A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals

Abstract

The aim of this work is to characterize one-dimensional homogeneous diffusion process, under the assumption that marginal density of the process is Gaussian. The method considers the forward Kolmogorov equation and Fourier transform operator approach. The result establishes the necessary characteristic equation between drift and diffusion coefficients for homogeneous and nonhomogeneous diffusion processes. The equation for homogeneous diffusion process leads to characterize the possible diffusion processes that can exist. Two well-known examples using the necessary characteristic equation are also given.

1. Introduction and Preliminaries

The characterization of drift and diffusion coefficients by considering the marginal density of stochastic process is one of the known methods of estimating the actual underlying process and to construct new diffusion processes [1–6]. In this paper, we will consider the forward Kolmogorov equation and apply the Fourier transform operator to prove necessary characteristic equation between drift and diffusion coefficients for a homogeneous one-dimensional diffusion process when the marginal density is Gaussian. This equation characterizes the possible diffusion processes. A similar equation is proved for a nonhomogeneous diffusion process. However, the idea is motivated by Hamza and Klebaner [6], who constructed an entire family of non-Gaussian martingales given that the marginals are Gaussian.

Before stating our main result, we introduce some of the common notations, assumptions, and key facts used in this work.(i)𝑎(𝑧) and 𝑎(𝑧,𝑡) denote the drift coefficient of homogeneous and nonhomogeneous one-dimensional Itô-diffusion process 𝑋={𝑋_𝑡,𝑡≥0} given 𝑋_𝑡=𝑧. Similarly, 𝜎(𝑧) and 𝜎(𝑧,𝑡) denote the diffusion coefficient of homogeneous and non-homogeneous one-dimensional Itô-diffusion process 𝑋={𝑋_𝑡,𝑡≥0} given 𝑋_𝑡=𝑧.(ii)𝑝_𝑡,𝑡0(𝑧∣𝑦) denotes the density function of (𝑋_𝑡=𝑧∣𝑋_𝑡0=𝑦) with mean 𝜇_𝑡 and variance 𝛾_𝑡.(iii)𝐶₀(ℝ) denotes the set of all continuous functions with domain ℝ, 𝐶²₀(ℝ) denotes the set of all continuous twice-differentiable functions with domain ℝ, and 𝐶₀^2,1(ℝ) denotes the set of all continuous twice-differentiable functions in the first variable and continuous differentiable functions in the second variable with domain ℝ².(iv)𝑓^′_𝑥 or 𝑓^′(𝑥) denotes the first derivative of function 𝑓 with respect to 𝑥.(v)The conditional density 𝑝_𝑡,𝑡0(𝑧∣𝑦) is bounded by 𝑒^−𝑘𝑧2.(vi)The assumption 𝑡₀>0 is considered due to the continuity problem at 𝑡=0. It means that if the conditional probability distribution of the increments is considered from time 0, then the method we use is not able to identify those drift and diffusion coefficients that lead us to construct continuous diffusion process.(vii)𝑓(𝑧)∶=𝑒^𝑖𝑧𝑥 is considered to use the uniqueness property of the characteristic function.

We need the following standard forward Kolmogorov equations introduced in [7].

Definition 1.1. Let the homogeneous diffusion process 𝑋={𝑋_𝑡,𝑡≥0} satisfy the one-dimensional stochastic differential equation (SDE) 𝑑𝑋_𝑡=𝑎𝑋_𝑡𝑑𝑡+𝜎𝑋_𝑡𝑑𝐵_𝑡, (1.1) where {𝐵_𝑡,𝑡≥0} is a standard Brownian motion. Let 𝑓(𝑥) be bounded twice continuously differentiable in 𝑥 (i.e., 𝐶²₀(ℝ)) function, then 𝑑𝐸𝑓𝑋𝑑𝑡^𝑡∣𝑋_𝑡0=𝑦=𝐸𝑓^′𝑋_𝑡𝑎𝑋_𝑡+𝐸𝑓^′′𝑋_𝑡𝜎²𝑋_𝑡2, (1.2) where 𝑦 is any fixed point.

Definition 1.2. Let the non-homogeneous diffusion process 𝑋={𝑋_𝑡,𝑡≥0} satisfy the one-dimensional stochastic differential equation (SDE) 𝑑𝑋_𝑡=𝑎𝑋_𝑡,𝑡𝑑𝑡+𝜎𝑋_𝑡,𝑡𝑑𝐵_𝑡. (1.3) Let 𝑓(𝑥,𝑡) be bounded twice continuously differentiable in 𝑥 and continuously differentiable in 𝑡 (i.e., 𝐶₀^2,1(ℝ)) function, then 𝜕𝐸𝑓𝑋𝜕𝑡^𝑡,𝑡∣𝑋_𝑡0𝑋=𝑦=𝐸𝜕𝑓_𝑡,𝑡𝜕𝑋_𝑡𝑎𝑋_𝑡𝜕,𝑡+𝐸²𝑓𝑋_𝑡,𝑡𝜕𝑋²_𝑡𝜎²𝑋_𝑡,𝑡2, (1.4) where 𝑦 is any fixed point.

The conditions that ensure the existence and uniqueness of any stochastic differential equation, following terminology of [8], are as follows.

Definition 1.3. Let the diffusion process 𝑋={𝑋_𝑡,𝑡≥0} satisfy (1.3), then(1)drift and diffusion coefficients are said to be locally Lipschitz in 𝑥 and uniformly in 𝑡 if for every 𝑇 and 𝑁, there is a constant 𝐾 depending only on 𝑇 and 𝑁 such that for all |𝑥|, |𝑦|≤𝑁 and all 0≤𝑡≤𝑇, 𝑎(𝑥,𝑡)−𝑎𝑦,𝑡+𝜎(𝑥,𝑡)−𝜎𝑦,𝑡<𝐾𝑥−𝑦, (1.5)(2)the coefficients satisfy the linear growth condition if |𝑎(𝑥,𝑡)|+|𝜎(𝑥,𝑡)|≤𝐾(1+|𝑥|). (1.6)
Definition 1.3 holds in the same way for homogeneous diffusion process.

2. Main Results

In Section 2.1, we obtain the characteristic equation between drift and diffusion coefficients under the assumption that marginals are Gaussian. In Section 2.2, we identify possible diffusion processes using the characteristic equation.

2.1. Characteristic Equation

We first prove the characteristic equation for homogeneous diffusion process and then extend our results when the starting position 𝑋_𝑡0 is arbitrary. Finally we prove the characteristic equation in case of non-homogeneous diffusion.

Theorem 2.1. Let the homogeneous diffusion process 𝑋={𝑋_𝑡,𝑡≥0} satisfy the stochastic differential equation (1.1), 𝑎(𝑋_𝑡)∈𝐶₀(ℝ), 𝜎(𝑋_𝑡)∈𝐶²₀(ℝ), and both 𝑎(𝑋_𝑡) and 𝜎(𝑋_𝑡) satisfy the linear-growth and local Lipschitz conditions:(i)assume (𝑋_𝑡=𝑧∣𝑋_𝑡0=𝑦)~𝑁(𝑦+𝜇_𝑡,𝛾_𝑡) where 𝑡₀>0 and 𝑡>𝑡₀, 𝜇_𝑡∈𝐶₀(ℝ) and 𝛾_𝑡∈𝐶₀(ℝ), then the necessary equation on drift and diffusion coefficients obtained by conditioning on 𝑋_𝑡=𝑧 is given by 𝑎(𝑧)=𝜇^′_𝑡+𝜎²(𝑧)^′_𝑧2+𝑧−𝑦−𝜇_𝑡𝛾_𝑡^′−𝜎²(𝑧)2𝛾_𝑡, (2.1)(ii)assume 𝑋_𝑡0 is arbitrary and 𝐸[𝑝_𝑡(𝑧∣𝑋_𝑡0)]~𝑁(𝜇_𝑡,𝛾_𝑡), where 𝑡₀>0 and 𝑡>𝑡₀, 𝜇_𝑡∈𝐶₀(ℝ) and 𝛾_𝑡∈𝐶₀(ℝ), then the necessary equation on drift and diffusion coefficients by conditioning on 𝑋_𝑡=𝑧 is (2.1).

Proof (i). Using (1.2), 𝑑𝑑𝑡^∞_−∞𝑓(𝑧)𝑝_𝑡,𝑡0𝑧∣𝑦𝑑𝑧=^∞_−∞𝑑𝑓(𝑧)𝑎(𝑧)𝑝𝑑𝑧^𝑡,𝑡0+𝑧∣𝑦𝑑𝑧_∞−∞𝑑²𝑓(𝑧)/𝑑𝑧²𝜎²(𝑧)2𝑝_𝑡,𝑡0𝑧∣𝑦𝑑𝑧. (2.2) Under the assumption 𝜉∶=𝑋_𝑡=𝑧∣𝑋_𝑡0=𝑦∼𝑁𝑦+𝜇_𝑡,𝛾_𝑡, (2.3) which implies 𝑝_𝑡(𝜉)=𝑝_𝑡,𝑡01𝑧∣𝑦∶=2𝜋𝛾_𝑡𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}, (2.4) consider a function 𝑓∶ℝ→ℝ, that is, 𝑓(𝑧)∶=𝑒^𝑖𝑧𝑥. (2.5) Thus ^∞_−∞𝑒^𝑖𝑧𝑥𝑝_𝑡,𝑡0𝑧∣𝑦𝑑𝑧=𝑒^{(𝑖𝑥(𝑦+𝜇𝑡))−((𝑥2𝛾𝑡)/2)}. (2.6) Substituting in (2.2) and simplification yields 𝑥22𝜋𝛾_𝑡^∞_−∞𝜎²(𝑧)−𝛾^′_𝑡𝑒^𝑖𝑧𝑥𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}𝑖𝑑𝑧=2𝜋𝛾_𝑡^∞_−∞𝑎(𝑧)−𝜇^′_𝑡𝑒^𝑖𝑧𝑥𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}𝑑𝑧. (2.7) Consider left-hand side of (2.7) and define 𝐺(𝑧)∶=𝜎²(𝑧)−𝛾^′_𝑡𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}. (2.8) Clearly 𝐺(𝑧) is dominated by 𝑒^{−𝐾𝑧2/2}. Thus, integration by parts and using property of characteristic function, we obtain 𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}𝜎(𝑧)(𝜎(𝑧))^′_𝑧−𝑧−𝑦−𝜇_𝑡2𝛾_𝑡𝜎²(𝑧)−𝛾^′_𝑡=𝑎(𝑧)−𝜇^′_𝑡𝑒^{−(𝑧−𝑦−𝜇𝑡)𝑡2/2𝛾}, (2.9) and it further implies that 𝑎(𝑧)=𝜇^′_𝑡+(𝜎²(𝑧))^′_𝑧2−𝑧−𝑦−𝜇_𝑡2𝛾_𝑡𝜎²(𝑧)−𝛾^′_𝑡. (2.10)

Proof (ii). Consider 𝐸𝐸𝑓𝑋_𝑡+Δ𝑡∣𝑋_𝑡0=𝐸^∞_−∞𝐸𝑓𝑋_𝑡+Δ𝑡∣𝑋_𝑡=𝑧𝑝_𝑡𝑧∣𝑋_𝑡0=𝑑𝑧_∞−∞𝐸𝑓𝑋_𝑡+Δ𝑡∣𝑋_𝑡=𝑧𝑝_𝑡𝑧∣𝑋_𝑡0=𝑦𝑝_𝑡0𝑦𝑑𝑧𝑑𝑦, (2.11) provided that 𝐸[𝑓(𝑋_𝑡+Δ𝑡)]<∞. By Fubini’s theorem, interchanging the integrals gives 𝐸𝐸𝑓𝑋_𝑡+Δ𝑡∣𝑋_𝑡0=^∞_−∞𝐸𝑓𝑋_𝑡+Δ𝑡∣𝑋_𝑡=𝑧^∞_−∞𝑝_𝑡𝑧∣𝑋_𝑡0=𝑦p_t0𝑦=𝑑𝑦𝑑𝑧_∞−∞𝐸𝑓𝑋_𝑡+Δ𝑡∣𝑋_𝑡=𝑧𝐸𝑝_𝑡𝑧∣𝑋_𝑡0𝑑𝑧. (2.12) Now, using Taylor’s expansion around 𝑋_𝑡=𝑧 and then Itô’s formula, we reach the same forward Kolmogorov equation, that is, 𝑑𝐸𝐸𝑓𝑋𝑑𝑡^𝑡∣𝑋_𝑡0=^∞_−∞𝑓^′(𝑧)𝑎(𝑧)+𝑓^′′(𝑧)2𝜎²(𝑧)𝐸𝑝_𝑡𝑧∣𝑋_𝑡0𝑑𝑧, (2.13) where 𝐸𝐸𝑓𝑋_𝑡∣𝑋_𝑡0=𝐸^∞_−∞𝑓(𝑧)𝑝_𝑡𝑧∣𝑋_𝑡0=𝑑𝑧_∞−∞𝑓(𝑧)𝑝_𝑡𝑧∣𝑋_𝑡0=𝑦𝑝_𝑡0𝑦=𝑑𝑧𝑑𝑦_∞−∞𝑓(𝑧)^∞_−∞𝑝_𝑡𝑧∣𝑋_𝑡0=𝑦𝑝_𝑡0𝑦=𝑑𝑦𝑑𝑧_∞−∞𝑓(𝑧)𝐸𝑝_𝑡𝑧∣𝑋_𝑡0𝑑𝑧, (2.14) and therefore, 𝑑𝑑𝑡^∞_−∞𝑓(𝑧)𝐸𝑝_𝑡𝑧∣𝑋_𝑡0𝑑𝑧=^∞_−∞𝑓^′(𝑧)𝑎(𝑧)+𝑓^′′(𝑧)2𝜎²(𝑧)𝐸𝑝_𝑡𝑧∣𝑋_𝑡0𝑑𝑧, (2.15) under the assumption 𝐸𝑝_𝑡𝑧∣𝑋_𝑡0=12𝜋𝛾_𝑡𝑒^{−(𝑧−𝜇𝑡)2/2𝛾𝑡}. (2.16) The remaining proof follows the same arguments as (i).

Theorem 2.2. Let the non-homogeneous diffusion process 𝑋={𝑋_𝑡,𝑡≥0} satisfy the stochastic differential equation (1.3), 𝑎(𝑋_𝑡,𝑡)∈𝐶₀(ℝ), 𝜎(𝑋_𝑡)∈𝐶₀^2,1(ℝ), and both 𝑎(𝑋_𝑡,𝑡) and 𝜎(𝑋_𝑡,𝑡) satisfy the linear-growth and local Lipschitz conditions. Assuming (𝑋_𝑡=𝑧∣𝑋_𝑡0=𝑦)∼𝑁(𝑦+𝜇_𝑡,𝛾_𝑡) where 𝑡₀>0 and 𝑡>𝑡₀, 𝜇_𝑡∈𝐶₀(ℝ) and 𝛾_𝑡∈𝐶₀(ℝ), then the necessary equation on drift and diffusion coefficients by conditioning on 𝑋_𝑡=𝑧 is given by 𝑎(𝑧,𝑡)=𝜇^′_𝑡+𝜎²(𝑧,𝑡)^′_𝑧2+𝑧−𝑦−𝜇_𝑡𝛾_𝑡^′−𝜎²(𝑧,𝑡)2𝛾_𝑡. (2.17)

Proof. Consider (1.4) and treat it in an analogue manner as proof of Theorem 2.1 (i). We have omitted further details.

2.2. Characterization of Possible Diffusion Processes

We characterize the possible diffusion processes using (2.1). The following result explains possible solutions of (2.1).

Theorem 2.3. Suppose that the mean parameter of marginal density satisfies 𝜇_𝑡=0, then under the assumption that marginals are Gaussian, (2.1) characterizes that the only possible solution exists when it is a diffusion coefficient, that is, 𝜎²(𝑧)=𝐾 or variance parameter of marginal density, that is, 𝛾_𝑡=𝛾 or 𝜎²(𝑧)=𝐾 and drift coefficient, that is, 𝑎(𝑧)=1/(𝑧−𝑦).

Proof. Equation (2.1) can be written as (𝜎²(𝑧))^′_𝑧−2𝑎(𝑧)2+𝛾𝑧−𝑦2𝛾^𝑡′_𝑡−𝜎²(𝑧)2𝛾_𝑡=0. (2.18) Define 𝐺(𝑧)𝜎∶=²(𝑧)^′_𝑧−2𝑎(𝑧)2(𝑡)𝛾𝑧−𝑦,𝜉∶=^𝑡′2𝛾_𝑡(𝑡)1,𝐾∶=2𝛾_𝑡,𝐻(𝑧)∶=𝜎²(𝑧), (2.19) which yields 𝐺(𝑧)+𝜉(𝑡)−𝐾(𝑡)𝐻(𝑧)=0. (2.20)Case 1. Differentiating (2.20) with respect to 𝑡 gives 𝜉^′(𝑡)−𝐾^′(𝑡)𝐻(𝑧)=0. Clearly, 𝜉^′(𝑡) and 𝐾^′(𝑡) are not equal to zero. Hence, 𝜉^′(𝑡)/𝐾^′(𝑡)=𝐻(𝑧) which indicates that 𝐻(𝑧)=𝜎²(𝑧)=𝐾, where 𝐾 is any arbitrary constant.Case 2. Differentiating (2.20) with respect to 𝑧 gives 𝐺^′(𝑧)−𝐾(𝑡)𝐻^′(𝑧)=0.(i)If 𝐻(𝑧)=𝜎²(𝑧)=𝐾, then 𝐻^′(𝑧)=0. Clearly 𝐺^′(𝑧) is equal to zero if 𝑎(𝑧)=1/(𝑧−𝑦).(ii)If 𝐻^′(𝑧) and 𝐺^′(𝑧) are not equal to zero, then 𝐺^′(𝑧)/𝐻^′(𝑧)=𝐾(𝑡). This implies that 𝐾(𝑡)=𝐾, which yields 𝛾_𝑡=𝛾, where 𝛾 is any arbitrary constant which does not depend on 𝑡.

Example 2.4. Equation (2.1) characterizes the martingale case, that is, when drift coefficient 𝑎(𝑧)=0, the process is given by 𝑋_𝑡=𝑋_𝑡0+^𝑡_𝑡0𝐾𝑑𝐵_𝑠, (2.21) where 𝐾 is any arbitrary constant.

Proof. Assuming 𝑎(𝑧)=0 in (2.7) gives 𝑥22𝜋𝛾_𝑡^∞_−∞𝜎²(𝑧)−𝛾^′_𝑡𝑒^𝑖𝑧𝑥𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}𝑑𝑧=𝜇^′_𝑡−𝑖2𝜋𝛾_𝑡^∞_−∞𝑒^𝑖𝑧𝑥𝑒^{−(𝑧−𝑦−𝜇𝑡)2/2𝛾𝑡}𝑑𝑧. (2.22) Clearly 𝑥 is arbitrary, which implies that 𝜇^′_𝑡=0,or𝜇_𝑡=𝐾. (2.23) Hence, from (2.1), we obtain (𝜎²(𝑧))^′_𝑧2+𝛾𝑧−𝑦−𝐾2𝛾^𝑡′_𝑡−𝜎²(𝑧)2𝛾_𝑡=0. (2.24) The remaining arguments follow from proof of Theorem 2.3 of Case 1.

Example 2.5. Suppose that variance parameter of the marginal density 𝛾_𝑡 has two values, mean ofmarginal density 𝜇_𝑡=0, starting position of diffusion process 𝑋_𝑡0=𝑦 and diffusion coefficient 𝜎²(𝑧)=𝐾, then (2.1) characterizes the Vasicek model given by 𝑋_𝑡𝐾=𝑦+2^𝑡_𝑡0𝑋_𝑠−𝑦𝑑𝑠+𝐾^𝑡_𝑡0𝑑𝐵_𝑠. (2.25)

Proof. Equation (2.1) can be written as 2𝑎(𝑧)−𝜎(𝑧)𝜎^′(𝑧)𝛾𝑧−𝑦^𝑡=𝛾^′_𝑡−𝜎²(𝑧). (2.26) Define 𝐿(𝑧)2𝑎(𝑧)(𝑧)𝜎∶=−𝜎^′(𝑧),𝑧−𝑦 (2.27) and assume two different values of 𝛾_𝑡, that is, 𝛾_𝑡=𝑎_𝑡 and 𝛾_𝑡=𝑏_𝑡, which gives 𝐿(𝑧)𝑎_𝑡+𝜎²(𝑧)=𝑎^′_𝑡, (2.28)𝐿(𝑧)𝑏_𝑡+𝜎²(𝑧)=𝑏^′_𝑡. (2.29) Subtracting (2.28) from (2.29), we obtain 𝐿(𝑧)=𝑏^′_𝑡−𝑎^′_𝑡𝑏_𝑡−𝑎_𝑡=𝐾whichdoesnotdependon𝑧. (2.30) Therefore, 𝜎²(𝑧)=𝛾^′_𝑡−𝐾𝛾_𝑡=𝐶thus(𝜎²(𝑧))^′_𝑧=0. (2.31) By substituting values in (2.27), we obtain 𝑎(𝑧)=𝐾𝑧−𝑦2when𝜎²(𝑧)=𝐶. (2.32) Hence, 𝑋_𝑡𝐾=𝑦+2^𝑡_𝑡0𝑋_𝑠−𝑦𝑑𝑠+𝐾^𝑡_𝑡0𝑑𝐵_𝑠. (2.33)

Remark 2.6. The variance of the marginal density, that is, 𝛾_𝑡, can have at most two different values.

3. Conclusion

We have proved the necessary characteristic equation for homogeneous and non-homogeneous one-dimensional diffusion processes, under the assumption that marginal density of the process is Gaussian. The possible diffusion processes that can exist are also characterized. Two well-known examples using our characteristic equation are given. However, we are still working on sufficient condition of the characteristic equation and about the construction of a new diffusion process whose diffusion coefficients do not satisfy usual Lipschitz and linear-growth conditions, but still it exists.

Acknowledgments

The author would like to thank the referee for his valuable comments and Dr. Sergey Utev for his invaluable guidance and support. Thanks are also due to Higher Education Commission of Pakistan and Lahore School of Economics, Pakistan for funding this research.