Corrigendum

This article refers to:

Stochastic Motion Under G-Framework: I. Nelson Stochastic Derivatives

Article Title: Stochastic Motion Under G-Framework:

I. Nelson Stochastic Derivatives

Authors(s): Hong Zhang and D. Kannan

Journal: Stochastic Analysis and Applications

Bibliometrics: Published in Volume 32, Issue 6, pp. 1067–1096.

DOl: 10.1080/07362994.2014.964870

Publisher: Taylor & Francis

In our earlier paper, we were working on stochastic motion under the sublinear expectation framework. However, in the proof of Theorem 3.6 there, we made a careless mistake in working with , when . Subsequently, we need to redefine the function ξ_θ and this affects the definitions of and Consequently, expressions/formulas in two other results and two examples change, and the corrected formulas are given below.

Assumption K-2:

For each , let p_θ be the probability density function of X_t that satisfies Equation (3.5) under P_θ. Let Define Assume that the following functions are continuous and in L²_G(Ω):

Note: Theorem 3.6, Theorem 3.7, Proposition 3.11, Example 4.4, and Example 4.5 (given below) all discuss the one-dimensional G-Itô process X_t satisfying under the Assumptions (K-1) and (K-2). Here, X₀ ∈ L²_G(Ω₀), and α, β, η ∈ M²_G(0, T) are bounded processes. We will not include this hypothesis in the statements of the said theorems and examples below.

Theorem 3.6

Let X_t be the one dimensional G −Itô process where X₀ ∈ L²_G(Ω₀), α, β, η ∈ M²_G(0, T) are bounded processes. Assume the conditions (K-1) and (K-2). Then (1) (2) (3) (4)

Proof.

For t ∈ (0, T), and h > 0, satisfying 0 < t − h < t < t + h < T, we have Observing that the process ∫^{t + h}_tη(u) d⟨B⟩_u − 2∫^{t + h}_tG(η(u)) du is a G-martingale with respect to the filtration , and that we have Hence, Similarly, Recall now that a one-dimensional G-Itô process X_t admits the following representation under the probability measure P_θ, and that the time-reversed process of X(t) under P_θ satisfies here, is a -Brownian motion under P_θ in which and, with p_θ(t) denoting the probability density of X_t under P_θ, For fixed t ∈ [0, T] and , set It is an -martingale for each θ. For 0 ≤ s < t ≤ T, and

For each fixed , and is an -martingale under P_θ with and

Notice that for h > 0, [t − h, t]⊂[0, T], and fixed , we have Letting h → 0⁺, we now get and, hence, Consequently,

Similarly, we obtain Noticing now that, we obtain (5)

and It is easily seen that D_iX(t), i = 1, 2, 3, 4, are continuous on (0, T) in L²_G(Ω).

Recall the hypothesis in the Note above. Under that hypothesis, we have the following Proposition as a special case of the above theorem.

Proposition 3.7

now reads as: If α(t) = α, η(t) = η, β(t) = β are bounded in L²_G(Ω₀), η is deterministic, and X₀ ∈ L²_G(Ω₀), then D_iX, i = 1,…, 4, exist, are continuous on (0, T), and are given by:

Theorem 3.11

now reads as: Subject to the Note above, if with we have

Proof.

Note from Itô’s formula that, for s, t ∈ [0, T] with s < t, and The results follow from Theorem 3.6.

Now the and of Example 4.2, are given by:

Example 4.4:

Keeping in mind the Note supra, the forward and backward drifts are given by respectively, and the current and osmotic velocities of X(t) are given by respectively.

Example 4.5:

Now, for with , we have the forward and backward drifts of ϕ(X(t)) given by respectively, and the current and osmotic velocities of ϕ(X(t)) are given by respectively.

Next we make the corresponding changes in the multidimensional case. Let B be a d-dimensional G-Brownian motion and X = (X¹,…, Xⁿ)^T be an n-dimensional process on [0, T] in the form (6) Here, for ν = 1, 2,…, n, i, j = 1, 2,…, d, α^ν, η^ν_ij, β^ν_j ∈ M²_G(0, T) are bounded and continuous processes, X(0) is a given n-dimensional random column vector in L²_G(Ω),β = (β¹, β²,…, βⁿ) is an d × n-matrix with d-dimensional column vectors β^ν,ν = 1, 2,…, n, and β^* is the transpose of β.

Assumption K-3:

For each , let p_θ be the probability density function of X_t under P_θ. For the d × d matrices we set Assume that are continuous in L²(Ω) .

Theorem 5.1:

Let B be an d-dimensional G-Brownian motion and X = (X¹,…, Xⁿ)^T be an n-dimensional process on [0, T] in the form (7)

Here, for ν = 1, 2,…, n, i, j = 1, 2,…, d, α^ν, η^ν_ij, β^ν_j ∈ M²_G(0, T) are bounded and continuous processes, and X(0) = (X¹(0),…, Xⁿ(0))^T is a given random vector in L²_G(Ω). β = (β¹, β²,…, βⁿ) is an d × n-matrix with d-dimensional column vectors β^ν,ν = 1, 2,…, n,, and β^* is the transpose of β. We then have the following:

1.	DiX(t), i = 1, 2, exist, are continuous on (0, T), and are given by (8) (9)
2.	If in addition, the Assumptions (K-1) and (K-3) also hold, then DiX(t), i = 3, 4, exist, are continuous on (0, T), and are given by (10) (11)

Proof.

(2) Notice that the time reversed process of X(t) under P_θ satisfies Here, is a d-dimensional -Brownian motion under P_θ, and where p_θ(t) is the probability density of X_t under P_θ.

For h > 0, t, t − h ∈ [0, T], we have and For each , is an -martingale, and We have Therefore, and, consequently, Similarly,

Theorem 5.2

now reads: Assume that with for μ, ν = 1,…, n. Then,

Proof.

This follows from Itô’s formula and Theorem 5.1.