An addendum to ’An Introduction to Malliavin Calculus with Applications to Economics’

- Exercises with solutions and miscellaneous notes -

Fred Espen Benth

1 Introduction

In May 1998 professor Bernt Øksendal gave an advanced course in Malliavin Calculus with applications to economics at the Centre for Mathematical Physics and Stochastics (MaPhySto). The course consisted of lectures and tutorial classes.

This note contains the exercises with suggested solutions discussed in the classes. Notation and necessary theoretical background to understand the exercises can be found in the Lecture Notes, [Ø]. Also included is a section with some remarks and theoretical results discussed in the lectures, but not treated in [Ø].

2 Exercises

The “chapters” below refer to the Lecture Notes [Ø].

Chapter 1 The Wiener-Ito chaos expansion

Exercise 1. Exercises 1.1–1.3 in the Lecture Notes.

Exercise 2. Find the chaos expansion of

∫_{0}^{T} W (s)dW (s)

and calculate its variance.

Exercise 3. Find the chaos expansion of

exp (W (t) − \frac{1}{2}t)

Exercise 4. Calculate the L^{2}(Ω)-norm of

exp (W (t) − \frac{1}{2}t)

Exercise 5. Find the chaos expansion to the solution X_{t} of the stochastic differential equation

dX_{t} = μX_{t}dt + σX_{t}dW_{t}, t > 0
X_{0} = x

¿From its chaos expansion, calculate E[X_{t}] and E[X_{t}^{2}].

Chapter 2 The Skorohod integral

Exercise 1. Exercise 2.1 in the notes. Exercise 2. Use the definition of the Skorohod integral to calculate

∫_{0}^{T} e^{(W (t)− \frac{1}{2} t)}δW (t)

Exercise 3. Give conditions on g_{t}(·) ∈ L^{2}([0, T]) such that

X_{t} = exp (∫_{0}^{T} g_{t}(s)dW (s) − \frac{1}{2} ∫_{0}^{T} g_{t}^{2}(s)ds)

is F_{t}-adapted.

Chapter 4 Differentiation

Exercise 1. Exercise 4.1–4.3 in the Lecture Notes. Exercise 2. Find the Malliavin derivative of X_{t}, where X_{t} solves the stochastic differential equations

a) dX_{t} = μX_{t}dt + σX_{t}dW_{t}, X_{0} = x

b) dX_{t} = μX_{t}dt + σdW_{t}, X_{0} = x Exercise 3. Consider the Ornstein-Uhlenbeck process

X_{t} = x + μ ∫_{0}^{t} X_{u}du + ∫_{0}^{t} dW (u)

Apply the Malliavin derivative D_{s} directly to this equation to derive that

D_{s}X_{t} = \{ \begin{matrix}0 & if s > t \\ e^{μ(t−s)} & if s ≤ t \end{matrix}

Exercise 4. Consider the solution X_{t} of the stochastic differential equation

X_{t} = x + ∫_{0}^{t} b(X_{u})du + σ ∫_{0}^{t} dW (u)

Find D_{s}X_{t} for s ∈ [0, T]. (You can assume that b is continuously differentiable)

Exercise 5. Let u_{s} be a stochastic process. Show that

D_{t} ∫_{0}^{T} u_{s}δW (s) = u_{t} + ∫_{0}^{T} D_{t}u_{s}δW (s)

You can assume that all the objects involved are well-defined.

Exercise 6. Find the Malliavin derivative D_{s}X_{t}, where X_{t} is the solution of

X_{t} = x + ∫_{0}^{t} σ(X_{u})dW (u)

You can assume that σ is continuously differentiable. (Hint: Make use of exercise 5 above.)

Apply your result to the case σ(x) = σx and σ(x) = σ. (Check your results with exercise 2).

Chapter 5 The Clark-Ocone formula and its generalization

Exercise 1. Exercises 5.1–5.4 in the Lecture Notes.

3 Solutions to exercises

Chapter 1 The Wiener-Ito Chaos Expansion

Exercise 1:

Solutions to the exercises can be found in the Lecture Notes.

Exercise 2:

Writing W (s) = ∫_{0}^{s} dW (t_{1}) we get

∫_{0}^{T} W (s)dW (s) = ∫_{0}^{T} ∫_{0}^{t_{2}} dW (t_{1})dW (t_{2})
= J_{2}(f)

where f(t_{1}, t_{2}) = 1. f is obviously symmetric. Using the identity n!J_{n}(f_{n}) = I_{n}(f_{n}) for n = 2, we have the chaos representation

∫_{0}^{T} W (s)dW (s) = I_{2}(\frac{1}{2})

Since the expectation is zero the variance is given by the L^{2}(P)-norm of ∫_{0}^{T} W (s)dW (s). But by the Chaos Representation Theorem we have

∥ ∫_{0}^{T} W (s)dW (s)∥_{L^{2}(P)}^{2} = 2!∥\frac{1}{2}∥_{L^{2}([0,T]^{2})}^{2}

= 2 · \frac{1}{4} · T^{2}

= \frac{1}{2}T^{2}

Exercise 3: Using Exercise 1.1.d in the Lecture Notes we have

exp (W (t) − \frac{1}{2}t) = ∑_{n=0}^{∞} \frac{t^{n/2}}{n!} h_{n} (\frac{W (t)}{\sqrt{t}})

= ∑_{n=0}^{∞} ∫_{0}^{T} ∫_{0}^{t_{n}} ⋯ ∫_{0}^{t_{2}} 1_{[0,t]}(t_{1}) ⋯ 1_{[0,t]}(t_{n})dW (t_{1}) ⋯ dW (t_{n})

In the second equality we have used equation (1.14) in the Lecture Notes with g(s) = 1_{[0,t]}(s). Let

f_{n}(t_{1}, …, t_{n}) = 1_{[0,t]}(t_{1}) ⋯ 1_{[0,t]}(t_{n}) = 1_{[0,t]}^{⊗n}(t_{1}, …, t_{n})

The function f_{n} is symmetric. Thus

exp (W (t) − \frac{1}{2}t) = ∑_{n=0}^{∞} J_{n}(f_{n})

= ∑_{n=0}^{∞} \frac{1}{n!}I_{n}(f_{n})

= ∑_{n=0}^{∞} I_{n} (\frac{1_{[0,t]}^{⊗n}}{n!} )

Exercise 4: Recall from Exercise 3 above that exp(W (t) − t/2) has chaos expansion

exp (W (t) − \frac{1}{2}t) = ∑_{n=0}^{∞} I_{n} (\frac{1_{[0,t]}^{⊗n}}{n!} )

The Chaos Representation Theorem gives us the L^{2}(P)-norm of this object:

∥ exp (W (t) − \frac{1}{2}t) ∥_{L^{2}(P)}^{2} = ∑_{n=0}^{∞} n! ∥\frac{1_{[0,t]}^{⊗n}}{n!} ∥_{L^{2}([0,T]^{n})}^{2}

= ∑_{n=0}^{∞} \frac{1}{n!}∥1_{[0,t]}^{⊗n}∥_{L^{2}([0,T]^{n})}^{2}

Since ∥1_{[0,t]}^{⊗n}∥_{L^{2}([0,T]^{n})}^{2} is the volume of the n-dimensional cube in IR^{n} with edges of length t, we get

∥ exp (W (t) − \frac{1}{2}t) ∥_{L^{2}(P)}^{2} = ∑_{n=0}^{∞} \frac{1}{n!}t^{n} = e^{t}

Exercise 5:

We recognize the stochastic differential equation as geometric Brownian motion,

X_{t} = x exp ((μ − \frac{1}{2}σ^{2})t + σW (t))

Write X_{t} like

X_{t} = x exp(μt) exp (σW (t) − \frac{1}{2}σ^{2}t)

Use g(s) = σ1_{[0,t]}(s) in Exercise 1.1. c in the notes to derive

exp (σW (t) − \frac{1}{2}σ^{2}t) = ∑_{n=0}^{∞} \frac{∥g∥^{n}}{n!} h_{n} (\frac{σW (t)}{∥g∥})

But

∥g∥ = (∫_{0}^{t} σ^{2}ds)^{1/2} = σ\sqrt{t}

Hence,

X_{t} = x exp(μt) ∑_{n=0}^{∞} \frac{σ^{n}t^{n/2}}{n!} h_{n} (\frac{W (t)}{\sqrt{t}})

= x exp(μt) ∑_{n=0}^{∞} σ^{n}I_{n} (\frac{1_{[0,t]}^{⊗n}}{n!} )

= ∑_{n=0}^{∞} I_{n} (x exp(μt)\frac{σ^{n}}{n!} 1_{[0,t]}^{⊗n})

In the second equality we have made use of Exercise 3 above.

To calculate the expectation of X_{t} note that all the I_{n}’s have zero expectation except I_{0}. Thus

E[X_{t}] = I_{0} (x exp(μt)) = x exp(μt)

The L^{2}(P)-norm of X_{t} is calculated by applying the Chaos Representation Theorem again:

E[X_{t}^{2}] = ∥X_{t}∥_{L^{2}(P)}^{2} = ∑_{n=0}^{∞} n!∥x exp(μt)\frac{σ^{n}}{n!} 1_{[0,t]}^{⊗n}∥_{L^{2}([0,T]^{n})}^{2}

= ∑_{n=0}^{∞} n!\frac{x^{2} exp(2μt)}{(n!)^{2}}σ^{2n}t^{n}

= x^{2} exp(2μt) ∑_{n=0}^{∞} \frac{1}{n!}(σ^{2}t)^{n}

= x^{2} exp(2μt + σ^{2}t)

Chapter 2 The Skorohod Integral

Exercise 1:

Solutions to the exercises can be found in the Lecture Notes.

Exercise 2:

¿From previous exercises we have seen that exp(W (t)−t/2) has chaos expansion,

exp (W (t) − \frac{1}{2}t) = ∑_{n=0}^{∞} I_{n} (\frac{1_{[0,t]}^{⊗n}}{n!} )

Put

f_{n}(t_{1}, …, t_{n}; t) := \frac{1}{n!}1_{[0,t]}^{⊗n}(t_{1}, …, t_{n})

We calculate the symmetrization f̃_{n} of f_{n}:

f̃_{n}(t_{1}, …, t_{n+1}) = \frac{1}{n + 1} (f_{n}(t_{1}, …, t_{n}; t_{n+1}) + … + f_{n}(t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n+1}; t_{i})+
… + f_{n}(t_{2}, …, t_{n+1}; t_{1}))

where

f_{n}(t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n+1}; t_{i}) = 1_{[0,t]}(t_{1}) ⋯ 1_{[0,t]}(t_{i−1})1_{[0,t]}(t_{i+1}) ⋯ 1_{[0,t]}(t_{n+1})
= 1, iff t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n+1} < t_{i}

Note that for a given tuple t_{1}, …, t_{n+1} we can always find a t_{i} such that t_{i} > t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n+1}, namely t_{i} = max_{1≤j≤n+1} t_{j}. Thus

f̃_{n}(t_{1}, …, t_{n+1}) = \frac{1}{(n + 1)!}

We use the definition of the Skorohod integral to obtain

∫_{0}^{T} exp (W (t) − \frac{1}{2}t) δW (t) = ∑_{n=0}^{∞} \frac{1}{n!}I_{n+1}(f̃_{n}) = ∑_{n=0}^{∞} \frac{1}{(n + 1)!}I_{n+1}(1)

= exp (W (t) − \frac{1}{2}t) − 1

Exercise 3:

¿From the exercises in chapter 1 we know the chaos expansion of X_{t},

X_{t} = ∑_{n=0}^{∞} I_{n} (\frac{g_{t}^{⊗n}}{n!} )

The adaptedness of X_{t} can be characterized via its chaos functions by Lemma 2.5 in the notes:

X_{t} adapted ⇔ \frac{1}{n!}g_{t}^{⊗n}(t_{1}, …, t_{n}) = 0, if t < max_{1≤j≤n} t_{i}

But this is equivalent of saying that

g_{t}(t_{1}) ⋯ g_{t}(t_{n}) = 0, if t < max_{1≤j≤n} t_{i}

which again is equivalent to

g_{t}(s) = 0, if s > t (3.1)

Thus, to ensure adaptedness of X_{t} we need to impose the condition (3.1).

Chapter 4 Differentiation

Exercise 1: Solutions to the exercises can be found in the Lecture Notes.

Exercise 2:

a) The solution of the stochastic differential equation is the geometric Brownian motion,

X(t) = x exp ((μ − \frac{1}{2}σ^{2})t + σW (t))

Applying the Malliavin derivative D_{s} to X(t) gives,

D_{s}X(t) = x exp((μ − \frac{1}{2}σ^{2})t) · D_{s} exp(σW (t))

= x exp((μ − \frac{1}{2}σ^{2})t) exp(σW (t))σD_{s}W (t)

= σX(t) · 1_{[0,t]}(s)

We see that the Malliavin derivative is

D_{s}X(t) = \{ \begin{matrix}σX(t), & s ≤ t \\ 0, & s > t \end{matrix}

b) The solution X(t) is the Ornstein-Uhlenbeck process:

X(t) = exp(μt) (x + σ ∫_{0}^{t} exp(−μu)dW (u))

For our purposes it is convenient to rewrite this as

X(t) = exp(μt) (x + σ ∫_{0}^{T} 1_{[0,t]}(u) exp(−μu)dW (u))

Application of the Malliavin derivative yields,

D_{s}X(t) = exp(μt) (σD_{s} ∫_{0}^{T} 1_{[0,t]}(u) exp(−μu)dW (u))
= exp(μt)σ1_{[0,t]}(s) exp(−μs)
= σ exp(μ(t − s))1_{[0,t]}(s)

Note that the Malliavin derivative is deterministic in the case of an Ornstein-Uhlenbeck process, while it is stochastic for the geometric Brownian motion.

Exercise 3: By using the rules of the Malliavin derivative we get,

D_{s}X(t) = D_{s}x + μ ∫_{0}^{t} D_{s}X(u)du + D_{s} ∫_{0}^{T} 1_{[0,t]}(u)dW (u)
= μ ∫_{0}^{t} D_{s}X(u)du + 1_{[0,t]}(s)

Consider the case when s > t: We prove that D_{s}X(t) = 0 (confer Corollary 5.7 in the notes for a general statement). The equation becomes

D_{s}X(t) = μ ∫_{0}^{t} D_{s}X(u)du

Let ρ_{s}(t) = D_{s}X(t) for t ∈ [0, s). It is easily seen that ρ_{s}(t) satisfies the linear differential equation

\frac{dρ_{s}(t)}{dt} = μρ_{s}(t)

with initial condition ρ_{s}(0) = 0. This implies

ρ_{s}(t) = 0, for all t ∈ [0, s)

and we conclude

D_{s}X(t) = 0, s > t

Consider now the case s ≤ t: The equation for the Malliavin derivative becomes

D_{s}X(t) = 1 + μ ∫_{0}^{t} D_{s}X(u)du
= 1 + ∫_{s}^{t} D_{s}X(u)du

by using that D_{s}X(u) = 0 whenever s > u. Introduce again the function ρ_{s}(t) = D_{s}X(t), but this time for t ≥ s. We see that ρ_{s}(t) is the solution of the differential equation

\frac{dρ_{s}(t)}{dt} = μρ_{s}(t), t ≥ s

with initial condition ρ_{s}(s) = 1. The solution of this differential equation is known to be

ρ_{s}(t) = exp(μ(t − s))

which gives

D_{s}X(t) = exp(μ(t − s)), t ≥ s

Exercise 4:

First of all note that since X(t) is adapted, D_{s}X(t) is zero when s > t by Corollary 5.7 in the notes. Consider s ≤ t and apply the Malliavin derivative directly to the stochastic differential equation:

D_{s}X(t) = D_{s}x + ∫_{0}^{t} b′(X(u))D_{s}X(u)du + D_{s} ∫_{0}^{T} 1_{[0,t]}(u)σdW (u)
= ∫_{s}^{t} b′(X(u))D_{s}X(u)du + σ1_{[0,t]}(s)

Define

ρ_{s}(t) = D_{s}X(t)

It is easily seen that ρ_{s}(s) = σ and

\frac{dρ_{s}(t)}{dt} = b′(X(t))ρ_{s}(t), t > s

This differential equation has the solution

ρ_{s}(t) = σ exp (∫_{s}^{t} b′(X(u))du)

Thus we have proved that the Malliavin derivative D_{s}X(t) is

D_{s}X(t) = \{ \begin{matrix}σ exp(∫_{s}^{t} b′(X(u))du) & , & s ≤ t \\ 0 & , & s > t \end{matrix}

Exercise 5:

The proof of the result can be found in the Lecture Notes, Theorem 5.12. The following argument is, however, a direct calculation showing the relation: Let

u_{t} = ∑_{n=0}^{∞} I_{n}(u_{n}(·; t))

¿From the definition of the Malliavin derivative and the Skorohod integral, we have

D_{t} ∫_{0}^{T} u_{s}δW (s) = D_{t} (∑_{n=0}^{∞} I_{n+1}(ũ_{n}))

= ∑_{n=0}^{∞}(n + 1)I_{n}(ũ_{n}(·, t))

= ∑_{n=0}^{∞}(n + 1)I_{n} (\frac{1}{n + 1}\{u_{n}(t_{1}, …, t_{n}; t) + … + u_{n}(t_{2}, …, t_{n}, t; t_{1})\})

= ∑_{n=0}^{∞} I_{n}(u_{n}(·; t)) + ∑_{n=0}^{∞} I_{n} (∑_{i=1}^{n} u_{n}(t_{t}, …, t̂_{i}, …, t_{n}, t; t_{i}))

= u_{t} + ∑_{n=0}^{∞} nI_{n}(symm u_{n}(·, t; ·))

where the notation t̂_{i} means that t_{i} is removed and symm u_{n}(·, t; ·) is the symmetrization of the function u_{n}(t_{1}, …, t_{n−1}, t; t_{n}) for fixed t. Consider ∫_{0}^{T} D_{t}u_{s}δW (s):

∫_{0}^{T} D_{t}u_{s}δW (s) = ∫_{0}^{T} (∑_{n=0}^{∞} nI_{n−1}(u_{n}(·, t; s))) δW (s)

= ∑_{n=0}^{∞} nI_{n}(symm u_{n}(·, t; ·))

which proves the relation.

Exercise 6:

Corollary 5.7 gives that D_{s}X(t) = 0 whenever s > t since X(t) is adapted. Consider the case s ≤ t. Applying the Malliavin derivative directly to the stochastic differential equation and then using the relation from exercise 5 above, we get

D_{s}X(t) = D_{s}x + D_{s} ∫_{0}^{T} 1_{[0,t]}(u)σ(X(u))dW (u)
= σ(X(s))1_{[0,t]}(s) + ∫_{0}^{T} D_{s} (1_{[0,t]}(u)σ(X(u))) δW (u)
= σ(X(s)) + ∫_{0}^{t} σ′(X(u))D_{s}X(u)δW (u)
= σ(X(s)) + ∫_{s}^{t} σ′(X(u))D_{s}X(u)δW (u)

By corollary 5.7 again, we know that D_{s}X(u) is adapted with respect to the variable u. Thus

D_{s}X(t) = σ(X(s)) + ∫_{s}^{t} σ′(X(u))D_{s}X(u)dW (u)

We observe that the equation for D_{s}X(t) is a linear stochastic differential equation with a stochastic coefficient and initial variable. Moreover, σ(X(s)) is independent of W (u) when u ≥ s. Hence we get

D_{s}X(t) = σ(X(s)) exp (−\frac{1}{2} ∫_{s}^{t} σ′(X(u))^{2}du + ∫_{s}^{t} σ′(X(u))dW (u))

Letting σ(x) = σx or σ(x) = σ we reobtain the results from exercise 2 above.

Chapter 5 The Clark-Ocone Formula and its Generalization

Exercise 1:

Solutions to the exercises can be found in the Lecture Notes.

4 Miscellaneous Results and Remarks presented in the lectures

Proof of Ito’s identity between n-fold Wiener integrals and Hermite polynomials (eq. (1.14) in the Lecture Notes.)

The following argument was suggested by F. Oertel from University of Bonn, Germany.

Define the “normalized” Hermite polynomial

H_{n}(x, λ) := λ^{n/2}h_{n} (\frac{x}{\sqrt{λ}})

¿From the series representation

exp(tx − tλ^{2}/2) = ∑_{n=0}^{∞} \frac{t^{n}}{n!} H_{n}(x, λ)

we derive the following properties:

\frac{∂}{∂x}H_{n}(x, λ) = nH_{n−1}(x, λ)

\frac{∂^{2}}{∂x^{2}} H_{n}(x, λ) + \frac{∂}{∂λ}H_{n}(x, λ) = 0

The proof of the identity

I_{n}(g^{⊗n}) = H_{n}(∫_{0}^{T} g(s)dW (s), ∥g∥^{2})

goes by induction on n:

I_{n+1}(g^{⊗n+1}) = (n + 1)!J_{n+1}(g^{⊗n+1}|S_{n+1})
= (n + 1) ∫_{0}^{T} g(s)I_{n}(g^{⊗n})dW (s)

where we have restricted the I_{n} inside the integral to [0, s]^{n}. Using the Induction Assumption, we have

I_{n+1}(g^{⊗n+1}) = (n + 1) ∫_{0}^{T} g(s)H_{n}(∫_{0}^{s} g(u)dW (u), ∥g∥_{L^{2}([0,s])}^{2})dW (s)

= ∫_{0}^{T} \frac{∂}{∂x}H_{n+1}(∫_{0}^{s} g(u)dW (u), ∥g∥_{L^{2}([0,s])}^{2})dW (s)

= H_{n+1}(∫_{0}^{T} g(s)dW (s), ∥g∥^{2})

In the last equality we have used the Itos Formula and the second property of H_{n} stated above.

Remark to the chaos representation theorem

In the Chaos Representation Theorem it was proved that every ϕ ∈ L^{2}(P) has a representation

ϕ = ∑_{n=0}^{∞} I_{n}(f_{n})

where f_{n}(·) ∈ L̂^{2}([0, T]^{n}). We here prove that this representation is unique:

Proof. Let ϕ = ∑_{n=0}^{∞} I_{n}(g_{n}) be another expansion of ϕ. Then by linearity of the n-fold Wiener integrals we have

0 = ∑_{n=0}^{∞} I_{n}(f_{n} − g_{n})

But this expansion will have L^{2}(P)-norm equal to

0 = ∑_{n=0}^{∞} n!∥f_{n} − g_{n}∥_{L^{2}([0,T]^{n})}^{2}

Hence, f_{n} = g_{n} a.e.

How to calculate the functions in the chaos expansion

The following characterization of the chaos functions is taken from Ustunel, [U]: Let ϕ ∈ L^{2}(P) have the chaos representation ϕ = ∑_{n=0}^{∞} I_{n}(f_{n}). Then

f_{n}(t_{1}, …, t_{n}) = \frac{1}{n!}E [D_{t_{1}} ⋯ D_{t_{n}}ϕ]

D_{t} denotes the Malliavin Derivative.

On the adjointness of Malliavin differentiation and Skorohod integration

The Malliavin derivative transforms random variables into stochastic processes, while the Skorohod integral maps stochastic processes into random variables,

D :L^{2}(Ω) → L^{2}([0, T] × Ω)
δ :L^{2}([0, T] × Ω) → L^{2}(Ω)

(we suppress the domain of definition to illustrate the action of the two operators). We have the following adjointness relation (see e.g. Nualart, [N]): If F ∈ D_{1,2} and u_{t} ∈ Dom(δ),

E [F δ(u)] = E [(D_{t}F, u_{t})_{L^{2}([0,T])}]

where (·, ·)_{L^{2}([0,T])} is the inner product of L^{2}([0, T]).

Generalization of the Clark-Ocone Formula

The following extension of the Clark-Ocone Formula has been proved by Aase, Øksendal and Ubøe, [AaØU]:

Define the spaces G_{λ} for λ ∈ IR as the set of random variables F ∈ L^{2}(Ω) where

∥F ∥_{2,λ}^{2} := ∑_{n=0}^{∞} n!e^{2λn}∥f_{n}∥_{L^{2}([0,T]^{n})}^{2}

The f_{n}’s are the chaos functions of F. Defining

G := ∩_{λ≥0}G_{λ}

and

G^{∗} := ∪_{λ≥0}G_{−λ}

we have the following triplet of smooth and generalized random variables:

G ⊂ L^{2}(μ) ⊂ G^{∗}

Note that G and G^{∗} are dual spaces, where G is equipped with the projective limit topology and G^{∗} the inductive limit topology.

The Malliavin derivative is extended to the space G^{∗} such that the following holds:

D_{t}Φ = ∑_{n=1}^{∞} nI_{n−1}(Φ_{n}(·, t))

whenever D_{t}Φ ∈ G^{∗}. In the sense of random distributions, the Clark-Ocone formula is proved for Malliavin differentiable elements in G^{∗}:

Φ = E[Φ] + ∫_{0}^{T} E[D_{t}Φ|F_{t}]dW (t)

The formula also has a multi-dimensional analogue. Conditional expectation and Wiener integration can be defined for distributions in G^{∗}. We refer to [AaØU] for details.

More information about the spaces G and G^{∗} can be found in [PT].

References

[AaØU] K. Aase, B. Øksendal and J. Ubøe: White Noise Generalizations of the Clark-Ocone Formula. Manuscript, University of Oslo, Norway, 1998.

[N] D. Nualart: “The Malliavin Calculus and Related Topics”. Springer-Verlag, 1995.

[Ø] B. Øksendal: An Introduction to Malliavin Calculus with Applications to Economics. Working Paper No. 3/96, Norwegian School of Economics, Bergen, Norway, 1996. (Lecture Notes).

[PT] J. Potthoff and M. Timpel: On a Dual Pair of Spaces of Smooth and Generalized Random Variables. Potential Analysis, 4, 1995. pp. 637-654.

[U] A. S. Ustunel: “An Introduction to Analysis on Wiener Space”. Lecture Notes in Mathematics 1610, Springer-Verlag, 1995.