Complement: density of simple processes in H 2

The proof of Proposition 5.2 is a consequence of the following Lemmas. Through-out this section, tⁿ_i := i2⁻ⁿ, i ≥ 0 is the sequence of dyadic numbers.

Lemma 5.15. Let φ be a bounded F−adapted process with continuous sample paths. Then φ can be approximated by a sequence of simple processes in H². Proof. Define the sequence

φ⁽ⁿ⁾_t := φ01_{0}(t) + X

tⁿ_i≤T

φtⁿ_i1(tⁿ_i,tⁿ_i+1](t), t ≤ T.

Then, φ⁽ⁿ⁾ is a simple process for each n ≥ 1. By the dominated convergence theorem, Eh

RT

0 |φ⁽ⁿ⁾_t − φt|²i

−→ 0 as n → ∞. ♦

Lemma 5.16. Let φ be a bounded F−progressively measurable process. Then φ can be approximated by a sequence of simple processes in H².

Proof. Notice that the process ψ^(k)_t := k

Z t 0∨(t−¹_k)

φsds, 0 ≤ t ≤ T,

is progressively measurable as the difference of two adapted continuous pro-cesses, see Proposition 3.2, and satisfies

ψ^(k)− φ H²

−→ 0 as k → ∞, (5.8)

by the dominated convergence theorm. For each k ≥ 1, we can find by Lemma 5.15 a sequence ψ^(k,n)

n≥0of simple processes such that kψ^(k,n)−ψ^(k)k_H² −→ 0 as n → ∞. Then, for each k ≥ 0, we can find n_k such that

the process φ^(k):= ψ^(k,nk) satisfies

φ^(k)− φ H²

−→ 0 as k → ∞.

♦ Lemma 5.17. Let φ be a bounded measurable and F−adapted process. Then φ can be approximated by a sequence of simple processes in H².

Proof. In the present setting, the process ψ^(k), defined in the proof of the previous Lemma 5.16, is measurable but is not known to be adapted. For each ε > 0, there is an integer k ≥ 1 such that ψ^ε:= ψ^(k) satisfies kψ^ε− φk_H2 ≤ ε.

Then, with φt= φ0 for t ≤ 0:

kφ − φ.−hk_H2 ≤ kφ − ψ^εk_H2+

ψ^ε− ψ_.−h^ε H²+

ψ_.−h^ε − φ.−h

H²

≤ 2ε +

ψ^ε− ψ_.−h^ε H².

5.4. Complement 65

By the continuity of ψ^ε, this implies that lim sup

h&0

kφ − φ_.−hk

H² ≤ 4ε². (5.9)

We now introduce

ϕn(t) := 1_{0}(t) +X

i≥1

tⁿ_i1_(tⁿ

i−1,tⁿ_i], and

φ^(n,s)_t := φ_ϕn(t−s)+s, t ≥ 0, s ∈ (0, 1].

Clearly φ^n,sis a simple adapted process, and

E

"

Z T 0

Z 1 0

|φ^(n,s)_t − φt|²dsdt

#

= 2ⁿE

"

Z T 0

Z 2⁻ⁿ 0

|φt− φt−h|²dhdt

#

= 2ⁿ Z 2⁻ⁿ

0

E

"

Z T 0

|φ_t− φ_t−h|²dt

# dh

≤ max

0≤h≤2⁻ⁿE

"

Z T 0

|φt− φt−h|²dt

#

which converges to zero as n → ∞ by (5.9). Hence

φ^(n,s)_t (ω) −→ φ_t(ω) for almost every (s, t, ω) ∈ [0, 1] × [0, T ] × Ω, and the required result follows from the dominated convergence theorem. ♦ Lemma 5.18. The set of simple processes S is dense in H².

Proof. We only have to extend Lemma 5.17 to the case where φ is not neces-sarily bounded. This is easily achieved by applying Lemma 5.17 to the bounded process φ ∧ n, for each n ≥ 1, and passing to the limit as n → ∞. ♦

Chapter 6

Itˆ o Differential Calculus

In this chapter, we focus on the differential properties of the Brownian motion.

To introduce the discussion, recall that EW_t² = t for all t ≥ 0. If standard differential calculus were valid in the present context, then one would expect that W_t² be equal to M_t = 2Rt

0W_sdW_s. But the process M is a square integrable martingale on every finite interval [0, T ], and therefore E[M_t] = M₀ = 0 6=

E[Wt²] !

So, the standard differential calculus is not valid in our context. We should not be puzzled by this small calculation, as we already observed that the Brow-nian motion sample path has very poor regularity properties, has infinite total variation, and finite quadratic variation.

We can elaborate more on the above example by considering a discrete-time approximation of the stochastic integral

n

X

i=1

2Wti−1 Wti− Wti−1

= −

n

X

i=1

Wti− Wti−1

2

+

n

X

i=1



W_t²_i− W_t²_i−1

= W_t²−

n

X

i=1

W_ti− Wti−1

² ,

where 0 = t0 < t1 < . . . , tn = t. We know that the latter sum converges in L² towards t, the quadratic variation of the Brownian motion at time t (the convergence holds even P−a.s. if one takes the dyadics as (tⁱ)i). Then, by sending n to infinity, this shows that

Z t 0

2W_sdW_s = W_t²− t, t ≥ 0. (6.1)

In particular, there is no contradiction anymore by taking expectations on both sides.

67

6.1 Itˆ o’s formula for the Brownian motion

The purpose of this section is to prove the Itˆo formula for the change of variable.

Given a smooth function f (t, x), we will denote by ft, Df and D²f , the partial gradients with respect to t, to x, and the partial Hessian matrix with respect to x.

Proof. 1 We first fix T > 0, and we show that the above Itˆo’s formula holds with probability 1. By possibly adding a constant to f we may assume that f (0, 0) = 0. Let πⁿ = (tⁿ_i)_i≥0 be a partition of R+ with tⁿ₀ = 0, and denote

By a Taylor expansion, this provides:

I_Tⁿ(Df ) := X

i+1 for some random variable λⁿ_i with values in [0, 1].

1.b Since a.e. sample path of the Brownian motion is continuous, and there-fore uniformly continuous on the compact interval [0, T + 1], it follows that

f

6.1. Itˆo’s formula, Brownian motion 69 Next, using again the above uniform continuity together with Proposition 4.23 and the fact that the L²−convergence implies the a.s. convergence along some subsequence, we see that:

1.c. For the last term in the decomposition (6.2), we estimate:



by an immediate extension of (6.1) to the multidimensional setting. Since

E

it follows that (6.4) converges to zero P−a.s. along some subsequence, and it follows from (6.3) that along some subsequence:

X

1.d In order to complete the proof of Itˆo’s formula for fixed T > 0, it remains to prove that

I_Tⁿ(Df ) −→

Z T 0

Df (t, W_t) · dW_tP − a.s. along some subsequence.(6.5)

Notice that I_Tⁿ(Df ) = I_T⁰ φ⁽ⁿ⁾
where φ⁽ⁿ⁾is the simple process defined by φ⁽ⁿ⁾_t = X

tⁿ_i≤T

Df tⁿ_i, Wtⁿ_i
1[^tni,tⁿ_i+1)(t), t ≥ 0.

Since Df is continuous, it follows from the proof of Proposition 5.2 that φ⁽ⁿ⁾−→

φ in H²with φ_t:= Df (t, W_t). Then I_Tⁿ(Df ) −→ I_T(φ) in L², by the definition of the stochastic integral in Theorem 5.3, and (6.5) follows from the fact that the L²convergence implies the a.s. convergence along some subsequence.

2. From the first step, we have the existence of subsets Nt⊂ F for every t ≥ 0 such that P[N^t] = 0 and the Itˆo’s formula holds on N_t^c, the complement of Nt. Of course, this implies that the Itˆo’s formula holds on the complement of the set N := ∪t≥0Nt. But this does not complete the proof of the theorem as this set is a non-countable union of zero measure sets, and is therefore not known to have zero measure. We therefore appeal to the continuity of the Brownian motion and the stochastic integral, see Proposition 3.15. By usual approximation along rational numbers, it is easy to see that, with probability 1, the Itˆo formula holds

for every T ≥ 0. ♦

Remark 6.2. Since, with probability 1, the Itˆo formula holds for every T ≥ 0, it follows that the Itˆo’s formula holds when the deterministic time T is replaced by a random time τ .

Remark 6.3. (Itˆo’s formula with generalized derivatives) Let f : R^d−→ R be C^1,2(R^d\ K) for some compact subset K of R^d. Assume that f ∈ W²(K), i.e.

there is a sequence of functions (fⁿ)n≥1such that

fⁿ= f on R^d\ K, fⁿ ∈ C²(K) and kf_xⁿ− f_x^mk_L2(K)+ kf_xxⁿ − f_xx^mk_L2(K)−→ 0.

Then, Itˆo’s formula holds true:

f (Wt) = f (0) + Z t

0

Df (Ws)dWs+1 2

Z t 0

D²f (Ws)ds,

where Df and D²f are the generalized derivatives of f . Indeed, Itˆo’s formula holds for fⁿ, n ≥ 1, and we obtain the required result by sending n → ∞.

A similar statement holds for a function f (t, x).

Exercise 6.4. Let W be a Brownian motion in R^d and consider the process X_t := X₀+ bt + σW_t, t ≥ 0,

where b is a vector in R^d and σ is an (d × d)−matrix. Let f be a C^1,2(R+, R^d) function. Show that

df (t, X_t) = ∂f

∂t(t, X_t)dt +∂f

∂x(t, X_t) · dX_t+1 2Tr

 ∂²f

∂x∂x^T(t, X_t)σσ^T

 .

6.2. Extension to Itˆo processes 71

In document Stochastic Calculus in Finance (Page 64-71)