Dynamic Systems and Applications 15 (2006) 241-268

STOCHASTIC α-CALCULUS, A FUNDAMENTAL THEOREM AND BURKHOLDER-DAVIS-GUNDY-TYPE ESTIMATES

HENRI SCHURZ

Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, Illinois 62901-4408, USA. and Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042, USA.

ABSTRACT. Aspects of stochastic α-integration calculus are in the center of this presentation. A discrete multidimensional version of the fundamental theorem of stochastic α-calculus is presented. It allows to verify the well-known formulas of Itô, Stratonovich, Dynkin, Tanaka, integration-by-parts for Wiener processes and Burkholder-type estimates among many applications. For this purpose, we generalize the notion of stochastic integral to the α-integral as linear combinations of known Itô forward and backward integrals. Elementary properties and discrete versions of well-known continuous counterparts are proven. As the main result, we obtain more elementary and very transparent proofs of fundamental tools and inequalities which are very well-known in the analysis of stochastic dynamical systems (e.g. the continuous and discrete Itô formula).

AMS (MOS) Subject Classification. 34F05, 37H10, 60H05, 60H10, 65C30, 93E03.

1. INTRODUCTION

Stochastic integration calculus needs to be studied to understand the qualitative behavior of stochastic dynamical systems generated by difference or differential equations. The fundamental theorem of deterministic calculus plays an essential role in all calculus courses taught throughout the world. Due to this apparent role there is a necessity to transfer this theorem and related integration concepts to corresponding stochastic versions. We are also longing for an easy-to-handle explanation to students and researchers with lesser experienced background in stochastic calculus to confront them with important material as early as possible. This paper aims at such a presentation which should ease the way to teach related topics to our students as simple as possible. At the same time we are going to maintain an innovative character of this research paper.

Received April 30, 2004           1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

To our knowledge, a thorougly presented approach to the multidimensional discrete formulas of Itô, Stratonovich, Dynkin, Tanaka, the integrationby-parts formulas for discretized Wiener processes and semimartingales, and discrete Burkholder-type estimates originating from one and the same fundamental theorem is not published in the literature. There are some trials to present a discrete stochastic calculus in the textbook of Shiryaev [25] and lecture notes of Schurz [24]. Moreover, Kannan and Zhang [15] presented a discrete version of Itô formula recently. Their approaches are further developed in a more general context and from a slightly different point of view here, leading to further applications and, in particular, to discrete Burkholder-type estimations for stochastic dynamical systems.

Recall that the deterministic fundamental theorem of calculus for line integrals can be presented as follows. For all absolutely continuous functions F : ℝ^{d} → ℝ^{1} with L_{loc}^{1}(ℝ^{d}, B(ℝ^{d}), μ)-integrable gradient ∇F : ℝ^{d} → ℝ^{d} (where B(ℝ^{d}) is the σ-algebra of Borel sets of ℝ^{d}), we have

(1.1)  F (b) = F (a) + ∫_{a}^{b} < ∇F (x), dx >_{d}

for all rectangles [a, b] ∈ ℝ^{d}, where < x, y >_{d} represents the Euclidean scalar product for vectors x, y ∈ ℝ^{d}. We are searching for a discrete version of (1.1) and, in particular, for a stochastic counterpart of (1.1) along the path of a given sequence X = (X_{n})_{n∈ℕ} of random variables X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) with the stochastic basis (Ω, F, (F_{n})_{n∈ℕ}, ℙ) equipped with a complete, nondecreasing filtration (F_{n})_{n∈ℕ} and F_{n} ⊂ F, where all F_{n}, F have the structure of σ-algebras. Throughout the paper, let ∇ be the gradient in ℝ^{d} and H = ∇^{2} be the Hessian matrix of the function F : ℝ^{d} → ℝ^{1} whenever these expressions exist. C^{k}(ℝ^{d}, ℝ^{l}) denotes the space of k-times continuously differentiable functions. I_{d} represents the unit matrix of ℝ^{d×d}. μ denotes the Lebesgue measure and B(Q) the set of all Borel-sets of the inscribed set Q.

The paper is organized as follows. In Section 2 we present the notion of stochastic α-integral and related properties as direct generalizations of known laws of deterministic integration. Section 3 proves a version of the fundamental theorem of stochastic α-calculus. In Section 4 we shall study the behavior of the remainder term occurring in that theorem. Simple examples and first applications illustrate the large potential of applicability of our main results in Section 5. Among them we present discrete versions of Itô, Stratonovich, Dynkin and Tanaka formula. There Burkholder-type estimates for local martingales and integration-by-parts formulas for Wiener processes and semimartingales are discussed. Section 6 closes this paper with a summary and final remarks.

STOCHASTIC α-CALCULUS, FTC AND BDG-ESTIMATES 243

2. THE CONCEPT OF STOCHASTIC α-INTEGRATION

This section generalizes the idea of stochastic integration originated from Itô [11] - [13]. For this purpose, we consider certain linear combinations of discrete Itô forward and backward integrals. Set ΔX_{k} = X_{k} − X_{k−1} and ΔY_{k} = Y_{k} − Y_{k−1}.

2.1. Basic definitions and the role of quadratic covariation.

Definition 2.1. Let X = (X_{n})_{n∈ℕ},Y = (Y_{n})_{n∈ℕ} be two sequences of random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and Y_{n} : (Ω, F_{n}, ℙ) → (ℝ^{l}, B(ℝ^{l})), and f : ℝ^{d} → ℝ^{l} be a Borel-measurable function. Then, the expression

(2.1)  I_{n}^{F} (f(X), Y) := ∑_{k=1}^{n} < f(X_{k−1}), ΔY_{k} >_{l}

is called the discrete forward integral (or discrete Itô integral) of f(X) with respect to Y, and

(2.2)  I_{n}^{B}(f(X), Y) := ∑_{k=1}^{n} < f(X_{k}), ΔY_{k} >_{l}

the discrete backward integral of f(X) with respect to Y, and

(2.3) I_{n}^{α}^{∘} (f(X), Y) := ∑_{k=1}^{n} < f(αX_{k} + (I_{d} − α)X_{k−1}), ΔY_{k} >_{l}

the discrete inner α-integral of f(X) with respect to Y with α ∈ ℝ^{d×d} as the d × d matrix of constant integration parameters, and

(2.4) I_{n}^{α}(f(X), Y) := ∑_{k=1}^{n} < αf(X_{k}) + (I_{l} − α)f(X_{k−1}), ΔY_{k} >_{l}

the discrete (outer) α-integral of f(X) with respect to Y with α ∈ ℝ^{l×l} as the l × l matrix of constant integration parameters. Furthermore, the stochastic process (< f(X), Y >_{n})_{n∈ℕ} with

(2.5) < f(X), Y >_{n} := ∑_{k=1}^{n} < f(X_{k}) − f(X_{k−1}), Y_{k} − Y_{k−1} >_{l}

is said to be the process of quadratic covariation of f(X) with Y, and the process < X >= (< X >_{n})_{n∈ℕ} with

(2.6)  < X >_{n} := ∑_{k=1}^{n} < ΔX_{k}, ΔX_{k} >_{d}

the process of quadratic variation of X.

Convention: We set I_{0}^{F} (., .) = 0, I_{0}^{B}(., .) = 0, I_{0}^{α}^{∘} (., .) = 0, I_{0}^{α}(., .) = 0.

Trivially, if l = d then I_{n}^{O}^{∘} (f(X), Y) = I_{n}^{O}(f(X), Y) = I_{n}^{F} (f(X), Y) and

I_{n}^{I_{d}}^{∘} (f(X), Y) = I_{n}^{I_{d}}(f(X), Y) = I_{n}^{B}(f(X), Y) where O ∈ ℝ^{d×d} is the zero

matrix.

The following basic transformation rule between forward and backward integrals can be established. Moreover, this rule explains the role of the process of quadratic covariation and a major difference to deterministic calculus.

Lemma 2.2. ∀n ∈ ℕ ∀ Borel-measurable f : ℝ^{d} → ℝ^{l} and ∀X, Y

(2.7)  < f(X), Y >_{n} = I_{n}^{B}(f(X), Y) − I_{n}^{F} (f(X), Y)

Proof. Consider

(2.8)  I_{n}^{B}(f(X), Y) − I_{n}^{F} (f(X), Y) = ∑_{k=1}^{n} < f(X_{k}), ΔY_{k} >_{l}

− ∑_{k=1}^{n} < f(X_{k−1}), ΔY_{k} >_{l}= ∑_{k=1}^{n} < f(X_{k})

− f(X_{k−1}), ΔY_{k} >_{l}=< f(X), Y >_{n},

hence the proof is complete.

2.2. Simple transformation rules. Let X^{B} = (X_{n}^{B})_{0≤n≤N} with X_{n}^{B} = X_{N−n} be the reversed sequence belonging to X = (X_{n})_{n∈ℕ}, where N is a fixed integer.

Theorem 2.3. Fix N ≥ 1. Let X = (X_{n})_{n∈ℕ} and Y = (Y_{n})_{n∈ℕ} be two sequences of random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and Y_{n} : (Ω, F_{n}, ℙ) → (ℝ^{l}, B(ℝ^{l})). Then ∀n : 0 ≤ n ≤ N and ∀ Borel-measurable f : ℝ^{d} → ℝ^{l}

(2.9)  I_{N}^{B}(f(X), Y) = −I_{N}^{F}(f(X^{B}), Y^{B})
(2.10)  I_{n}^{B}(f(X), Y) = −[I_{N}^{F}(f(X^{B}), Y^{B}) − I_{N−n}^{F}(f(X^{B}), Y^{B})]
(2.11)  < f(X), Y >_{N} = −[I_{N}^{F}(f(X^{B}), Y^{B}) + I_{N}^{F}(f(X), Y)]
(2.12)  < f(X), Y >_{n} = −[I_{N}^{F}(f(X^{B}), Y^{B}) − I_{N−n}^{F}(f(X^{B}), Y^{B})]
− I_{n}^{F} (f(X), Y).

Proof. First, the proof of (2.9). Calculate

I_{N}^{F}(f(X^{B}), Y^{B}) = ∑_{k=1}^{N} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l}

= ∑_{k=1}^{N} < f(X_{N−k+1}), ΔY_{N−k} >_{l}

= ∑_{k=1}^{N} < f(X_{N−k+1}), Y_{N−k} − Y_{N−k+1} >_{l}

= ∑_{r=N}^{1} < f(X_{r}), Y_{r−1} − Y_{r} >_{l}

= − ∑_{k=1}^{N} < f(X_{k}), ΔY_{k} >_{l}= −I_{N}^{B}(f(X), Y)

using a change of the summation index r = N − k + 1. Second, the proof of (2.10).

I_{N}^{F}(f(X^{B}), Y^{B}) − I_{N−n}^{F}(f(X^{B}), Y^{B})

= ∑_{k=1}^{N} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l} − ∑_{k=1}^{N−n} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l}

= ∑_{k=N−n+1}^{N} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l}

= ∑_{k=N−k+1}^{N} < f(X_{N−k+1}), Y_{N−k} − Y_{N−k+1} >_{l}

= ∑_{r=1}^{n} < f(X_{r}), Y_{r−1} − Y_{r} >_{l}= − ∑_{r=1}^{n} < f(X_{r}), ΔY_{r} >_{l}

= −I_{n}^{B}(f(X), Y)

using a change of the summation index r = N − k + 1. Third, the claim of (2.11) follows from (2.12) by setting n = N and recalling the convention that I_{0}^{F} (., .) = 0. Finally, the proof of (2.12).

I_{N}^{F}(f(X^{B}), Y^{B}) − I_{N−n}^{F}(f(X^{B}), Y^{B})

= ∑_{k=1}^{N} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l} − ∑_{k=1}^{N−n} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l}

= ∑_{k=N−n+1}^{N} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l}

= ∑_{k=N−n+1}^{N} < f(X_{N−k+1}), Y_{N−k} − Y_{N−k+1} >_{l}

= ∑_{r=n}^{1} < f(X_{r}), Y_{r−1} − Y_{r} >_{l}= − ∑_{r=1}^{n} < f(X_{r}), ΔY_{r} >_{l}

= − ∑_{r=1}^{n} < f(X_{r}) − f(X_{r−1}), ΔY_{r} >_{l} − ∑_{r=1}^{n} < f(X_{r−1}), ΔY_{r} >_{l}

= − < f(X), Y >_{n} −I_{n}^{F} (f(X), Y).

Thus, the proof is complete.

Remark 2.4. The properties (2.9) - (2.12) can be interpreted as discrete stochastic counterparts of the following well-known rules of deterministic integral calculus. For all real numbers a < b and L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ)-integrable functions f : ℝ^{1} → ℝ^{1}, we have

(i) Rule (2.9) corresponds to ∫_{a}^{b} f(x)dx = − ∫_{b}^{a} f(x)dx, (ii) Rule (2.10) to ∀c ∈ [a, b] ∫_{a}^{c} f(x)dx = − [∫_{b}^{a} f(x)dx − ∫_{b}^{c} f(x)dx], (iii) Rule (2.11) to 0 = ∫_{b}^{a} f(x)dx + ∫_{a}^{b} f(x)dx and (iv) Rule (2.12) to 0 = − [∫_{b}^{a} f(x)dx − ∫_{b}^{c} f(x)dx].

From the rules (2.11) and (2.12) we clearly recognize a significant change in the setting of stochastic calculus compared to deterministic one, caused by the nonvanishing quadratic covariation in the stochastic case. The proof of (2.12) also becomes clear from an application of (2.10) and Lemma 2.2. This leads to the chain of conclusions

I_{N}^{F}(f(X^{B}), Y^{B}) − I_{N−n}^{F}(f(X^{B}), Y^{B})

= ∑_{k=1}^{N} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l} − ∑_{k=1}^{N−n} < f(X_{k−1}^{B}), ΔY_{k}^{B} >_{l}
= −I_{n}^{B}(f(X), Y) = −(I_{n}^{F} (f(X), Y)+ < f(X), Y >_{n}).

2.3. A general transformation rule between calculi.

Theorem 2.5. Let X = (X_{n})_{n∈ℕ} and Y = (Y_{n})_{n∈ℕ} be two sequences of random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and Y_{n} : (Ω, F_{n}, ℙ) → (ℝ^{l}, B(ℝ^{l})). Then ∀n ∈ ℕ and ∀ Borel-measurable f : ℝ^{d} → ℝ^{l} ∀α ∈ ℝ^{l×l}

(2.13) I_{n}^{α}(f(X), Y) = I_{n}^{B}(αf(X), Y) + I_{n}^{F} ((I_{l} − α)f(X), Y)
(2.14) I_{n}^{α}(f(X), Y) = I_{n}^{F} (f(X), Y)+ < αf(X), Y >_{n}

Moreover, in the scalar parameter case, if α = βI_{l} with β ∈ ℝ^{1} then ∀n ∈ ℕ and ∀ Borel-measurable f : ℝ^{d} → ℝ^{l}

(2.15) I_{n}^{β}(f(X), Y) = βI_{n}^{B}(f(X), Y) + (1 − β)I_{n}^{F} (f(X), Y)
(2.16) I_{n}^{β}(f(X), Y) = I_{n}^{F} (f(X), Y) + β < f(X), Y >_{n}

Proof. Fix α ∈ ℝ^{l×l}. Elementary calculations lead to

I_{n}^{α}(f(X), Y) = ∑_{k=1}^{n} < αf(X_{k}) + (I_{l} − α)f(X_{k−1}), ΔY_{k} >_{l}

= ∑_{k=1}^{n} < αf(X_{k}), ΔY_{k} >_{l} + ∑_{k=1}^{n} < (I_{l} − α)f(X_{k−1}), ΔY_{k} >_{l}

= I_{n}^{B}(αf(X), Y) + I_{n}^{F} ((I_{l} − α)f(X), Y) and

I_{n}^{α}(f(X), Y) = ∑_{k=1}^{n} < α[f(X_{k}) − f(X_{k−1})], ΔY_{k} >_{l}

+ ∑_{k=1}^{n} < f(X_{k−1}), ΔY_{k} >_{l}

=< αf(X), Y >_{n} +I_{n}^{F} (f(X), Y),

hence the equations (2.13) and (2.14) hold. Suppose that α = βI_{l} with β ∈ ℝ^{1}. Then, the claimed assertions (2.15) and (2.16) are trivial conclusions due to the homogeneity of the scalar product < ., . >_{l}.

Remark 2.6. Thus, the α-integral is completely described by the sum of the forward integral and the quadratic covariation (see (2.14) and (2.16)). It can be also represented by the sum of certain forward and backward integrals (see (2.13) and (2.15)). Taking into account the general stochastic case, these latter facts are not true for the inner α-integral. However, for linear functions f or for continuous versions with vanishing quadratic covariation (as in the deterministic case), an identity of inner and outer α-integrals can be found. If α = \frac{1}{2}I_{l} then we have the case of the classical Stratonovich calculus (Stratonovich integral, see Stratonovich [26] for its origin). In the scalar parameter case, the α-integral I^{α} is a linear combination of forward I^{F} and backward integral I^{B} as seen in (2.15).

3. A FUNDAMENTAL THEOREM OF STOCHASTIC α-CALCULUS

Recall from deterministic calculus that, along any curve connecting the points x_{k} ∈ ℝ^{d} (k = 0, 1, …, n) for line integrals, we have

F (x_{n}) = F (x_{0}) + ∫_{x_{0}}^{x_{n}} < f(y), dy >_{d}

≐ F (x_{0}) + ∑_{k=1}^{n} < f(x_{k}^{∗}), Δx_{k} >_{d} +R_{n}(f)

for absolutely continuous function F : ℝ^{d} → ℝ^{1} with gradient f(x) = ∇F (x), where R_{n}(f) = R_{n}(f)(x_{k} : 0 ≤ k ≤ n) is an appropriate remainder term and x_{k}^{∗} = x_{k−1} + θ_{k}Δx_{k} with θ_{k} ∈ [0, 1] and Δx_{k} = x_{k} − x_{k−1}.

An analogous representation as the fundamental theorem of stochastic α-calculus abbreviated by FTSC(α) is obtained as follows.

Theorem 3.1. Let X = (X_{n})_{n∈ℕ} be a sequence of (F_{n})-adapted random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})). Assume that F : ℝ^{d} → ℝ^{1} is an absolutely continuous function with L_{loc}^{1}(ℝ^{d}, B(ℝ^{d}), μ)-integrable gradient f : ℝ^{d} → ℝ^{d}. Then there is a (F_{n}, B(ℝ^{1}))-measurable representation

(3.1)  F (X_{n}) = F (X_{0}) + I_{n}^{α}(f(X), X) + \frac{1}{2} < (I_{d} − 2α)f(X), X >_{n}
+ R_{n}(f(X), X)

where

(3.2) R_{n}(f(X), X) = ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < f(x) − \frac{1}{2}[f(X_{k}) + f(X_{k−1})], dx >_{d}

for all n ∈ ℕ and all parameter matrices α ∈ ℝ^{d×d}.

Definition 3.2. The expression R_{n}(f(X), X) occurring at the right hand side of equation (3.1) is called the remainder term of F (X) at time n.

Proof. First, we notice the following elementary fact. For all Borel-measurable functions g : ℝ^{d} → ℝ^{d} and for all parameters α ∈ ℝ^{d×d} we have that the vector

Z_{k} = αg(X_{k}) + (I_{d} − α)g(X_{k−1})

+ \frac{1}{2}(I_{d} − 2α)[g(X_{k}) − g(X_{k−1})] − \frac{1}{2}[g(X_{k}) + g(X_{k−1})]

is equal to 0_{d} ∈ ℝ^{d}, hence < Z_{k}, ΔX_{k} >_{d}= 0. Therefore, by telescoping F, adding this zero expression Z_{k} = 0_{d} to the summation and separating the involved terms correspondingly, one arrives at

F (X_{n}) = F (X_{0}) + ∑_{k=1}^{n}[F (X_{k}) − F (X_{k−1})]

= F (X_{0}) + ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < f(x), dx >_{d}

= F (X_{0}) + ∑_{k=1}^{n} [ < αg(X_{k}) + (I_{d} − α)g(X_{k−1}), ΔX_{k} >_{d} +

+ \frac{1}{2} < (I_{d} − 2α)[g(X_{k}) − g(X_{k−1})], ΔX_{k} >_{d} +

+ ∫_{X_{k−1}}^{X_{k}} < f(x), dx >_{d} −\frac{1}{2} < g(X_{k}) + g(X_{k−1}), ΔX_{k} >_{d} ]

= F (X_{0}) + I_{n}^{α}(g(X), X) + \frac{1}{2} < (I_{d} − 2α)g(X), X >_{n}

+ ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < f(x) − \frac{1}{2}[g(X_{k}) + g(X_{k−1})], dx >_{d} .

It remains to set g(x) = f(x) = ∇F (x) in order to verify the formula (3.1) with the remainder term R_{n}(f(X), X) satisfying (3.2). Thus, this completes the proof.

4. ESTIMATION OF THE REMAINDER TERMS IN FTSC(α)

An estimation of the remainder term R with compatible matrix norm ||.||_{d×d} and vector norm ||.||_{d} induced naturally by the Euclidean scalar product < ., . >_{d} is provided as follows.

Theorem 4.1. Let X = (X_{n})_{n∈ℕ} be a sequence of (F_{n})-adapted random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})). Assume that F ∈ C^{2}(ℝ^{d}, ℝ^{1}) with gradient f = ∇F and Hessian matrix H = ∇^{2}F. Then the remainder term R(f(X), X) from the representation (3.1) satisfies

(4.1)  R_{n}(f(X), X) = \frac{1}{4} ∑_{k=1}^{n} < [H(η_{k}^{1}) − H(η_{k}^{2})]ΔX_{k}, ΔX_{k} >_{d}

with intermediate vector values η_{k}^{1} = X_{k−1} + θ_{k}^{1}ΔX_{k},η_{k}^{2} = X_{k−1} + θ_{k}^{2}ΔX_{k} where θ_{k}^{1}, θ_{k}^{2} ∈ [0, 1]. Moreover, the remainder term can be estimated by

|R_{n}(f(X), X)| ≤ \frac{1}{4} ∑_{k=1}^{n} ||H(η_{k}^{1}) − H(η_{k}^{2})||_{d×d} · ||ΔX_{k}||_{d}^{2} and (4.2)

|R_{n}(f(X), X)| (4.3)

≤ \frac{1}{4} (∑_{k=1}^{n} ||H(η_{k}^{1}) − H(η_{k}^{2})||_{d×d}^{2} · ||ΔX_{k}||_{d}^{2})^{1/2} · (< X >_{n})^{1/2}.

Remark 4.2. If F ∈ C^{3}(ℝ^{d}, ℝ^{1}) then the remainder term can be controlled by the third partial derivatives of F and the cubic variation (as known from estimates of the trapezoidal rule as used in deterministic quadrature rules, cf. Shiryaev [25], p. 556).

4.1. An auxiliary lemma. The following lemma represents a mean value theorem (MVT) for deterministic integrals and is needed to prove certain estimates of the remainder term R in the FTSC(α).

Lemma 4.3. Assume that

(i) g, h ∈ C^{0}([a, b], ℝ^{1}),
(ii) either ∀t ∈ [a, b] : g(t) ≥ 0 or ∀t ∈ [a, b] : g(t) ≤ 0.

Then ∃c ∈ [a, b] such that

(4.4)  ∫_{a}^{b} h(t)g(t) dt = h(c) ∫_{a}^{b} g(t) dt.

Proof. Without loss of generality, we may suppose that ∀t ∈ [a, b] : g(t) ≥ 0. One encounters with the trivial estimate

(4.5) ( min_{t∈[a,b]} h(t)) ∫_{a}^{b} g(s) ds ≤ ∫_{a}^{b} h(s)g(s)ds ≤ ( max_{t∈[a,b]} h(t)) ∫_{a}^{b} g(s) ds.

Using the intermediate value theorem (IVT) applied to F : [a, b] → ℝ^{1} which is defined by

F (t) = ∫_{a}^{b} h(s)g(s) ds − h(t) ∫_{a}^{b} g(s) ds, t ∈ [a, b]

we get to the claimed asssertion as follows. We know that F is continuous on [a, b] and, thanks to (4.5) and the continuity of h, that

∃t_{+}, t_{−} ∈ [a, b] : F (t_{+}) ≥ 0, F (t_{−}) ≤ 0.

Therefore, there is c ∈ [min(t_{−}, t_{+}), max(t_{−}, t_{+})] such that F (c) = 0 which is equivalent to (4.4).

4.2. The proof of Theorem 4.1.

Proof. Recall formula (3.2) for the remainder term R. We conclude that

R_{n}(f(X), X) = ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < f(x) − \frac{1}{2}[f(X_{k}) + f(X_{k−1})], dx >_{d}

= \frac{1}{2} ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < f(x) − f(X_{k}), dx >_{d}

+ \frac{1}{2} ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < f(x) − f(X_{k−1}), dx >_{d}

= \frac{1}{2} ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < H(ξ_{k}^{1}(x))(x − X_{k}), dx >_{d} +

+ \frac{1}{2} ∑_{k=1}^{n} ∫_{X_{k−1}}^{X_{k}} < H(ξ_{k}^{2}(x))(x − X_{k−1}), dx >_{d}

= \frac{1}{2} ∑_{k=1}^{n} ∫_{0}^{1} < H(ξ_{k}^{1}(z))ΔX_{k}, ΔX_{k} >_{d} (z − 1) dz+

+ \frac{1}{2} ∑_{k=1}^{n} ∫_{0}^{1} < H(ξ_{k}^{2}(z))ΔX_{k}, ΔX_{k} >_{d} z dz

with intermediate values ξ_{k}^{1} and ξ_{k}^{2} between X_{k−1} and X_{k} (using the transformation x = X_{k−1} + zΔX_{k}). Notice that there is no sign change of z and z − 1 for z ∈ [0, 1]. Apply Lemma 4.3 as the mean value theorem (MVT) for integrals to both integrals in the previous sum. We obtain

R_{n}(f(X), X) = \frac{1}{2} ∑_{k=1}^{n} < H(η_{k}^{2})ΔX_{k}, ΔX_{k} >_{d} ∫_{0}^{1}(z − 1) dz

+ \frac{1}{2} ∑_{k=1}^{n} < H(η_{k}^{1})ΔX_{k}, ΔX_{k} >_{d} ∫_{0}^{1} z dz

= \frac{1}{4} ∑_{k=1}^{n} < H(η_{k}^{1})ΔX_{k}, ΔX_{k} >_{d}

− \frac{1}{4} ∑_{k=1}^{n} < H(η_{k}^{2})ΔX_{k}, ΔX_{k} >_{d}

= \frac{1}{4} ∑_{k=1}^{n} < [H(η_{k}^{1}) − H(η_{k}^{2})]ΔX_{k}, ΔX_{k} >_{d}

with intermediate values η_{k}^{1}, η_{k}^{2} between X_{k} and X_{k−1}. Thus, this verifies the identity (4.1). To confirm the estimates (4.2), one uses the triangle inequality. For the verification of (4.3), the Cauchy-Bunyakovski-Schwarz inequality is applied to the occurring random scalar product < Z, ΔX >_{L^{2}(Ω,F_{n},ℙ)}= 𝔼 ∑_{k=1}^{n} < Z_{k}, ΔX_{k} >_{d}. This completes the proof.

Remark 4.4. One might also substitute by

Z_{k} = αX_{k} + (I_{d} − α)X_{k−1} − (αX_{k} + (I_{d} − α)X_{k−1}) = 0_{d}

in the proof above. However, then the quadratic covariation part is hidden in the remainder term R_{n}. Hence, the form of representation of remainder terms is not unique.

4.3. An application to derive the continuous FTSC(α). The proved estimate of the remainder term R allows to derive the continuous time fundamental theorem of stochastic α-calculus as a first application.

Definition 4.5. Let X = (X_{t})_{t_{0}≤t≤T} be an a.s. continuous (F_{t})-adapted stochastic process with X_{t} : (Ω, F_{t}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and f : ℝ^{d} → ℝ^{d} any Borel-measurable function. Then the continuous quadratic covariation on [s, t] ⊆ [t_{0}, T] for given parameter α ∈ ℝ^{d×d} is defined by

< (I_{d} − 2α)f(X), X >_{s}^{t}:= (4.6)

lim_{n→+∞} ∑_{k=1}^{n} < (I_{d} − 2α)[f(X_{t_{k}}) − f(X_{t_{k−1}})], X_{t_{k}} − X_{t_{k−1}} >_{d}

provided that this limit uniquely exists (a.s.) along any partition s = t_{0} < t_{1} < ⋯ < t_{n} = t. If f(X) = X, s = 0 and α = O then < X >_{t}:=< X, X >_{0}^{t}.

The expression

α ∫_{s}^{t} < f(X_{u}), dX_{u} >_{d}:= (4.7)

lim_{n→+∞} ∑_{k=1}^{n} < αf(X_{t_{k}}) + (I_{d} − α)f(X_{t_{k−1}}), X_{t_{k}} − X_{t_{k−1}} >_{d},

provided that the above limit taken along any partition s = t_{0} < t_{1} < ⋯ < t_{n} = t for a fixed parameter α ∈ ℝ^{d×d} uniquely exists (a.s.), is called the continuous (outer) α-integral from s to t.

Remark 4.6. If f ∈ C^{1}(ℝ^{d}, ℝ^{d}), X a.s. continuous on [0, T] and < X >_{T} < +∞ then obviously

| < (I_{d} − 2α)f(X), X >_{0}^{T} | ≤
||I_{d} − 2α||_{d×d} · ∫_{0}^{T} ||∇f(X_{s})||_{d×d} d < X >_{s}< +∞

with probability one, where ||.||_{d×d} is a compatible matrix norm, thanks to the continuity of ∇f(X_{s}) and finite variation of < X >_{s}. Thus, in this case the continuous quadratic covariation is well-defined and finite. Then, the continuous α-integral is also well-defined and finite as this fact follows from the following theorem.

Theorem 4.7. Assume that

(i) F ∈ C^{2}(ℝ^{d}, ℝ^{1}) with gradient f = ∇F, (ii) X = (X_{t})_{0≤t≤T} is an a.s. continuous, (F_{t})-adapted stochastic process

with X_{t} : (Ω, F_{t}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and < X >_{T} < +∞.

Then ∀0 ≤ s ≤ t ≤ T ∀α ∈ ℝ^{d×d}

(4.8)  F (X_{t}) = F (X_{s}) + α ∫_{s}^{t} < f(X_{u}), dX_{u} >_{d}

+ \frac{1}{2} < (I_{d} − 2α)f(X), X >_{s}^{t} .

Proof. Let F ∈ C^{2}(ℝ^{d}, ℝ^{1}), f = ∇F and H = ∇^{2}F. Set X̃_{k} = X_{t_{k}} for any partition of the form s = t_{0} < t_{1} < ⋯ < t_{n} = t. As in the Remark 4.3 noted, we have the existence and finiteness of the continuous quadratic covariation part < (I_{d} − 2α)f(X), X >_{0}^{T} (a.s.) since

∫_{0}^{T} ||H(X_{s})||_{d×d} d < X >_{s} < +∞.

It trivially follows that (X̃_{k})_{0≤k≤n} converges to X = (X_{u})_{s≤u≤t} (a.s.) as n tends to +∞. Now, apply the discrete FTSC(α) to the random sequence (X̃_{k})_{0≤k≤n}. The occurring remainder term R_{n}(f(X̃), X̃) converges to zero since

|R_{n}(f(X̃), X̃)| ≤ \frac{1}{4} (∑_{k=1}^{n} ∥H(η_{k}^{1}) − H(η_{k}^{2})∥_{d×d}^{2} · ∥ΔX̃_{k}∥_{d}^{2})^{1/2} · (< X̃ >_{n})^{1/2}

thanks to Theorem 4.1, the continuity of both X and the Hessian matrix H = ∇^{2}F, and the fact that < X̃ >_{n} tends to < X >_{t} as n tends to +∞. Finally, the discrete outer α-integral converges to the continuous outer α-integral due to the convergence of the quadratic covariation processes, cf. Remark 4.3. Thus, the limit must be given by (4.8) and the proof is complete.

Remark 4.8. Obviously, this version includes the well-known Itô formula by taking α = O and the Stratonovich formula by putting α = \frac{1}{2}I_{d}. Actually, the requirement F ∈ C^{2}(ℝ^{d}, ℝ^{1}) is not fully needed. However, if F ∈ C^{1}(ℝ^{d}, ℝ^{1}) with a Hessian H = ∇^{2}F which is integrable with respect to (< X >_{t})_{0≤t≤T} then one needs to add the following assumptions

(A1) ∫_{0}^{T} ||H(X_{s}))||_{d×d}d < X >_{s}< +∞ (a.s.) (A2) ∀η_{k}^{1} = X_{t_{k−1}} + θ_{k}^{1}(X_{t_{k}} − X_{t_{k−1}}), η_{k}^{2} − X_{t_{k−1}} + θ_{k}^{2}(X_{t_{k}} − X_{t_{k−1}})

(θ_{k}^{i} ∈ [0, 1]) lim_{n→+∞} ∑_{k=1}^{n} < [H(η_{k}^{1}) − H(η_{k}^{2})](X_{t_{k}} − X_{t_{k−1}}), X_{t_{k}} − X_{t_{k−1}} >_{d}= 0 (a.s.)

to establish the representation (4.8) too. If one is willing to add some limit corrections to the FTSC above then the requirement (A2) can be replaced by the assumption

(A2′)  ∃! lim_{n→+∞} |R_{n}(f(X), X) + \frac{1}{2} < (I_{d} − 2α)f(X), X >_{n} | < +∞

along any partitions of X. For further continuous time generalizations, see also Föllmer, Protter and Shiryaev [10], Föllmer and Protter [9], which are taylored to the case of Brownian motions instead of more general terms of X, and also Russo and Vallois [21].

5. SOME FURTHER APPLICATIONS

The following distinct series of applications shows the significant role of FTSC(α) within stochastic calculus.

5.1. A simple application to quadratic forms. For simplicity, consider F : ℝ^{1} → ℝ^{1} with

F (x) = ax^{2} + bx + c.

Then the trivial identities F′(x) = f(x) = 2ax + b and F′′(x) = f′(x) = 2a hold. Hence

R_{n}(f(X), X) = \frac{1}{4} ∑_{k=1}^{n}(f′(η_{k}^{1}) − f′(η_{k}^{2}))(ΔX_{k})^{2} = 0

by applying Theorem 4.1. Therefore, Theorem 3.1 applied to the random sequence X = (X_{n})_{n∈ℕ} leads to the identities

F (X_{n}) = aX_{n}^{2} + bX_{n} + c = F (X_{0}) + I_{n}^{α}(2aX + b, X)

+ \frac{1 − 2α}{2} < 2aX + b, X >_{n}

= F (X_{0}) + b(X_{n} − X_{0}) + 2aI_{n}^{α}(X, X) + a(1 − 2α) < X >_{n}

= F (X_{0}) + b(X_{n} − X_{0}) + 2aI_{n}^{F} (X, X) + a < X >_{n}

= F (X_{0}) + b(X_{n} − X_{0}) + 2aI_{n}^{B}(X, X) − a < X >_{n} .

The quadratic form F (X) is decomposed into a constant part, a linear part, a discrete integral part and a part controlled by the quadratic variation of X. If a ≠ 0 and the quadratic variation process < X > does not vanish then we apparently get an infinite number of different, but equivalent representations of the quadratic form F (X). Not suprisingly, only in the Stratonovich case α = \frac{1}{2} we have the same calculus rule as in the deterministic situation. In passing we note that here the concepts of outer and inner integrals coincide for quadratic forms F (X) since F′ = f is linear. Suppose that a > 0. We can also recognize that we have a stabilizing effect caused by the nonnegative and nondecreasing quadratic variation while increasing the parameter α.

5.2. A discrete Tanaka formula and symmetric random walks. Let F : ℝ^{1} → ℝ^{1} with F (x) = |x|. Thus, we have

f(x) = sign(x) = \{ \begin{matrix}1 & if & x > 0 \\ 0 & if & x = 0 \\ −1 & if & x < 0 \end{matrix}

Consider a sequence X = (X_{n})_{n∈ℕ} of i.i.d. real-valued X_{n} with ℙ(X_{k} = ±1) = \frac{1}{2}. Set S_{n} = ∑_{k=1}^{n} X_{k} for n ≥ 1 and S_{0} = 0. Using the FTSC with α = 0, we find the distance of S_{n} to the starting point S_{0} = 0

(5.1)  |S_{n}| = |S_{0}| + ∑_{k=1}^{n} sign(S_{k−1})ΔS_{k}

+ \frac{1}{2} < sign(S), S >_{n} +R_{n}(sign(S), S)

which can be interpreted as a version of the discrete Tanaka formula. Let us briefly discuss this formula.

(i) M_{n} = ∑_{k=1}^{n} sign(S_{k−1})ΔS_{k} forms a (F_{n})-martingale since S = (S_{n})_{n∈ℕ} is a (F_{n})-martingale.

(ii) N_{n} := R_{n}(sign(S), S) + \frac{1}{2} < sign(S), S >_{n} is an increasing process of bounded variation since we have the unique Doob-Meyer decomposition of the form |S_{n}| = |S_{0}| + M_{n} + N_{n} with M_{0} = N_{0} = 0, martingale M = (M_{n})_{n∈ℕ} and compensator N = (N_{n})_{n∈ℕ}.

(iii) 𝔼|S_{n}| = 𝔼N_{n} since S_{0} = 0 and 𝔼M_{n} = 0.

(iv) N_{n} = = ∥ {k : 0 ≤ k < n, S_{k} = 0} = = ∥ of sign-changes of the path of the random walk S up to time n. This fact can be seen from

R_{n}(sign(S), S) = ∑_{k=1}^{n} ∫_{S_{k−1}}^{S_{k}} (sign(x) − \frac{sign(S_{k−1}) + sign(S_{k})}{2}) dx

= ∑_{k=1}^{n} (|S_{k}| − |S_{k−1}| − \frac{sign(S_{k−1}) + sign(S_{k})}{2}ΔS_{k})

= ∑_{k=1}^{n} \frac{sign(S_{k})S_{k} − sign(S_{k−1})S_{k−1} − sign(S_{k−1})S_{k} + sign(S_{k})S_{k−1}}{2}

= ∑_{k=1}^{n} (sign(S_{k}) − sign(S_{k−1}))S_{k} − \frac{1}{2} < sign(S), S >_{n} .

Therefore

N_{n} = ∑_{k=1}^{n} (sign(S_{k}) − sign(S_{k−1}))S_{k}

which counts the number of sign-changes of the path of S up to n as claimed.

(v) \sqrt{\frac{t}{n}}N_{n} ⟶^{ℙ−(a.s.)} 2L_{t}(0) = ∫_{0}^{t} δ_{0}(W_{s}) ds as n tends to +∞ by strong laws of large numbers, where δ_{0} denotes the dirac distribution concentrated at 0 and W = (W_{t})_{t≥0} represents a standard Wiener process. Thus, N_{n}/\sqrt{n} can be interpreted as discrete version of the local time of the random walk S at 0.

(vi) 𝔼|S_{n}| = 𝔼N_{n} ≈ \sqrt{\frac{n}{2π}} for large n and, for all t ≥ 0, we have

|\sqrt{\frac{t}{n}}S_{n}| = |\sqrt{\frac{t}{n}}S_{0}| + \sqrt{\frac{t}{n}} ∑_{k=1}^{n} sign(S_{k−1})ΔS_{k−1} + \sqrt{\frac{t}{n}}N_{n} .

As n tends to +∞, cf. Karatzas and Shreve [14], this expression converges (a.s.) to

|W_{t}| = |W_{0}| + ∫_{0}^{t} sign(W_{s}) dW_{s} + 2L_{t}(0)

which replicates the continuous Tanaka formula, where W = (W_{t})_{t≥0} is a Wiener process.

5.3. A formula of integration by parts for the discretized Wiener process. Consider partitions of the form 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{n} = T as discretizations of intervals [0, T]. For simplicity, take t_{k} = \frac{k}{n}T. Set X_{k} = W_{\frac{k}{n} T} where W is a standard Wiener process with independent increments W_{t} − W_{s} ∈ N (0, |t − s|). X = (X_{k})_{k∈ℕ} represents the (F_{t})-adapted discretized Wiener process.

Corollary 5.1. Let F : ℝ^{1} → ℝ^{1} be an absolutely continuous function with integrable derivative f = F′ ∈ L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ). Then ∀α ∈ ℝ^{1}

(5.2)  F (W_{T}) = F (W_{0}) + I_{n}^{α}(f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T})

+ \frac{1 − 2α}{2} < f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T} >_{n} +

+ R_{n}(f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T})

where R_{n} is given by (4.1).

Proof. Just an application of FTSC(α) to the sequence X = (X_{k})_{0≤k≤n}.

Remark 5.2. Corollary 5.1 with formula (5.2) corresponds to the FTSC (α) for discretized Wiener processes. If α = 0 then we have the Itô formula for discretized Wiener processes and if α = \frac{1}{2} the Stratonovich formula for discretized Wiener processes. It can also be understood as discrete integration-by-parts formula for Wiener processes.

Corollary 5.3. Let F ∈ C^{1}(ℝ^{1}, ℝ^{1}) have f′ = F′′ ∈ L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ). Assume that

(5.3)  ∫_{0}^{T} |f′(W_{s})|ds < +∞ (a.s.) .

Then ∀α ∈ ℝ^{1}

(5.4)  F (W_{T}) = F (W_{0}) + α ∫_{0}^{T} f(W_{s}) dW_{s} + \frac{1 − 2α}{2} ∫_{0}^{T} f′(W_{s}) ds.

Proof. At first, apply the FTSC(α) to X = (X_{k})_{0≤k≤n}. Second, to control the limit behavior as n → +∞, one needs to prove two auxiliary lemmas as done below. Consequently, the limits are given by those lemmas and the proof is complete.

Remark 5.4. Formula (5.4) corresponds to the integration-by-parts formula of Wiener processes related to the concept of α-integration.

Lemma 5.5. If ∫_{0}^{T} |f′(W_{s})| ds < +∞ (a.s.) and f′ ∈ L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ) then

lim_{n→+∞} < f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T} >_{n} = ∫_{0}^{T} f′(W_{s}) ds (a.s.).

Proof. Suppose f ∈ C^{1}(ℝ^{1}, ℝ^{1}). Set ΔW_{k} = W_{\frac{k}{n} T} − W_{\frac{k−1}{n} T}. Then one estimates

lim_{n→+∞} < f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T} >_{n}= lim_{n→+∞} ∑_{k=1}^{n} [f(W_{\frac{k}{n} T}) − f(W_{\frac{k−1}{n} T})]ΔW_{k}

= lim_{n→+∞} ∑_{k=1}^{n} \frac{f(W_{\frac{k}{n} T}) − f(W_{\frac{k−1}{n} T})}{W_{\frac{k}{n} T} − W_{\frac{k−1}{n} T}}(ΔW_{k})^{2}

= lim_{n→+∞} ∑_{k=1}^{n} f′(η_{k})[ < W_{\frac{.}{n} T} >_{k} − < W_{\frac{.}{n} T} >_{k−1} ]

= ∫_{0}^{T} f′(W_{s}) d < W >_{s}= ∫_{0}^{T} f′(W_{s}) ds (a.s.)

since < W >_{s}= s (a.s.), W is continuous (a.s.) and f′ is continuous (a.s.). The same result must be true for f′ ∈ L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ) by the dominated convergence theorem since C^{0} is dense in L^{1} (or by using Sobolev’s mollifying procedures). This completes the proof.

Lemma 5.6. If ∫_{0}^{T} |f′(W_{s})| ds < +∞ (a.s.) and f′ ∈ L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ) then

lim_{n→+∞} |R_{n}(f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T})| = 0 (a.s.).

Proof. Use localizing arguments with Sobolev mollifiers. Thus, we may confine us to the case of f ∈ C^{1}(ℝ^{1}, ℝ^{1}). From Theorem 4.1 we know about the estimates of the remainder term

|R_{n}(f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T})| ≤ \frac{1}{4} ∑_{k=1}^{n} |f′(η_{k}^{1}) − f′(η_{k}^{2})|(W_{t_{k}} − W_{t_{k−1}})^{2}.

Thus, it follows that lim_{n→+∞} |R_{n}(f(W_{\frac{.}{n} T}), W_{\frac{.}{n} T})| = 0 (a.s.), thanks to the continuity of W, < W >_{T} = T < +∞ and the assumption ∫_{0}^{T} |f′(W_{s})| ds < +∞. This fact carries over to the case f′ ∈ L_{loc}^{1}(ℝ^{1}, B(ℝ^{1}), μ) while taking the limit in the mollified version of f. Thus, the proof is complete.

Example 5.1. For all a ∈ ℝ^{1}, decompose

|W_{t} − a| = |W_{0} − a| + ∫_{0}^{t} sign(W_{s} − a) dW_{s} + 2L_{t}(a)

by the help of Corollary 5.3, where L_{t}(a) = \frac{1}{2} ∫_{0}^{t} δ_{0}(W_{s} − a) ds is the related local time. This is the more general form of Tanaka formula for Wiener processes, and it is obtained as an application of FTSC to (x − a)_{+} and (x − a)_{−}. Example 5.2. Apply the FTSC to F (W_{t}) = (W_{t})^{2} (i.e. Corollary 5.3). Then

(W_{t})^{2} = W_{0}^{2} + 2α ∫_{0}^{t} W_{s+} dW_{s} + 2(1 − α) ∫_{0}^{t} W_{s−} dW_{s} + (1 − 2α)t.

Therefore, the α-integral of the Wiener process with itself is given by

α ∫_{0}^{t} W_{s} dW_{s} = α ∫_{0}^{t} W_{s+} dW_{s} + (1 − α) ∫_{0}^{t} W_{s−} dW_{s}

= \frac{W_{t}^{2} − W_{0}^{2} − (1 − 2α)t}{2}

which can be interpreted as a transformation formula for double αintegrals of Wiener processes.

5.4. A formula of integration by parts for α-integrals and semimartingales. More general, we can establish the discrete version of integration-by-parts formula for α-integrals involving general sequences of random variables.

Theorem 5.7. Assume that X = (X_{n})_{n∈ℕ}, Y = (Y_{n})_{n∈ℕ} are two sequences of random variables X_{n}, Y_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) . Then ∀α ∈ ℝ^{d×d} ∀n ∈ ℕ

(5.5) I_{n}^{α}(X, Y) =< X_{n}, Y_{n} >_{d} − < X_{0}, Y_{0} >_{d} − I_{n}^{α}(Y, X)

− \frac{1}{2}( < (I − 2α)Y, X >_{n} + < (I − 2α)X, Y >_{n} ).

Proof. Define the new parameter matrix

α̃ := ( \begin{matrix}α & O \\ O & α \end{matrix} ) ∈ ℝ^{2d×2d}

for α ∈ ℝ^{d×d}. Apply the FTSC(α̃) to F : ℝ^{2d} → ℝ^{1} defined by F (x, y) =< x, y >_{d}. For this purpose, calculate the gradient f(x, y) = ∇F (x, y) = (y^{T} x^{T})^{T} and the Hessian

H(x, y) = ∇^{2}F (x, y) = ( \begin{matrix}O & I_{d} \\ I_{d} & O \end{matrix} ) .

Therefore, the remainder term R vanishes in view of its representation (4.1) with constant Hessian. Now, evaluate the quadratic covariation

< (I_{2d} − 2α̃)f(X, Y), (X^{T} Y^{T})^{T} >_{n}
=< (I_{d} − 2α)Y, X >_{n} + < (I_{d} − 2α)X, Y >_{n}

with α ∈ ℝ^{d×d}. The related α̃-integrals satisfy

Iα̃ ; n(f(X, Y), (X^{T} Y^{T})^{T}) = I_{n}^{α}(Y, X) + I_{n}^{α}(X, Y)

for all n ∈ ℕ. Finally, putting all the side calculations together and returning to the discrete FTSC(α̃) leads to the claimed integration-by-parts formula. Thus, the proof is complete.

Similarly one can find the continuous version of integration-by-parts formula for α-integrals involving continuous semimartingales.

Theorem 5.8. Assume that X = (X_{t})_{0≤t≤T} , Y = (Y_{t})_{0≤t≤T} are two continuous d-dimensional (F_{t})-adapted semimartingales with Doob-Meyer decompositions X_{t} = X_{0} + M_{t} + A_{t} and Y_{t} = Y_{0} + N_{t} + C_{t} with unique martingale parts M = (M_{t})_{0≤t≤T} and N = (N_{t})_{0≤t≤T}, where X_{t}, Y_{t} : (Ω, F_{t}, ℙ) → (ℝ^{d}, B(ℝ^{d})) for 0 ≤ t ≤ T. Then ∀α ∈ ℝ^{d×d} ∀t : 0 ≤ t ≤ T

(5.6)  α ∫_{0}^{t} < X_{s}, dY_{s} >_{d}=< X_{t}, Y_{t} >_{d} − < X_{0}, Y_{0} >_{d}

− α ∫_{0}^{t} < Y_{s}, dX_{s} >_{d}

− \frac{1}{2}( < (I − 2α)N, M >_{0}^{t} + < (I − 2α)M, N >_{0}^{t} ).

Proof. Take F (x, y) =< x, y >_{d} and α̃ as in the proof before. Either one uses the discrete FTSC(α̃) applied to F along the sequences X̃ = (X_{t_{n}})_{n∈ℕ}, Ỹ = (Y_{t_{n}})_{n∈ℕ} and partitions (t_{k})_{0≤k≤n} of [0, T] and takes the corresponding limits or one directly uses the continuous FTSC(α̃). It remains to note that the continuous quadratic covariation simplifies to

< (I_{2d} − 2α̃)f(X, Y), (X^{T} Y^{T})^{T} >_{s}^{t} =< (I_{d} − 2α)N, M >_{s}^{t}
+ < (I_{d} − 2α)M, N >_{s}^{t}

since the bounded variation part A and C vanishes in it. Thus, the proof is completed.

5.5. A discrete Dynkin formula for semimartingales. Taking the expected values in the FTSC(α) provides us with the discrete Dynkin formula for α-integrals.

Theorem 5.9. Let X = (X_{n})_{n∈ℕ} be a semimartingale sequence of (F_{n})- adapted random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and Doob-Meyer decomposition X_{n} = X_{0} + M_{n} + A_{n} where M_{n} is the unique (F_{n})- martingale part and A_{n} the unique predictable compensator. Assume that F : ℝ^{d} → ℝ^{1} is absolutely continuous with L_{loc}^{1}(ℝ^{d}, B(ℝ^{d}), μ)-integrable gradient f : ℝ^{d} → ℝ^{d}. Then there is the representation

(5.7) 𝔼F (X_{n}) = 𝔼F (X_{0}) + 𝔼[I_{n}^{B}(αf(X), X) + I_{n}^{F} ((I_{d} − α)f(X), A)]

+ 𝔼[R_{n}(f(X), X) + \frac{1}{2} < (I_{d} − 2α)f(X), X >_{n} ]

for all n ∈ ℕ, where R_{n} is given by (3.2).

Proof. Take the expectation in FTSC(α) as stated in Theorem 3.1.

Remark 5.10. The discrete version of the classical Dynkin formula (see Dynkin [8])

(5.8)  𝔼F (X_{n}) = 𝔼F (X_{0}) + 𝔼[I_{n}^{F} (f(X), A)]

+ 𝔼[R_{n}(f(X), X) + \frac{1}{2} < f(X), X >_{n} ]

is included in (5.7) with the choice α = O. Continuous versions can be obtained by taking the limit while controlling the moment behavior of the remainder terms. For this purpose, weaker conditions for the convergence are expected, compared to the situation with the continuous Itô formula. A related analysis is delicate and omitted.

5.6. A discrete Burkholder-type estimate from above. Burkholdertype estimates which essentially originate from Paley [18], Burkholder [2], Burkholder and Gundy [5], Burkholder [4], Davis [6] are important tools to control the moment behavior of stochastic difference equations and stochastic differential equations. Here we present an alternative proof of a certain generalization. For a given process X = (X_{n})_{n∈ℕ}, define the maximum process X^{∗} = (X_{n}^{∗})_{n∈ℕ} related to X by

X_{n}^{∗} = max_{0≤k≤n} ||X_{k}||_{d}.

Let q be the conjugate exponent to p, i.e. \frac{1}{p} + \frac{1}{q} = 1.

Theorem 5.11. Let p ≥ 2. Assume that X = (X_{n})_{n∈ℕ} is a local martingale sequence of (F_{n})-adapted random variables with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and finite initial p-th moment 𝔼||X_{0}||_{d}^{p} < +∞. Then there exists an universal constant C = C(p) such that

(5.9)  ∥X_{n}^{∗}∥_{L^{p}(Ω,F,ℙ)} = (𝔼(X_{n}^{∗})^{p})^{1/p}
≤ (𝔼∥X_{0}∥_{d}^{p})^{1/p} + C(p)(𝔼(< X >_{n})^{p/2})^{1/p}.

This universal constant C(p) can be estimated by

(5.10)  C(p) ≤ q^{p/2} \frac{\sqrt{3p(p − 1)}}{2} = \frac{\sqrt{3}}{2} \frac{p^{(p+1)/2}}{(p − 1)^{(p−1)/2}} .

Proof. If 𝔼(< X >_{n})^{p/2} = +∞ then the inequality (5.9) trivially holds. Suppose that 𝔼(< X >_{n})^{p/2} < +∞. Recall that

(5.11)  ||X_{n}^{∗}||_{L^{p}(Ω,F,ℙ)}^{p} = 𝔼(X_{n}^{∗})^{p} ≤ q^{p}𝔼||X_{n}||_{d}^{p}

from Doob’s L^{p}-maximum inequality with q = p/(p − 1) for p > 1. Now, for simplicity, we may suppose that X is a (F_{n})-martingale since the proof of the more general local martingale case works similarly (only the formalism of localizing sequences τ_{n} need to be added and one may take the limit in the estimates afterwards anyway). Apply our FTSC (α) with α = 0 in conjunction with (4.2). For this purpose, calculate

∇||x||_{d}^{p} = px||x||_{d}^{p−2}, H(x) = ∇^{2}||x||_{d}^{p} = p||x||_{d}^{p−2}I_{d} + p(p − 2)||x||_{d}^{p−4}xx^{T} ,

||H(x)||_{d×d} ≤ p(p − 1)||x||_{d}^{p−2}

and

||H(x) − H(y)||_{d×d} ≤ p (p − 1) ( max(||x||_{d}.||y||_{d}))^{p−2}

for all x, y ∈ ℝ^{d}, due to the positive-definiteness of xx^{T} and p ≥ 2. Consequently,

||X_{n}||_{d}^{p} = ||X_{0}||_{d}^{p} + pI_{n}(||X||_{d}^{p−2}X, X)

+ \frac{p}{2} < ||X||_{d}^{p−2}X, X >_{n} +pR_{n}(||X||_{d}^{p−2}X, X)

holds, where

|pR_{n}(||X||_{d}^{p−2}X, X)| ≤ \frac{1}{4} ∑_{k=1}^{n} (∥H(η_{k}^{1}) − H(η_{k}^{2})∥_{d×d})||ΔX_{k}||_{d}^{2}

≤ \frac{p(p − 1)}{4}(X_{n}^{∗})^{p−2} ∑_{k=1}^{n} ||ΔX_{k}||_{d}^{2}

= \frac{p(p − 1)}{4}(X_{n}^{∗})^{p−2} < X >_{n}

since

||η_{k}^{i}||_{d} = ||X_{k−1} + θ_{k}^{i}ΔX_{k}||_{d} = ||(1 − θ_{k}^{i})X_{k−1} + θ_{k}^{i}X_{k}||_{d}
≤ (1 − θ_{k}^{i})||X_{k−1}||_{d} + θ_{k}^{i}||X_{k}||_{d} ≤ max(||X_{k−1}||_{d}, ||X_{k}||_{d}) ≤ X_{n}^{∗}

for 1 ≤ k ≤ n and i = 1, 2. Furthermore, we estimate the quadratic covariation

|p < ||X||_{d}^{p−2}X, X >_{n} |

= |p ∑_{k=1}^{n} < ||X_{k}||_{d}^{p−2}X_{k} − ||X_{k−1}||_{d}^{p−2}X_{k−1}, ΔX_{k} >_{d} |

= | ∑_{k=1}^{n} < H(η_{k})ΔX_{k}, ΔX_{k} >_{d} |

≤ max_{1≤k≤n} ||H(η_{k})||_{d×d} < X >_{n}≤ p(p − 1)(X_{n}^{∗})^{p−2} < X >_{n}

with intermediate vectors η_{k} = X_{k−1} + θ_{k}ΔX_{k} where θ_{k} ∈ [0, 1]. Next, taking expectation in the FTSC applied to ||X||_{d}^{p} and assembling all auxiliary estimates yields

𝔼||X_{n}||_{d}^{p} = 𝔼||X_{0}||_{d}^{p} + \frac{p}{2}𝔼 < ||X||_{d}^{p−2}X, X >_{n} +p𝔼R_{n}(||X||_{d}^{p−2}X, X)

≤ 𝔼||X_{0}||_{d}^{p} + \frac{3p(p − 1)}{4}𝔼((X_{n}^{∗})^{p−2} < X >_{n} )

≤ 𝔼||X_{0}||_{d}^{p} + \frac{3p(p − 1)}{4}(𝔼(X_{n}^{∗})^{p})^{(p−2)/p} · (𝔼(< X >_{n})^{p/2})^{2/p}

by Hölder inequality with p̃ = p/(p − 2) and q̃ = p/2. Now, divide this estimate by (𝔼(X_{n}^{∗})^{p})^{1−2/p}. This gives

\frac{𝔼||X_{n}||_{d}^{p}}{(𝔼(X_{n}^{∗})^{p})^{1−2/p}} ≤ \frac{𝔼||X_{0}||_{d}^{p}}{(𝔼(X_{n}^{∗})^{p})^{1−2/p}} + \frac{3p(p − 1)}{4}(𝔼(< X >_{n})^{p/2})^{2/p}.

Combine this estimate with the mentioned Doob’s L^{p}-maximum inequality and use the monotonicity of the maximum norm to conclude that

(𝔼(X_{n}^{∗})^{p})^{2/p} ≤ q^{p} \frac{𝔼||X_{0}||_{d}^{p}}{(𝔼(X_{n}^{∗})^{p})^{1−2/p}} + q^{p} \frac{3p(p − 1)}{4}(𝔼(< X >_{n})^{p/2})^{2/p}

≤ q^{p}(𝔼||X_{0}||_{d}^{p})^{2/p} + q^{p} \frac{3p(p − 1)}{4}(𝔼(< X >_{n})^{p/2})^{2/p}.

Eventually, taking the square root leads to

(𝔼(X_{n}^{∗})^{p})^{1/p} ≤ q^{p/2}(𝔼||X_{0}||_{d}^{p})^{1/p} + q^{p/2} \frac{\sqrt{3p(p − 1)}}{2}(𝔼(< X >_{n})^{p/2})^{1/p}.

This estimate can be improved to (5.9) by using the inverse triangle inequality for norms and applying the obtained latter estimate to the sequence X̂ = (X̂_{n})_{n∈ℕ} with X̂_{n} = X_{n} − X_{0} instead of X = (X_{n})_{n∈ℕ}. Therefore

||X_{n}^{∗}||_{L^{p}(Ω,F_{n},ℙ)} − ||X_{0}||_{L^{p}(Ω,F_{0},ℙ)} ≤ ||(X_{n} − X_{0})^{∗}||_{L^{p}(Ω,F_{n},ℙ)}

≤ q^{p/2} \frac{\sqrt{3p(p − 1)}}{2} (𝔼(< X >_{n})^{p/2})^{1/p} .

Thus, one arrives at the claimed assertion (5.9) with the universal estimate (5.10) for C(p), and the proof is complete.

Remark 5.12. We do not claim that the estimate (5.10) is the most efficient one. However, its proof seems to be rather transparent and short. It is interesting to compare this estimate with those in Protter [19] or Krylov [16] in case of continuous time processes and Shiryaev [25] in case of discrete processes. Obviously, those estimates do not coincide with (5.10). Our estimate is close to that of Protter [19] (p. 175) with

C(p) ≤ q^{p/2} \frac{\sqrt{2p(p − 1)}}{2}

since our proof represents a discrete version of his proof in the continuous time case. Both estimates cannot be the same since there is the remainder term in the discrete case, which additionally needs to be estimated, compared to the continuous case. Despite this minor difference, our Burkholdertype estimate from above is still much better than that stated in Shiryaev [25] (p. 499) with

C(p) ≤ 18\frac{p^{5/2}}{(p − 1)^{3/2}}

for the discrete martingale case with p ≥ 2. In passing we note that all these estimates grow linearly in p as p → +∞. Similar estimates of Burkholdertype can be derived if

𝔼 max_{1≤k≤n;θ_{k}^{1},θ_{k}^{2}∈[0,1]}∥H(X_{k−1} + θ_{k}^{1}ΔX_{k}) − H(X_{k−1} + θ_{k}^{2}ΔX_{k})∥_{d×d}^{p/(p−2)}
+ 𝔼(<X >_{n})^{p/2} < +∞.

These estimates would be asymptotically more efficient in the continuous case. For estimates in the case 1 ≤ p < 2, see Burkholder [3].

5.7. Estimations of L^{p}-norms from below. The FTSC(α) in form of integration-by-parts formula provides us with a simple way to obtain estimations of the L^{p}-norms of sequences X = (X_{n})_{n∈ℕ} from below.

Theorem 5.13. Assume that X = (X_{n})_{n∈ℕ} with X_{n} : (Ω, F_{n}, ℙ) → (ℝ^{d}, B(ℝ^{d})) is a local (F_{n})-martingale and has finite moments

𝔼( < X >_{n} +||X_{0}||_{d}^{2}) < +∞.

Then ∀p ≥ 2

∥X_{n}∥_{L^{p}(Ω,F_{n},ℙ)} ≥ [𝔼( < X >_{n} +||X_{0}||_{d}^{2})]^{1/2}
≥ [𝔼( < X >_{n} +||X_{0}||_{d}^{2})^{2/p}]^{p/4}.

In the case p = 2 we have

||X_{n}||_{L^{2}(Ω,F_{n},ℙ)} = [𝔼( < X >_{n} +||X_{0}||_{d}^{2})]^{1/2}.

Proof. For simplicity, we may suppose that X = (X_{n})_{n∈ℕ} is a (F_{n})-martingale (otherwise we can apply standard localization procedures by sequences (τ_{k})_{k∈ℕ} of finite stopping times τ_{k}). From the FTSC(α) applied to F (x) = ||x||_{d}^{2} with α = O (i.e. the integration-by-parts formula, see Theorem 5.7) it follows that

||X_{n}||_{d}^{2} − 2 ∑_{k=1}^{n} < X_{k−1}, ΔX_{k} >_{d} = ||X_{0}||_{d}^{2}+ < X >_{n} .

Note that the process Z = (Z_{n})_{n∈ℕ} with Z_{n} = ∑_{k=1}^{n} < X_{k−1}, ΔX_{k} >_{d} represents a (F_{n})-martingale started at Z_{0} = 0. Therefore

𝔼||X_{n}||_{d}^{2} = 𝔼(||X_{0}||_{d}^{2}+ < X >_{n} ).

By twice application of Lyapunov’s inequality (see Shiryaev [25]) we verify the claim of Theorem 5.13, hence the proof is complete.

5.8. On the relation between inner and outer α-integrals. As we already know, the inner and outer integrals coincide for α = O, I_{d} or linear functions f (with d = l). So what happens when α ≠ O, I_{d} and f is nonlinear? Obviously, in the discrete case they differ in general, whereas the situation is more complicated in the continuous case. The following supplement gives a partial answer on the relation between the inner and outer α-integrals with respect to strong convergence (i.e. in the sense of L^{1}(Ω, F, ℙ)) for continuous processes X.

Theorem 5.14. Let H be the Hessian matrix of the function F ∈ C^{2}(ℝ^{d}, ℝ^{1}), p ≥ 2, and \frac{1}{p} + \frac{1}{q} = 1. Assume that the process X = (X_{t})_{0≤t≤T} with X_{t} : (Ω, F_{t}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and finite quadratic variation < X >_{T} < +∞ (a.s.) satisfies

(5.12)  𝔼 (∫_{0}^{T} ||H(X_{s})||_{d×d}^{2} d < X >_{s})^{q/2} < + ∞ and
(5.13)  lim_{n→+∞} (n · 𝔼 max_{1≤k≤n} ||X_{t_{k}} − X_{t_{k−1}}||_{d}^{p}) = 0

along any nonrandom partitions 0 = t_{0} < t_{1} < ⋯ < t_{n−1} < t_{n} = T for the fixed constants p ≥ 2. Then ∀α ∈ ℝ^{d×d} with ∀x ∈ ℝ^{d} : αH(x) = H(x)α

lim_{n→+∞} 𝔼 max_{0≤k≤n} |I_{n}^{α}(f(X), X)− I_{n}^{α}^{∘} (f(X), X)| = 0.

Proof. Define X̃ = (X̃_{k})_{0≤k≤n} by X̃_{k} = X_{\frac{k}{n} T}. Consider

|I_{n}^{α}(f(X̃), X̃)− I_{n}^{α}^{∘} (f(X̃), X̃)| (5.14)

= | ∑_{k=1}^{n} ( < αf(X̃_{k}) + (I_{d} − α)f(X̃_{k−1}), ΔX̃_{k} >_{d}

− < f(αX̃_{k} + (I_{d} − α)X̃_{k−1}), ΔX̃_{k} >_{d} )|

= | ∑_{k=1}^{n} ( < α[f(X̃_{k}) − f(αX̃_{k} + (I_{d} − α)X̃_{k−1})], ΔX̃_{k} >_{d} +

+ < (I_{d} − α)[f(X̃_{k−1}) − f(αX̃_{k} + (I_{d} − α)X̃_{k−1})], ΔX̃_{k} >_{d} )|

= | ∑_{k=1}^{n} ( < αH(η_{k}^{1})(I_{d} − α)ΔX̃_{k}, ΔX̃_{k} >_{d}

− < (I_{d} − α)H(η_{k}^{2})αΔX̃_{k}, ΔX̃_{k} >_{d} )|

= | ∑_{k=1}^{n} < (αH(η_{k}^{1})(I_{d} − α) − (I_{d} − α)H(η_{k}^{2})α)ΔX̃_{k}, ΔX̃_{k} >_{d} |

where η_{k}^{1} and η_{k}^{2} are intermediate values between X̃_{k−1} and X̃_{k}. Define c = \sqrt{2} · ||α||_{d×d} · ||I_{d} − α||_{d×d}. Taking the expectation on both sides and applying the Hölder inequality leads to

𝔼|I_{n}^{α}(f(X̃), X̃)− I_{n}^{α}^{∘} (f(X̃), X̃)|

≤ c^{2} · (𝔼[ ∑_{k=1}^{n} ( max(||H(η_{k}^{1})||_{d×d}, ||H(η_{k}^{2})||_{d×d}))^{2}||ΔX̃_{k}||_{d}^{2}]^{q/2})^{1/q}

· (n𝔼 max_{1≤k≤n} ||ΔX̃_{k}||_{d}^{p})^{1/p}.

The right hand side is finite for sufficiently large n, due to the assumptions (5.12) and (5.13). Now, it remains to take the limit as n tends to +∞. Thus, the proof is completed.

Alternatively, the latter proof can slightly be modified to obtain the following.

Theorem 5.15. Let H be the Hessian matrix of the function F ∈ C^{2}(ℝ^{d}, ℝ^{1}) and p > 2. Assume that the process X = (X_{t})_{0≤t≤T} with X_{t} : (Ω, F_{t}, ℙ) → (ℝ^{d}, B(ℝ^{d})) and finite quadratic variation < X >_{T} < +∞ (a.s.) satisfies

(5.15) 𝔼 ∫_{0}^{T} ||H(X_{s})||_{d×d}^{2} d < X >_{s}< +∞, 𝔼( < X >_{T} )^{p/2} < +∞

and (5.16)

lim_{n→+∞} (𝔼 max_{1≤k≤n} sup_{X^{(i)}=X_{t_{k−1}}+θ_{k}^{i}(X_{t_{k}} −X_{t_{k−1}})^{θ^{i}_{k}∈[0,1],i=1,2}}||H(X^{(1)}) − H(X^{(2)})||_{d×d}^{p/(p−2)}) = 0

along any nonrandom partitions 0 = t_{0} < t_{1} < ⋯ < t_{n−1} < t_{n} = T for the fixed constant p > 2. Then ∀α ∈ ℝ^{d×d} with ∀x ∈ ℝ^{d} : αH(x) = H(x)α

lim_{n→+∞} 𝔼 max_{0≤k≤n} |I_{n}^{α}(f(X), X)− I_{n}^{α}^{∘} (f(X), X)| = 0.

Proof. Return to the proof before (i.e. (5.14)), and recall that

ε_{n} := |I_{n}^{α}(f(X̃), X̃)− I_{n}^{α}^{∘} (f(X̃), X̃)| (5.17)

= | ∑_{k=1}^{n} < (αH(η_{k}^{1})(I_{d} − α) − (I_{d} − α)H(η_{k}^{2})α)ΔX̃_{k}, ΔX̃_{k} >_{d} |

= | ∑_{k=1}^{n} < (αH(η_{k}^{1}) − H(η_{k}^{2})α + α(H(η_{k}^{2}) − H(η_{k}^{1}))α)ΔX̃_{k}, ΔX̃_{k} >_{d} |

= | ∑_{k=1}^{n} < (α(H(η_{k}^{1}) − H(η_{k}^{2})) + α(H(η_{k}^{2}) − H(η_{k}^{1}))α)ΔX̃_{k}, ΔX̃_{k} >_{d} |

Using the triangle inequality and taking the expectation yields

𝔼[ε_{n}] ≤ (||α||_{d×d} + ||α||_{d×d}^{2}) · 𝔼 ( max_{1≤k≤n} ||H(η_{k}^{1}) − H(η_{k}^{2})||_{d×d}· < X̃ >_{n}) .

Finally, apply the Hölder inequality with p′ = p/2 and q′ = p/(p − 2) to complete the proof.

6. CONCLUSIONS AND REMARKS

This paper has shown that many of the known fundamental results of stochastic analysis originate from one and the same discrete version of a theorem of stochastic α-calculus (abbreviated with FTSC(α)). Thus, we feel to be entitled to use the word combination of the fundamental theorem of stochastic calculus based on the analogy to deterministic calculus. Numerous applications of FTSC have been shown too. The special role of the concept of finite quadratic covariation is clearly visible as well. Obviously, with the analogy to the deterministic case in mind, the teaching of main tools of stochastic analysis seems to be easier while referring to the discrete version of FTSC and related calculus rules. Corresponding continuous versions come up very naturally by taking related limits applied to the discrete approach. The advantage of the discrete approach is obvious. One is able to explain fundamental ideas of stochastic analysis with a minimum of mathematical background from more advanced mathematics.

The presented calculus is so innovative such that plenty of further generalizations are conceivable. It would be interesting to get rid of the assumption of finite quadratic variation (covariation). It is possible to find more efficient estimates of Burkholder-Gundy-Davis type (i.e. from both above and below) for sequences of random variables without using the classical Itô formula. The relation between outer and inner integral needs to be studied further. Moreover, it is interesting to note that we originally intended to estimate the absolute difference of outer and inner integrals from below. However, apart from the trivial estimation by 0 from below, such an intention is meaningless in the asymptotic regime (i.e. for the continuous case) in general. This fact can be seen from the identity (5.14) where the additive commutator

αH(η)(I_{d} − α) − (I_{d} − α)H(η)α = αH(η) − H(η)α

which has always trace zero is involved. Furthermore, with the herein presented discrete approach to stochastic calculus, one can derive estimates for the extreme value behavior of semimartingales, stochastic differential and difference equations, and their approximations. See [22] - [24], and author’s future research contributions.

ACKNOWLEDGEMENTS

The author likes to express his gratitude to Prof. Dr. M. Sambandham and Prof. Dr. S. Sathananthan for their kind invitations to write a paper for this special volume of DYNAMIC SYSTEMS and APPLICATIONS in 2003.

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