w, w)∥_2α),

where we used that 1 +∥w∥_α+∥w∥²_α . 1 +∥w∥²_α. According to Corollary 4.4.17, the

speed of convergence can be estimated by

∥I(f,dg)−I(Sk−1f,dSk−1g)∥_∞.ε 2^{−k(3α−1−ε)}∥f∥_w,α∥g∥_w,α^1 +∥w∥_α+∥w∥²_α^ +∥f∥_α∥g∥_α∥L(S_k−1w, Sk−1w)−L(w, w)∥_2α.

Note that [w, w] is always a continuous function of bounded variation, but a priori it is not clear whether it is inC^2α. Under this additional assumption we have the following stronger result.

Corollary 4.5.2. In addition to the conditions of Theorem 4.5.1, assume that also [w, w]∈ C^2α. Then I^Itˆo(f,dg)∈ D^α_w with derivativef g^w, and

∥I^Itˆo(f,dg)∥_w,α.∥f∥_w,α^1 +∥g∥_w,α^1 +∥w∥²_α+∥L(w, w)∥_2α+∥[w, w]∥_2α^. Let moreover w˜ ∈ C^α(R^d) with Lévy area L(S_kw, S˜ _kw)˜ that converges uniformly and with uniformly bounded C^2α norm to L( ˜w,w), and with quadratic variation˜ [ ˜w,w]˜k that converges uniformly to [ ˜w,w]˜ ∈ C^2α. Letf ,˜ ˜g∈ D^α_w˜(R). Then

∥I^Itˆo(f,dg)−I^Itˆo( ˜f , d˜g)∥_α .^∥f−f˜∥_α+∥f^w−f˜^w˜∥_α+∥f^♯−f˜^♯∥_2α^∥g∥_w,α

×^1 +∥w∥²_α+∥L(w, w)∥_2α+∥[w, w]∥_2α^ +^∥g−˜g∥_α+∥g^w−˜g^w˜∥_α+∥g^♯−˜g^♯∥_2α^∥f˜∥_w,α˜

×^1 +∥w∥²_α+∥L(w, w)∥_2α+∥[w, w]∥_2α^ +^∥w−w∥˜ _α+∥L(w, w)−L( ˜w,w)∥˜ _2α+∥[w, w]−[ ˜w,w]∥˜ _2α^

× ∥f˜∥_w,α˜ ∥˜g∥_w,α˜ ^1 +∥w∥˜ _α+∥w∥_α^.

Proof. This is a combination of Theorem 4.4.15 and Corollary 4.4.16, and the explicit representation (4.28) for the quadratic variation. We also need continuity of the Young integral, Theorem 4.3.15, for example to estimate ∥[f, g]∥_2α.

The term I(Sk−1f,dSk−1g) has the pleasant property that if we want to refine our calculation by passing fromktok+ 1, then we can build on our existing calculation and only add the additional termsI(Sk−1f,d∆_kg)+I(∆kf,dSkg). For the quadratic variation [f, g]_k this is not exactly true. But note that [f, g]_k(m2^−k) = [Sk−1f, Sk−1g]_k(m2^−k) for m= 0, . . . ,2^k. And there is a recursive way of calculating [Sk−1f, Sk−1g]_k:

Lemma 4.5.3. Let f, g∈C([0,1],R). Then we have for all k≥1 and all t∈[0,1] that [S_kf, S_kg]_k+1(t) = 1

2[Sk−1f, Sk−1g]_k(t) + [Sk−1f,∆_kg]_k+1(t) + [∆_kf, S_kg]_k+1(t) +R_k(t), (4.29)

where

R_k(t) :=−1

2f_xtky,tg_xtky,t+f_xtky,pt^k+1q∧tg_xtky,pt^k+1q∧t+f_ptk+1q∧t,tg_ptk+1q∧t,t

and xt^ky:=⌊t2^k⌋2^−k and pt^kq:=xt^ky+ 2^−(k+1). In particular, we obtain for t= 1 that [f, g]_k+1(1) = 1

2[f, g]_k(1) + 1 2



m

f_kmg_km= 1 2^k+1



p≤k



m

2^pf_pmg_pm. (4.30) If moreover α∈(0,1)and f, g∈ C^α, then

∥[S_k−1f, Sk−1g]_k−[f, g]_k∥_∞.2^−2kα∥f∥_α∥g∥_α.

Proof. By subtracting [S_k−1f,∆_kg]_k+1(t) + [∆_kf, S_kg]_k+1(t) on both sides of (4.29), we see that it suffices to show [Sk−1f, Sk−1g]k+1= 1/2[Sk−1f, Sk−1g]k+Rk. Let us assume thatt=m2^−k. In that caseR_k(t) = 0, and for everyℓ≤2^k we obtain

([Sk−1f, Sk−1g]k+1)_t0

kℓ,t²_kℓ =^(Sk−1f)_t⁰

kℓ,t¹_kℓ(Sk−1g)_t⁰

kℓ,t¹_kℓ+ (Sk−1f)_t1

kℓ,t²_kℓ(Sk−1g)_t¹

kℓ,t²_kℓ



= 1

2(Sk−1f)_t0

kℓ,t²_kℓ(Sk−1g)_t0

kℓ,t²_kℓ = 1

2([Sk−1f, Sk−1g]_k)_t0 kℓ,t²_kℓ, where we used that Sk−1f and Sk−1g are linear on [t⁰_kℓ, t²_kℓ], and that the two intervals [t⁰_kℓ, t¹_kℓ] and [t¹_kℓ, t²_kℓ] have the same length 2^−k−1. The termR_kis now chosen exactly so that we also obtain the right expression fort∈[0,1] that is not of the formm2^−k.

The formula for [f, g]_k+1(1) follows because [f, g]_k+1(1) = [S_kf, S_kg]_k+1(1), and be-cause it is easy to see that [∆_pf,∆_qg]_k+1(1) = 0 unless p=q, and that [∆_kf,∆_kg]_k+1= 1/2^_mfkmgkm.

The estimate for∥[S_k−1f, Sk−1g]_k−[f, g]_k∥_∞holds because the two functions agree in all dyadic points of the form m2^−k, and because between two such points the quadratic variation can pick up mass of at most 2^−2kα∥f∥_α∥g∥_α.

Remark 4.5.4. The Cesàro mean formula (4.30) makes the study of existence of the quadratic variation accessible to ergodic theory. This was previously observed by Gan-tert [Gan94]. See also GanGan-tert’s thesis [Gan91], Beispiel 3.29, where it is shown that ergodicity alone (of the distribution of w with respect to suitable transformations on path space) is not sufficient to obtain convergence of ([w, w]_k(1)) as k tends to∞.

Recall that we definedI_k^Itˆo(f,dg)(t) =^_ℓf(t⁰_kℓ)g_t⁰

kℓ∧t,t²_kℓ∧t. Remark 4.5.5. Letα∈(0,1). If f ∈C([0,1]) and g∈ C^α, then





I_k^Itˆo(f,dg)−^I(Sk−1f,dSk−1g)−1

2[Sk−1f, Sk−1g]_k^_

∞.2^−kα∥f∥_∞∥g∥_α. This holds because both functions agree in all dyadic points of the form m2^−k, and because between those points the integrals can pick up mass of at most∥f∥_∞2^−kα∥g∥_α.

It follows from Remark 4.5.5 that our pathwise Itô type integral constructed in The-orem 4.5.1 is the limit of non-anticipating Riemann sums. Therefore, it would be more natural to assume that also for the controlling path w the non-anticipating Riemann sums converge, rather than assuming that (L(Skw, Skw))k and ([w, w]k) converge. Be-low we show that this is sufficient, as long as a uniform Hölder estimate is satisfied by the Riemann sums. In that case all the conditions of Theorem 4.5.1 and of Corollary 4.5.2 are satisfied.

We first show that the existence of the Itô iterated integrals implies the existence of the quadratic variation.

Lemma 4.5.6. Let α ∈(0,1/2) and let w∈ C^α(R^d). Assume that the non-anticipating Riemann sums (I_k^Itˆo(w,dw))_k converge uniformly to I^Itˆo(w,dw). Then also ([w, w]_k)_k converges uniformly to a limit[w, w]. Moreover, for all0≤s < t≤1

|[w, w]_k(t)−[w, w]_k(s)|.|I_k^Itˆo(w,dw)_s,t−w(s)w_s,t|+|w_s,t|². (4.31) If moreover

sup

k

sup

0≤ℓ<ℓ^′≤2^k

|I_k^Itˆo(w,dw)(ℓ^′2^−k)−I_k^Itˆo(w,dw)(ℓ2^−k)−w(ℓ2^−k)(w(ℓ^′2^−k)−w(ℓ2^−k))|

|(ℓ^′−ℓ)2^−k|^2α

=C <∞, then[w, w]∈ C^2α, and

∥[w, w]∥_2α .C+∥w∥²_α. (4.32) Proof. Lett∈[0,1] and 1≤i, j ≤d. Then

wⁱ(t)w^j(t)−wⁱ(0)w^j(0) =

2^k



ℓ=1



wⁱ(t²_kℓ∧t)w^j(t²_kℓ∧t)−wⁱ(t⁰_kℓ∧t)w^j(t⁰_kℓ∧t)^

=

2^k



ℓ=1



wⁱ(t⁰_kℓ)w_t^j0

kℓ∧t,t²_kℓ∧t+w^j(t⁰_kℓ)wⁱ_t0

kℓ∧t,t²_kℓ∧t+w_tⁱ0

kℓ∧t,t²_kℓ∧tw_t^j0

kℓ∧t,t²_kℓ∧t



=I_k^Itˆo(wⁱ,dw^j)(t) +I_k^Itˆo(w^j,dwⁱ)(t) + [wⁱ, w^j]_k(t), (4.33) which implies the convergence of ([w, w]k)_kasktends to∞. For 0≤s < t≤1 we obtain from (4.33) that

([wⁱ, w^j]_k)_s,t=^wⁱw^j_s,t−I_k^Itˆo(wⁱ,dw^j)_s,t−I_k^Itˆo(w^j,dwⁱ)_s,t

=^wⁱ(s)w^j_s,t−I_k^Itˆo(wⁱ,dw^j)_s,t^+^w^j(s)wⁱ_s,t−I_k^Itˆo(w^j,dwⁱ)_s,t^+wⁱ_s,tw_s,t^j , leading to (4.31). Given (4.31) it is now easy to estimate ∥[w, w]∥_2α. We estimate the classical Hölder norm, not the C^2α norm. Let 0 ≤ s < t ≤ 1. Using the continuity of [w, w], we choose k large enough such that there exist s < s_k = ℓ_s2^−k < t and

s < tk=ℓt2^−k< t with

|[w, w]_s,sk|+|[w, w]_tk,t|+∥[w, w]_k−[w, w]∥_∞≤ ∥w∥²_α|t−s|^2α. Since

|[w, w]_s,t| ≤ |[w, w]_s,sk|+|[w, w]_tk,t|+∥[w, w]_k−[w, w]∥_∞, we obtain (4.32) as a consequence of (4.31) and the hypothesis.

Let us show that convergence of (I_k^Itˆo(w,dw)) implies convergence of (L(Skw, Skw))k: Lemma 4.5.7. Let α∈(0,1/2), and let w∈ C^α(R^d). Assume that the non-anticipating integrals (I_k^Itˆo(w,dw))_k converge uniformly, and that

sup

k

sup

0≤ℓ<ℓ^′≤2^k

|I_k^Itˆo(w,dw)(ℓ^′2^−k)−I_k^Itˆo(w,dw)(ℓ2^−k)−w(ℓ2^−k)(w(ℓ^′2^−k)−w(ℓ2^−k))|

|(ℓ^′−ℓ)2^−k|^2α

=C <∞.

ThenL(S_kw, S_kw) converges uniformly as k→ ∞, and sup

k

∥L(S_kw, S_kw)∥_2α.C+∥w∥²_α.

Proof. Letk∈N and 0≤ℓ≤2^k, and write t=ℓ2^−k. Then we obtain from (4.27) that

L(Sk−1w, Sk−1w)(t) (4.34)

=I(Sk−1w,dSk−1w)(t)−π<(Sk−1w, Sk−1w)(t)−S(Sk−1w, Sk−1w)(t)

=I_k^Itˆo(w,dw)(t) + 1

2[w, w]_k(t)−π_<(Sk−1w, Sk−1w)(t)−S(Sk−1w, Sk−1w)(t).

Let now s, t ∈ [0,1]. We first assume that there exists ℓ such that t⁰_kℓ ≤ s < t ≤ t²_kℓ. Then we use that∥∂_t∆_qw∥_∞.2^q(1−α)∥w∥_α to obtain

|L(S_k−1w, Sk−1w)_s,t| ≤ ^

p<k



q<p









 _t

s

∆_pw(r)d∆_qw(r)−

 _t

s

d∆_qw(r)∆_pw(r)









(4.35) .^

p<k



q<p

|t−s|2^−pα2^q(1−α)∥w∥²_α .|t−s|2^{−k(2α−1)}∥w∥²_α≤ |t−s|^2α∥w∥²_α, where we used that 2α−1<0, and also that|t−s| ≤2^−k by assumption.

Combining (4.34) and (4.35), we obtain the uniform convergence of (L(Sk−1w, Sk−1w)) from Lemma 4.5.6 and from the continuity of π< and S.

Forsandtthat do not lie in the same dyadic interval of generationk, letps^kq=ℓs2^−k and xt^ky = ℓ_t2^−k be such that ps^kq−2^−k < s ≤ ps^kq and xt^ky ≤ t <xt^ky+ 2^−k. In

particular,ps^kq≤xt^ky. Moreover

|L(S_k−1w, Sk−1w)_s,t| ≤ |L(S_k−1w, Sk−1w)_s,pskq|+|L(S_k−1w, Sk−1w)_pskq,xt^ky| +|L(S_k−1w, Sk−1w)_xtky,t|.

According to (4.35), the first and third term on the right hand side can be estimated by (|ps^kq−s|^2α+|t−xt^ky|^2α)∥w∥²_α .|t−s|^2α∥w∥²_α. For the middle term we apply (4.34) to obtain

|L(S_k−1w, Sk−1w)_pskq,xt^ky| ≤^_I_k^Itˆo(w,dw)_pskq,xt^ky−w(ps^kq)(w(xt^ky)−w(ps^kq))^_ +^_w(ps^kq)w_pskq,xt^ky−π<(Sk−1w, Sk−1w)_pskq,xt^ky





 +1

2





([w, w]k)_pskq,xt^ky





+^_S(Sk−1w, Sk−1w)_ps^k_q,xt^k_y^_ .|xt^ky−ps^kq|^2αC+∥w∥²_α^≤ |t−s|^2αC+∥w∥²_α^, where we used Lemma 4.4.5, Lemma 4.5.6, and Lemma 4.3.14.

Combining Lemma 4.5.6 and Lemma 4.5.7 with Theorem 4.5.1, we see that uni-form convergence of (I_k^Itˆo(w,dw))_k to I^Itˆo(w,dw) implies the uniform convergence of (I_k^Itˆo(f,dg))k toI^Itˆo(f,dg) for f and g controlled by w:

Corollary 4.5.8. Let α∈(1/3,1/2)and letw∈ C^α(R^d) andf, g∈ D^α_w(R). Assume that the non-anticipating Riemann sums(I_k^Itˆo(w,dw))_kconverge uniformly toI^Itˆo(w,dw), and that furthermore

sup

k

sup

0≤ℓ<ℓ^′≤2^k

|I_k^Itˆo(w,dw)(ℓ^′2^−k)−I_k^Itˆo(w,dw)(ℓ2^−k)−w(ℓ2^−k)(w(ℓ^′2^−k)−w(ℓ2^−k))|

|(ℓ^′−ℓ)2^−k|^2α

=C <∞.

Then the non-anticipating Riemann sums (I_k^Itˆo(f,dg))_k converge to a limit I^Itˆo(f,dg) that satisfies

∥I^Itˆo(f,dg)∥_∞.∥f∥_w,α∥g∥_w,α(1 +∥w∥²_α+C).

Remark 4.5.9. Observe that we calculate the pathwise Itô integralI^Itˆo(f,dg) as limit of Riemann sums involving onlyf andg, and not the Lévy area ofL(w, w) or the quadratic variation [w, w]. The classical rough path integral, see Proposition 4.2.4, is obtained as a “compensated Riemann sum” that involves f and g, but also their derivatives with respect to w, as well as the iterated integrals of w. For applications in mathematical finance, it is more convenient to have an integral that is the limit of Riemann sums involving only f and g, because then this integral can be interpreted as capital process obtained by investing in g.

It follows from the work of Föllmer [Föl79] that our pathwise Itô integral satisfies Itô’s formula:

Corollary 4.5.10. Let α ∈ (1/3,1/2) and let w ∈ C^α(R^d) and f, g ∈ D^α_w(R). Assume that the non-anticipating Riemann sums(I_k^Itˆo(w,dw))_kconverge uniformly toI^Itˆo(w,dw), and that furthermore

sup

k

sup

0≤ℓ<ℓ^′≤2^k

|I_k^Itˆo(w,dw)(ℓ^′2^−k)−I_k^Itˆo(w,dw)(ℓ2^−k)−w(ℓ2^−k)(w(ℓ^′2^−k)−w(ℓ2^−k))|

|(ℓ^′−ℓ)2^−k|^2α

=C <∞.

Let F ∈C²(R^d,R). Then (I^Itˆo(DF(w),dw))_k converges to a limit I^Itˆo(DF(w),dw) that satisfies for allt∈[0,1]

F(w(t))−F(w(0)) =I^Itˆo(DF(w),dw)(t) +

 t 0

d



k,ℓ=1

∂_xk∂_xℓF(w(s))d[w^k, w^ℓ](s).

Proof. This is Remarque 1 of Föllmer [Föl79] in combination with Lemma 4.5.6.

Remark 4.5.11. Note that DF ∈C¹, and therefore DF(w) is not controlled by w. Just as in the Stratonovich case, see Remark 4.4.20, the symmetry of the derivative of DF leads to crucial cancellations that allow to take DF less regular than in the non-gradient case.

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