A Hilbert space approach to Wiener Chaos Decomposition and applications to finance

Work in Progress

Frank Oertel Zurich University of Applied Sciences ZHW Department T – Mathematics and Physics CH – 8401 Winterthur

                 frank.oertel@zhwin.ch

Abstract

The Malliavin calculus (also known as the stochastic calculus of variations) is an infinite–dimensional differential calculus on the Wiener space, initiated by Malliavin and further developed by Bismut, Stroock and others (cf. [9], [2], [14]) and which has been shown as a very important tool for mathematical physicists working in Quantum Field Theory.

We consider the first aspects of this calculus which have been used in financial applications, such as the pricing and hedging of path dependent options or the investigation of residual risk in incomplete markets. To be able to understand these applications, we have to work through the theory and methods of the underlying mathematical machinery. In this paper, we use only some elementary Hilbert space methods to obtain another description of the multiple Wiener integral (given a finite time horizon T > 0 and the natural filtration induced by the Wiener process W) and its relations to a decomposition of an arbitrary random variable in L^{2}(Ω, F_{T}^{W} , ℙ) in successive Wiener chaoses, which works in a similar way as a Taylor’s expansion but taking into account the stochastic dynamics. In particular we will present the main ideas, leading to this representation of L^{2}(Ω, F_{T}^{W} , ℙ). We recognize the appearance of the well known family of stochastic exponentials and its relations to the Itô representation of such L^{2}–random variables.

1 Introduction

Let T > 0 and (Ω, F, ℙ) be a probability space. Given a (one-dimensional) Wiener process W and its natural filtration 𝔽^{W} = (F_{t}^{W})_{0≤t≤T}, let us recall the standard example in the classical Black & Scholes-market (without transaction costs), where the stock price S is modelled as dS_{t} = μS_{t} dt + σS_{t} dW_{t} and the price of the riskless asset is given by dB_{t} = rB_{t} dt, where μ, σ ∈ ℝ, σ ≠ 0 and r ≥ 0. Then it is well known that there exists an eqivalent martingale measure (EMM) Q ∼ ℙ so that B_{t} := W_{t} + \frac{μ−r}{σ}t defines a standard Q-Wiener Process B (so that 𝔽^{W} = 𝔽^{B}) and S̃_{t} := exp(−rt) S_{t} defines a Q-martingale S̃, which implies that every contingent claim Z ∈ L^{2}(Ω, F_{T}^{B}, Q) can be replicated by a self-financing portfolio and that the value of any replicating self-financing portfolio Π at time t is given by

Π_{t} = 𝔼_{Q}(e^{−r(T −t)}Z | F_{t}^{B}) .

The existence of such a replicating portfolio follows by the Kunita-Watanabe representation theorem - applied to the the Q−square-integrable martingale Π (cf. [7]):

Theorem 1.1 Let M = (M_{t})_{0≤t≤T} be a square-integrable martingale, with respect to the filtration 𝔽^{B}. Then there exists an adapted process K so that 𝔼_{Q}(∫_{0}^{T} K_{s}^{2} ds) < ∞ and

M_{t} = M_{0} + ∫_{0}^{t} K_{s} dB_{s}

for all t ∈ [0, T].

Corrolary 1.1 Let Z ∈ L^{2}(Ω, F_{T}^{B}, Q). Then there exists an adapted process L so that 𝔼_{Q}(∫_{0}^{T} L_{s}^{2} ds) < ∞ and

Z = 𝔼_{Q}(Z) + ∫_{0}^{T} L_{s} dB_{s} .

Unfortunately, this is only a statement leading to the existence of a hedging strategy H := K/σS̃, which in general does not tell us, how this portfolio looks like. However, if the contingent claim Z can be written as Z = f(S_{T}), (e.g., if Z is a standard Call option) we can express the portfolio value Π_{t} as a sufficiently smooth function of t and S_{t}, Π_{t} = F (t, S_{t}), and an application of Itô’s formula leads to the concrete value H_{t} = \frac{∂F}{∂x} (t, S_{t}) - satisfying trader’s needs. Here, Malliavin calculus enters the scene. Due to this calculus, it is possible to transfer the Kunita-Watanabe representation to random variables which are functionals of Brownian motion. It turns out that the process L in the previous corollary can be identified as the optional projection of the Malliavin derivative process (D_{t}F)_{0≤t≤T} of F. In these lecture notes we finally explain the construction of such an derivative^{1} and prove and formulate precisely the following (cf. [5])

Theorem 1.2 Let Z = F (B) be a functional of the Brownian motion B. Then, under technical hypotheses on F

Z = 𝔼_{Q}(Z) + ∫_{0}^{T} 𝔼_{Q}(D_{s}F | F_{s}^{B}) dB_{s} .

1This means, that we want to ”differentiate” F with respect to the chance variable ω ∈ Ω - without assuming any topological structure on Ω.

This result, built on deep theorems in Malliavin calculus, directly lead to applications in finance, such as the construction of hedging portfolios of certain exotic options or the investigation of discontinuous path-dependent payoff functionals of multidimensional diffusion processes (cf. [1], [3], [4]). A general approach to the analysis of dynamic hedging portfolos is discussed in [8]. The common source of these papers (and the basics of the Malliavin calculus) is the nontrivial fact that option prices can be decomposed in Wiener chaoses. This chaotic decomposition works in a similar way as a Taylor’s power series expansion in complex analysis, taking additionally into account the stochastic dynamics of the underlying asset price process. As in complex analysis, we will see, that the Malliavin derivative allows us to differentiate piecewise ”power series” (in L^{2}(Ω)), and we will obtain the Itô integral as a special case of the Skorohod integral, which is the adjoint of the Malliavin derivative operator. Since the Wiener chaos decomposition plays the crucial role in the Malliavin calculus, we give a detailed introduction to it, including complete proofs. We only use elementary facts of Hilbert space theory. Nevertheless, our (analytic) proofs could lead to a transfer to more general semimartingales than Brownian motion.

2 The construction of the multiple Wiener integral

Let T > 0 fixed and n ∈ ℕ. Let f ∈ L^{2}(S_{n}(T)), where S_{n}(T) := {(t_{1}, t_{2,}..., t_{n}) ∈ [0, T]^{n} : t_{1} ≤ t_{2} ≤ ...t_{n}} denotes the n− simplex in the cube [0, T]^{n}. Let W be a Wiener process (Brownian motion) and 𝔽^{W} = (F_{t}^{W})_{0≤t≤T} the usual W -augmented filtration. We want to construct an iterated Itô-integral, in the following sense

J_{n}f = ∫_{0}^{T} (∫_{0}^{t_{n}} ...(∫_{0}^{t_{2}} f(t_{1}, t_{2}, ..., t_{n})dW_{t_{1}})...dW_{t_{n−1}})dW_{t_{n}} .  (1)

To prove and understand the existence of (1), we use the well known Hilbert space approach to the Itô-integral. So let us recall the basic Hilbert spaces which lead to the fundamental Itô isometry: L^{2}(Ω × [0, T], P^{W} , λ_{T} ⊗ ℙ) = : L_{T}^{2} (W) is the Hilbert space of all predictable processes X with ∥X∥_{L_{T}^{2} (W)}^{2} = 𝔼( ∫_{0}^{T} X_{s}^{2}ds) < ∞, where P^{W} denotes the predictable σ−algebra, i.e., the σ−algebra, generated by all left-continuous and 𝔽^{W} -adapted processes. Since W is (left-)continuous, 𝔽^{W} -adapted processes and 𝔽^{W} -predictable processes coincide. If X ∈ L_{T}^{2} (W), then it is well known, that the stochastic integral X • W = (∫_{0}^{t} X_{s} dW_{s})_{0≤t≤T} is a L^{2}(Ω)-bounded (and continuous) martingale: X • W ∈ M_{T}^{2}(W) := {Z : Z is a 𝔽^{W} - martingale and sup_{0≤s≤T} 𝔼(Z_{s}^{2}) < ∞}. Thus, it is an uniformly integrable and (in particular)

closable martingale with closing element (X • W)_{T} = ∫_{0}^{T} X_{s}dW_{s} ∈ L^{2}(Ω, F_{T}^{W} , ℙ). Let us recall the following

Theorem 2.1 (Itô Isometry) L_{T}^{2} (W) ↪^{1} L^{2}(Ω, F_{T}^{W} , ℙ), X ↦ (X • W)_{T} is an isometric linear embedding:

𝔼((X • W)_{T}^{2} ) = 𝔼( ∫_{0}^{T} X_{s}^{2}ds).

Let L_{0}^{2}(Ω, F_{T}^{W} , ℙ) := {X ∈ L^{2}(Ω) : 𝔼(X) = 0}. Then L_{0}^{2}(Ω, F_{T}^{W} , ℙ) obviously is a subspace of the Hilbert space L^{2}(Ω, F_{T}^{W} , ℙ), and the Kunita-Watanabe representation theorem, together with the Itô isometry now reveal the small ”size” of the space L_{T}^{2} (W) of all square integrable 𝔽^{W} -adapted processes:

Theorem 2.2 L_{T}^{2} (W) =^{∼} L_{0}^{2}(Ω, F_{T}^{W} , ℙ).

To construct the iterated Itô integral recursively, we first take a closer look at the finite dimensional simplexes. So, let k ≥ 2 be an arbitrary natural number and t > 0. Given a function g : ℝ^{k} ⟶ ℝ, we set g(·, s)(u) := g(u, s), where u ∈ ℝ^{k−1} and s ∈ ℝ. Since

1l_{S_{k}(t)}(u, s) = 1l_{S_{k−1}(s)}(u) · 1l_{[0,t]}(s)

for all (u, s) ∈ S_{k}(t), Fubini’s theorem directly leads to the following

Lemma 2.1 If g ∈ L^{2}(S_{k}(t)), then g(·, s) ∈ L^{2}(S_{k−1}(s)) for all s ∈ [0, t], and

∫_{S_{k}(t)} g(v) d^{k}v = ∫_{0}^{t} ( ∫_{S_{k−1}(s)} g(u, s) d^{k−1}u) ds .

The above isometry and Fubini-allowed interchanging of the operators 𝔼 and ∫_{0}^{t}, now lead us to a precise meaning of the iterated Itô-integral. Given an arbitrary t > 0, we will define linear isometries Y^{(k)} : L^{2}(S_{k}(t)) → L_{t}^{2}(W) recursively by^{2}

Y^{(1)}g(ω, s) := g(s) (s ∈ [0, t]) [deterministic]

and

Y^{(k)}g(ω, s) := (Y^{(k−1)}g(·, s) • W)_{s}(ω) (s ∈ [0, t])

for k > 1. The next proposition will show us that these operators are well defined.

Proposition 2.1 Let k, l ∈ ℕ and t > 0. Then Y^{(k)} : L^{2}(S_{k}(t)) → L_{t}^{2}(W) defines a linear isometry from the Hilbert space L^{2}(S_{k}(t)) into the Hilbert space L_{t}^{2}(W) and, if k ≠ l, then

(Y^{(k)}g | Y^{(l)}h)_{L_{t}^{2}(W)} = 0 for all g ∈ L^{2}(S_{k}(t)) and h ∈ L^{2}(S_{l}(t)).  (2)

Proof: Nothing is to show for k = 1. So let g ∈ L^{2}(S_{k+1}(t)) given and the linear isometry Y^{(k)} : L^{2}(S_{k}(s)) → L_{s}^{2}(W) already be constructed (for arbitrary s ∈ [0, T]). Let s ∈ [0, t]. Fubini’s theorem implies that g(·, s) ∈ L^{2}(S_{k}(s)), so that Y^{(k)}g(·, s) • W ∈ M_{s}^{2}(W) exists. Hence, setting (Y^{(k+1)}g)_{s} := (Y^{(k)}g(·, s)•W)_{s} = ∫_{0}^{s}(Y^{(k)}g(·, s))_{u} dW_{u} ∈ L^{2}(Ω, F_{s}^{W} , ℙ), the previous considerations show that

𝔼( ∫_{0}^{t}(Y^{(k+1)}g)_{s}^{2} ds) = ∫_{0}^{t} 𝔼((Y^{(k+1)}g)_{s}^{2}) ds = ∫_{0}^{t}(𝔼(∫_{0}^{s}(Y^{(k)}g(·, s))_{u}^{2}) du) ds.

By assumption, 𝔼(∫_{0}^{s}(Y^{(k)}g(·, s))_{u}^{2}) du = ∫_{S_{k}(s)} g^{2}(u, s) d^{k}u, and it follows that

∥Y^{(k+1)}g∥_{L_{t}^{2}(W)}^{2} = ∫_{S_{k+1}(t)} g^{2}(v) d^{k+1}v = ∥g∥_{L^{2}(S_{k+1}(t))}^{2} < ∞.

Now we will prove (2). Given m ∈ ℕ, we have to show that (Y^{(k)}g | Y^{(k+m)}h)_{L_{t}^{2}(W)} = 0 for all k ∈ ℕ. First, let k = 1. Then (Y^{(1)}g | Y^{(1+m)}h)_{L_{t}^{2}(W)} = 𝔼(∫_{0}^{t} g(s) · (Y^{(1+m)}h)_{s} ds) = ∫_{0}^{t} g(s) · 𝔼((Y^{(1+m)}h)_{s}) ds = 0, since Y^{(1)}g = g is deterministic and Y^{(m)}h(·, s) • W is a martingale which starts at 0. Let g ∈ L^{2}(S_{k+1}(t)) and h ∈ L^{2}(S_{k+1+m}(t)). Then, given the assumption that (2) is valid for k, the Itô isometry implies that in particular

2More precisely, we should use the symbol Y_{t}^{(k)} to denote the dependence on the parameter t.

𝔼((Y^{(k+1)}g)_{s} · (Y^{(k+1+m)}h)_{s}) = ((Y^{(k+1)}g)_{s} | (Y^{(k+1+m)}h)_{s})_{L^{2}(Ω,F_{s}^{W} ,ℙ)}
= (Y^{(k)}g(·, s) | Y^{(k+m)}h(·, s))_{L_{s}^{2}(W)}
= 0

for all s ∈ [0, t]. Integrating both sides over [0, t], finishes the proof. □

Now we define for arbitrary T > 0, n ∈ ℕ and f ∈ L^{2}(S_{n}(T))

J_{n}f := (Y^{(n)}f • W)_{T} = ∫_{0}^{T} (Y^{(n)}f)_{s} dW_{s} .

Evaluating this expression recursively, it follows by construction that

J_{n}f = ∫_{0}^{T} J_{n−1}f(·, s) dW_{s}

for all n > 1, and we obtain the above expression (1). Setting J_{0}c := c for c ∈ L^{2}(S_{0}(T)) := ℝ, we arrive at the following

Theorem 2.3 Let n ∈ ℕ_{0} and T > 0. Then

J_{n} : L^{2}(S_{n}(T)) ↪^{1} L^{2}(Ω, F_{T}^{W} , ℙ)

is a linear isometry, and if m ∈ ℕ_{0} with m ≠ n, then

𝔼(J_{n}f · J_{m}g) = (J_{n}f | J_{m}g)_{L^{2}(Ω,F_{T}^{W} ,ℙ)} = 0

for all f ∈ L^{2}(S_{n}(T) and g ∈ L^{2}(S_{m}(T)).

Proof: J_{n} is the composition of two linear isometries, namely

L^{2}(S_{n}(T)) ↪^{1} L_{T}^{2} (W) →^{≅} L^{2}(Ω, F_{T} , ℙ), f ↦ Y^{(n)}f ↦ (Y^{(n)}f • W)_{T} □

Until now, we have defined the multiple Wiener integral only for functions belonging to L^{2}(S_{n}(T)) with given T > 0. The definition of the simplex S_{n}(T) = {(t_{1}, t_{2}, ..., t_{n}) ∈ [0, T]^{n} : t_{1} ≤ t_{2} ≤ ... ≤ t_{n}} leads to the conclusion that the order structure of ℝ^{n} plays a fundamental role in the construction of the mutiple integral, but it is not really so. Let f ∈ L^{2}([0, T]^{n}) be a symmetric function; i.e.: f(t_{1}, t_{2}, ..., t_{n}) = f(t_{σ(1)}, t_{σ(2)}, ..., t_{σ(n)}) for all permutations σ ∈ 𝔖_{n}. Let L̂^{2}([0, T]^{n}) denote the (closed) linear subspace of all symmetric functions in L^{2}([0, T]^{n}).

Proposition 2.2 Let n ∈ ℕ and T > 0. Then^{3}

∥f∥_{L^{2}([0,T]^{n})}^{2} = n! · ∥f | S_{n}(T)∥_{L^{2}(S_{n}(T))}^{2}  (3)

for all f ∈ L̂^{2}([0, T]^{n}).

3Often, we will denote f | S_{n}(T) by the same symbol f, if no misunderstanding is possible.

Proof: Nothing is to show for n = 1. So, let (3) be true for n ∈ ℕ and let f ∈ L̂^{2}([0, T]^{n+1}). Set H_{k} = L^{2}([0, T]^{k}). Then, by induction assumption and Fubini

∥f∥_{H_{n+1}}^{2} = ∫_{0}^{T} ∥f(·, s)∥_{H_{n}}^{2} ds = n! ∫_{0}^{T} ∥f(·, s)∥_{S_{n}(T)}^{2} ds
= n! ∫_{S_{n}(T)×[0,T]} f^{2}(t) d^{n+1}t .

Now,

S_{n}(T) × [0, T] = {(τ, s) ∈ [0, T]^{n+1} : τ ∈ S_{n}(T)} = A_{n+1}(T) ∪ S_{n+1}(T) ,

where A_{n+1}(T) := {(τ, s) ∈ [0, T]^{n+1} : τ ∈ S_{n}(T), s < τ_{n}}. Since exactly n ordered positions for such an s are possible, A_{n+1}(T) equals to the disjoint junion ⋃_{k=1}^{n} Λ_{k}^{−1}(S_{n+1}(T)) (a.s.), where Λ_{k} ∈ O(n + 1; ℝ) denotes the matrix which maps the unit vector e_{l} to e_{l} if 1 ≤ l < k, e_{l} to e_{l+1} if k ≤ l < n + 1 and e_{n+1} to e_{k}. Since f is symmetric, it follows in particular that f^{2}(t) = f^{2}(Λ_{k}t) for all k ∈ {1, ..., n} and t ∈ [0, T]^{n+1}, so that the transformation formula implies that

∫_{S_{n+1}(T)} f^{2}(t) d^{n+1}t = ∫_{Λ_{k}^{−1}(S_{n+1}(T))} f^{2}(Λ_{k}t) d^{n+1}t = ∫_{Λ_{k}^{−1}(S_{n+1}(T))} f^{2}(t) d^{n+1}t,

and the proof is finished. □

Using this fact, we now transfer the definition of the multiple Wiener integral to functions f ∈ L̂^{2}([0, T]^{n}) in the following sense:

I_{n}f := n! · J_{n}(f | S_{n}(T)).

By the previous considerations, I_{n} : L̂^{2}([0, T]^{n}) → L^{2}(Ω, F_{T} , ℙ) is a linear continuous operator, and:

𝔼((I_{n}f)^{2}) = ∥I_{n}f∥_{L^{2}(Ω,F_{T} ,ℙ)}^{2} = n! · ∥f∥_{L^{2}([0,T]^{n})}^{2}.

3 Hermite Polynomials and Chaos Decomposition

In the following paragraph we consider functions in L̂^{2}([0, T]^{n}) of type

g^{⊗n}(x_{1}, ..., x_{n}) := ∏_{i=1}^{n} g(x_{i}) ,

where g ∈ L^{2}([0, T]). We will recognize that these symmetric products are the cornerstones of the factorization of L^{2}(Ω, F_{T} , ℙ) through multiple Wiener integrals. To that end, let us recall the well known Hermite polynomials (n ∈ ℕ_{0}, x ∈ ℝ) :

h_{n}(x) := (−1)^{n} · exp(\frac{x^{2}}{2} ) · g^{(n)}(x) ,

where g(x) := exp(−\frac{x^{2}}{2} ). The first Hermite polynomials are h_{0}(x) = 1, h_{1}(x) = x, h_{2}(x) = x^{2} − 1 and h_{3}(x) = x^{3} − 3x. Given a > 0, we put

H_{n}(x, a) := \sqrt{a^{n}} · h_{n}(\frac{x}{\sqrt{a}}).

Then we have the following Lemma 3.1 Let t ∈ ℝ, x ∈ ℝ and a > 0. Then

(i) exp(tx − \frac{t^{2}}{2} ) = ∑_{n=0}^{∞} \frac{t^{n}}{n!} · h_{n}(x)

(ii) exp(tx − \frac{at^{2}}{2} ) = ∑_{n=0}^{∞} \frac{t^{n}}{n!} · H_{n}(x, a).

Proof: Let x ∈ ℝ fixed. Since exp(tx − \frac{t^{2}}{2} ) = exp(\frac{x^{2}}{2} ) · (g ∘ τ_{x})(t), where τ_{x}(t) := x − t and g(y) := exp(−\frac{y^{2}}{2} ), Taylor’s formula applied to g ∘ τ_{x} leads to

exp(tx − \frac{t^{2}}{2} ) = exp(\frac{x^{2}}{2} ) · ∑_{n=0}^{∞} \frac{(g ∘ τ_{x})^{(n)}(0)}{n!} · t^{n}

= exp(\frac{x^{2}}{2} ) · ∑_{n=0}^{∞}(−1)^{n} · \frac{(g^{(n)} ∘ τ_{x})(0)}{n!} · t^{n}

= ∑_{n=0}^{∞} \frac{t^{n}}{n!} · h_{n}(x) ,

and (i) is proven. Now, (ii) follows directly, by setting exp(tx − \frac{at^{2}}{2} ) = exp(sy − \frac{s^{2}}{2} ), where s := t\sqrt{a} and y := \frac{x}{\sqrt{a}}. □

Due to partial differentiation of the function (x, a) ↦ exp(tx− \frac{at^{2}}{2} ), the corollary immediately implies the important

Remark 3.1 \frac{∂}{∂x}H_{n}(x, a) = n · H_{n−1}(x, a) and (\frac{1}{2} \frac{∂^{2}}{∂x^{2}} + \frac{∂}{∂a})H_{n}(x, a) = 0 on ℝ × ℝ_{+}^{∗} .

Now we can prove the following representation:

Theorem 3.1 Let T > 0 and g ∈ L^{2}([0, T]). Then g^{⊗n} ∈ L̂^{2}([0, T]^{n}) for all n ∈ ℕ and

I_{n}(g^{⊗n}) = H_{n}(X_{T} , ⟨X, X⟩_{T}) = H_{n}((g • W)_{T} , ∥g∥_{L^{2}([0,T]}^{2}) ,  (4)

where X := g • W.

Proof: We will prove the statement by induction on n. Nothing is to show for n = 1. Let (4) be valid for n ∈ ℕ and set ϕ_{n+1} := g^{⊗n+1} | S_{n+1}(T). Then ϕ_{n+1}(·, s) = (g | [0, s])^{⊗n} · g(s) on S_{n}(s) for all s ∈ [0, T], and the definition of I_{n+1} implies that

I_{n+1}g^{⊗n+1} = (n + 1)! · J_{n+1}ϕ_{n+1} = (n + 1)! · ∫_{0}^{T} (Y^{(n+1)}ϕ_{n+1})_{s} dW_{s}
= (n + 1)! · ∫_{0}^{T} (Y^{(n)}ϕ_{n+1}(·, s) • W)_{s} dW_{s}
= (n + 1)! · ∫_{0}^{T} g(s) · (Y^{(n)}(g | [0, s])^{⊗n} • W)_{s} dW_{s}
= (n + 1) · ∫_{0}^{T} g(s) · I_{n}(g | [0, s])^{⊗n} dW_{s}
= (n + 1) · ∫_{0}^{T} g(s) · H_{n}(X_{s} , ⟨X, X⟩_{s}) dW_{s} .

On the other hand, using the previous remark, Itô’s formula applied to H_{n+1}(X_{T} , ⟨X, X⟩_{T}) leads to

H_{n+1}(X_{T} , ⟨X, X⟩_{T}) = ∫_{0}^{T} D_{1}H_{n+1}(X_{s} , ⟨X, X⟩_{s}) dX_{s} + 0
= (n + 1) · ∫_{0}^{T} H_{n}(X_{s} , ⟨X, X⟩_{s}) dX_{s}
= (n + 1) · ∫_{0}^{T} H_{n}(X_{s} , ⟨X, X⟩_{s}) · g(s) dW_{s} ,

and the proof is finished. □

Let T > 0. To prove our main theorem in this paragraph - the Chaos representation of L^{2} -, we need a deeper investigation of the set {exp((g •W))_{T} : g ∈ L^{2}([0, T])}. Let g ∈ L^{2}([0, T]). Then X := g • W ∈ M_{T}^{2}(W) and exp(X_{T}) = E(X)_{T} · exp(\frac{1}{2}∥g∥_{L^{2}([0,T])}^{2}) is an element of L^{1}(Ω, F_{T} , ℙ) with 𝔼( exp(X_{T})) ≤ exp(\frac{1}{2}∥g∥_{L^{2}([0,T]}^{2}) < ∞ (since E(X) is a positive supermartingale^{4}), but in our case, we are able to show the following:

Lemma 3.2

(i) exp((g • W)_{T}) ∈ L^{2}(Ω, F_{T} , ℙ) for all g ∈ L^{2}([0, T]);

(ii) {exp((g • W)_{T}) : g ∈ L^{2}([0, T])} is total in L^{2}(Ω, F_{T} , ℙ).

Proof: \underline{ad (i)}: Let X = g • W. Due to Lemma 5 and the previous theorem, it follows (for t = 1) that

E(X)_{T} = ∑_{n=0}^{∞} \frac{1}{n!} · H_{n}(X_{T} , < X, X >_{T}) = ∑_{n=0}^{∞} \frac{1}{n!}I_{n}(g^{⊗n})

holds pointwise on Ω. Now we show that this convergence still holds in L^{2}(Ω). Let us

consider the finite sums Z_{n} := ∑_{k=0}^{n} \frac{1}{k!}I_{k}(g^{⊗k}) = ∑_{k=0}^{n} J_{k}(g^{⊗k} | S_{k}(T)) and let m < n. Each

Z_{n} belongs to L^{2}(Ω), and it follows (by orthogonality and isometry) that ∥Z_{n} − Z_{m}∥_{L^{2}(Ω)}^{2} =

∑_{k=m+1}^{n} \frac{1}{k!}∥g^{⊗k}∥_{L^{2}(S_{k}(T))}^{2} = ∑_{k=m+1}^{n} \frac{1}{k!}(∥g∥_{L^{2}([0,T])}^{2})^{k}. Hence (Z_{n}) is a Cauchy sequence in L^{2}(Ω),

and it follows the existence of a Z ∈ L^{2}(Ω) so that ∥Z_{n} − Z∥_{L^{2}(Ω)} → 0 for n → ∞. In particular, there exists a subsequence (Z_{n_{k}}) of (Z_{n}) so that Z = lim_{k→∞} Z_{n_{k}} pointwise in Ω.

Thus E(X)_{T} = Z ∈ L^{2}(Ω) and therefore exp(X_{T}) = Z · exp(\frac{1}{2}∥g∥_{L^{2}([0,T]}^{2}) ∈ L^{2}(Ω).

\underline{ad (ii)}: Let N be the L^{2}-closure of the linear hull M of the set \{exp((g • W)_{T}) : g ∈ L^{2}([0, T])}. Since L^{2}(Ω) = N ⊕ N^{⊥} and N^{⊥} = M^{⊥}, we only have to show that M^{⊥} = 0. So let ξ ∈ M^{⊥}. By linearity of the stochastic integral, it follows that

𝔼(ξ · exp(∑_{i=1}^{n} λ_{i} · (g_{i} • W)_{T})) = 0

4Indeed, Novikov’s condition implies that E(X) is still an uniformly integrable martingale.

for all n ∈ ℕ, λ_{1}, ..., λ_{n} ∈ ℝ and g_{1}, ..., g_{n} ∈ L^{2}([0, T]). In particular we obtain for arbitrary t_{1}, ..., t_{n} ∈ [0, T]

𝔼(ξ · exp((λ | Z(ω))_{ℝ^{n}})) = 0 ,  (5)

where λ := ( λ_{1}, ..., λ_{n}) and Z := ((1_{(0,t_{1}]} • W)_{T} , ...(1_{(t_{n−1},t_{n}]} • W)_{T}) = (W_{t_{1}}, ..., W_{t_{n}} − W_{t_{n−1}}). By construction of Z, w ↦ 𝔼(ξ · exp((w | Z)_{ℝ^{n}})) is an entire function on the connected open domain ℂ^{n}, so that (5) is true for all λ ∈ ℂ^{n}. Denoting m(A) := ∫_{A} ξ dℙ (A ∈ F_{T}^{W}), it follows in particular that

0 = 𝔼(ξ · exp(i (λ | Z(ω))_{ℝ^{n}})) = ∫_{Ω} exp(i (λ | Z(ω))_{ℝ^{n}}) m(dω)
= ∫_{ℝ^{n}} exp(i (λ | ρ)_{ℝ^{n}}) m_{Z}(dρ) = m̂_{Z}(λ)

for all λ ∈ ℝ^{n}. In other words, the Fourier transform of the measure m_{Z} vanishes on ℝ^{n}. Hence, by injectivity of the Fourier transform, m vanishes on σ(Z) = σ(W_{t_{1}}, ..., W_{t_{n}}) for all t_{1}, ..., t_{n} ∈ [0, T], which implies by definition of m, that 𝔼(1_{A} ·ξ) = m(A) = 0 for all A ∈ F_{T}^{W}. Hence, ξ = 0, and the proof of the lemma is finished. □

Now we are totally prepared to prove

Theorem 3.2 (Chaos decomposition) Let T > 0 and X ∈ L^{2}(Ω, F_{T}^{W} , ℙ). Then there exists a unique sequence (f_{n})_{n∈ℕ} of deterministic and symmetric functions f_{n} ∈ L̂^{2}([0, T]^{n}) so that

X = 𝔼(X)+ ∑_{n=1}^{∞} I_{n}f_{n}

and

∥X∥_{L^{2}(Ω)}^{2} = 𝔼^{2}(X)+ ∑_{n=1}^{∞} n! · ∥f_{n}∥_{L^{2}([0,T])}^{2} .

Proof: Let X ∈ L^{2}(Ω, F_{T}^{W} , ℙ) be given. Then, by the previous lemma, there exists a sequence (Z_{n}) belonging to the linear hull of the set {exp((g • W)_{T}) : g ∈ L^{2}([0, T])} so that ∥X − Z_{n}∥_{L^{2}(Ω)}^{2} → 0 . Each Z_{n} can be written as a finite sum of type ∑_{k=1}^{l_{n}} α_{k} exp((g_{k} • W)_{T}) with real α_{k} and square-integrable g_{k}. By theorem 6 and the previous considerations, each

stochastic exponential E(g_{k} • W)_{T} can be written as E(g_{k} • W)_{T} = ∑_{m=0}^{∞} \frac{1}{m!}I_{m}(g_{k}^{⊗m}), so that

Z_{n} = ∑_{m=0}^{∞} J_{m}ϕ_{m}^{n} with (ϕ_{m}^{n})_{n∈ℕ} ⊆ L^{2}(S_{m}(T)). Orthogonality and the isometry condition now lead to ∥Z_{i} − Z_{j}∥_{L^{2}(Ω)}^{2} = ∑_{m=0}^{∞} ∥ϕ_{m}^{i} − ϕ_{m}^{j}∥_{L^{2}(S_{m}(T))}^{2} for all i, j ∈ ℕ. Thus, (ϕ_{m}^{i})_{i∈ℕ} is a L^{2}(S_{m}(T))-Cauchy sequence for every m ∈ ℕ_{0}, implying the existence of a limit ϕ_{m} ∈ L^{2}(S_{m}(T)) with ∥ϕ_{m} − ϕ_{m}^{i}∥_{L^{2}(S_{m}(T))}^{2} → 0 for i → ∞, and we obtain that ∑_{m=0}^{∞} ∥ϕ_{m} − ϕ_{m}^{i}∥_{L^{2}(S_{m}(T))}^{2} → 0 for i → ∞. Hence, by orthogonality and the isometry condition again, it follows the existence of Z := ∑_{m=0}^{∞} J_{m}ϕ_{m} = ∑_{m=0}^{∞} J_{m}(ϕ_{m} − ϕ_{m}^{i_{0}}) + Z_{i_{0}} ∈ L^{2}(Ω) (for a suitable i_{0} ∈ ℕ), and we obtain that ∥Z − Z_{i}∥_{L^{2}(Ω)}^{2} = ∑_{m=0}^{∞} ∥ϕ_{m} − ϕ_{m}^{i}∥_{L^{2}(S_{m}(T))}^{2} → 0 for i → ∞. By uniqueness of the limits, X = Z = ∑_{m=0}^{∞} J_{m}ϕ_{m} with (ϕ_{m}) ⊆ L^{2}(S_{m}(T)).

To finish our proof we first extend each ϕ_{m} trivially to ψ_{m} ∈ L^{2}([0, T]^{m}) and consider then the symmetrization ψ̃_{m} of ψ_{m}:

ψ̃_{m} := \frac{1}{m!} · ∑_{σ∈𝔖_{n}} ψ_{m} ∘ A_{σ} ∈ L̂^{2}([0, T]^{m}) ,

where A_{σ}(t_{1}, ..., t_{m}) := (t_{σ(1)}, ..., t_{σ(m)}) for all (t_{1}, ..., t_{m}) ∈ [0, T]^{m}. Since A_{σ}(S_{m}(T)) has no common points with S_{m}(T) for all σ ≠ id, the definition of ψ_{m} implies that (ψ_{m} ∘ A_{σ}) | S_{m}(T) = 0 for all σ ≠ id, so that ψ̃_{m} | S_{m}(T) = \frac{1}{m!} · ϕ_{m}, and we obtain that X = ∑_{m=0}^{∞} J_{m}ϕ_{m} = ∑_{m=0}^{∞} I_{m}ψ̃_{m}. Moreover, due to proposition 4, it follows that

∥X∥_{L^{2}(Ω)}^{2} = ∑_{m=0}^{∞} ∥ϕ_{m}∥_{L^{2}(S_{m}(T))}^{2} = ∑_{m=0}^{∞} m! · ∥ψ̃_{m}∥_{L^{2}([0,T]^{m})}^{2} .

Since X = ∑_{m=0}^{∞} J_{m}ϕ_{m} = ϕ^{0} + ∑_{m=1}^{∞}(Y^{(m)}ϕ_{m} • W)_{T}, we have 𝔼(X) = ϕ^{0} + 0, and because of the orthogonal representation, uniqueness follows immediately. □

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