A new type of the Gronwall-Bellman inequality and its application to fractional stochastic differential equations

This paper presents a new type of Gronwall-Bellman inequality, which arises from a class of integral equations with a mixture of nonsingular and singular integrals. The new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order on (0, 1). This result generalizes the existence and uniqueness theorem related to fractional order (1/2 1) appearing in [1]. Finally, the fractional type Fokker-Planck-Kolmogorov equation associated to the solution of the fractional SDE is derived using It^o's formula.

ABOUT THE AUTHOR Qiong Wu received BS and MS in Mathematics from Harbin Institute of Technology, China. He is a full-time PhD student in Department of Mathematics, Tufts University in USA. His area of interest includes the theory of stochastic differential equations and their applications, mathematical biology, control theory and convex optimization.

PUBLIC INTEREST STATEMENT
Gronwall-Bellman inequality plays a significant role in mathematical modeling, particularly in applications of integral equations. For a mathematical model which arises from a class of integral equations with a mixture of nonsingular and singular integrals, there is lack of a powerful Gronwall-Bellman inequality to help researchers on this case. To derive such a Gronwall-Bellman inequality, the new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order 0 < < 1.
This result generalizes the known existence and uniqueness theorem related to fractional order 1 2 < < 1.

Introduction
It is well known that integral inequalities are instrumental in studying the qualitative analysis of solutions to differential and integral equations (Ames & Pachpatte, 1997). Among these inequalities, the distinguished Gronwall-Bellman type inequality from Bellman and Cooke (1963), and its associated extensions (Agarwal & Choi, 2016;Agarwal, Deng, & Zhang, 2005;Agarwal, Tariboon, & Ntouyas, 2016;Lipovan, 2000;Liu, Zhang, Agarwal, & Wang, 2016;Mao, 1989;Pachpatte, 1975;Wang, Agarwal, & Chand, 2014), are capable of affording explicit bounds on solutions of a class of linear differential equations with integer order. The following lemma concerns a standard Gronwall-Bellman inequality in Corduneanu (2008) for a differential equation with order one or equivalently an integral equation with nonsingular integrals.
In order to investigate the qualitative properties of solutions to differential equations of fractional order, there are several generalizations of Gronwall-Bellman inequalities developed by many researchers (Atıcı & Eloe, 2012;Lazarević & Spasić, 2009;Ye, Gao, & Ding, 2007;Zheng, 2013). Let us recall the following generalized Gronwall-Bellman inequality proposed in Ye et al. (2007) for a fractional differential equation with order > 0 or equivalently an integral equation with singular integrals.
Lemma 1.2 Suppose > 0, a(t) is a nonnegative function which is locally integrable on 0 ≤ t < T, 0 < T ≤ ∞, and g(t) is a nonnegative, nondecreasing continuous function defined on From many real applications, such as in physics, theoretical biology, and mathematical finance, there is substantial interest in a class of fractional SDEs (Jumarie, 2005a;Mandelbrot & Van Ness, 1968;Pedjeu & Ladde, 2012). The fractional SDEs take the form

Generalization of the Gronwall-Bellman inequality
In this section, we develop a new integral inequality, Equation (4) below, by verifying three claims. The first claim is established by using the method of induction and taking advantage of the binomial function; the second claim is verified by taking advantage of properties of the Gamma function; the third claim is verified by employing Gamma functions, Mittag-Leffler functions, and exponential functions. The established integral inequality is applicable to the fractional SDE (Equation (1)) or the stochastic integral equation (Equation (2)). Also this new integral inequality can be considered as a generalization of the integral inequalities in Lemmas 1.1 and 1.2.

This implies
In order to get the desired inequality, Equation (4), from Equation (5), there are three claims to be verified.
The first claim provides a general bound on B n (t): The method of induction will be used to verify the inequality in Equation (6). First let n = 1. Then the inequality in Equation (6) is true. Now, suppose that the inequality, Equation (6), holds for n = k, and then compute B n when n = k + 1,

Let and
Then, compute C(t) and G(t) term by term to reach the desired inequality (Equation (6)). Since b(t) and g(t) are nonnegative and nondecreasing functions, Combining Equations (7) and (8) yield This implies that for any n ∈ ℕ + , the first claim, Equation (6), holds.
The second claim shows that B n u(t) vanishes as n increases. For each t in [0, T), For the purpose of notation simplification during the proof of the second claim, define Note that Γ(x) is positive and decreasing on (0, 1] but positive and increasing on [2, ∞). Let This means x min i = n and x max i = n. Furthermore, for a fixed , there exists a large enough n 0 such that for any n > n 0 , there is n ≥ 2 . So the sequence satisfies x i ≥ 2 for any integer i ∈ [0, n] if n is large enough. Thus, Also for ∈ (0, 1), Γ( ) > 1. Therefore, Notice that b(t) and g(t) are both bounded by a positive constant M, i.e. b(t) ≤ M and g(t) ≤ M, and u(s) is locally integrable over 0 ≤ t < T. This means that from Equation (10), H n (t) → 0 as n → ∞ because the Gamma function, Γ(n ), is growing faster than a power function. Therefore, the second claim, Equation (9), is verified since B n u(t) ≤ H n (t) for any n ∈ ℕ + .
The third claim establishes that the right-hand side (RHS) of Equation (4) exists on 0 ≤ t < T. In order to show this statement, we first prove that for 0 ≤ t < T, the following infinite sum of sequences denoted by L(t; ) is convergent.
where (i + n − i) ⋯ (i + 1) is a product and it takes one if (i + n − i) < i + 1. Let k = n − i, then compute Substituting k = n − i and Equation (12) into Equation (11) gives which is finite for 0 ≤ t < T.
Since the Mittag-Leffler function E (t ) is an entire function in t , see Gorenflo, Loutchko, Luchko, and Mainardi (2002), the exponential function exp(t) is uniformly continuous in t, and both t −1 and a(t) are locally integrable over 0 ≤ t < T, the integral ∫

Existence and uniqueness of the solution to fractional SDEs
In this section, using the main results from Section 2, we investigate the existence and uniqueness of the solution to the fractional SDE (Equation (1)) with fractional order 0 < < 1. By application of the classical Picard-Lindelöf successive approximation scheme and the standard Gronwall-Bellman inequality, existence and uniqueness of the solution to Equation (1) with fractional order 1 2 < < 1 is discussed in Pedjeu and Ladde (2012). However, the case with 0 < ≤ 1 2 remains to be investigated. We can apply the generalized Gronwall-Bellman inequality developed in Section 2 to derive existence and uniqueness of the solution to Equation (1) when 0 < < 1.
for some constant K > 0 and the Lipschitz condition, for some constant L > 0. Let x 0 be a random variable, which is independent of the -algebra  t ⊂  ∞ generated by {B t , t ≥ 0} and satisfies |x 0 | 2 < ∞. Then, the fractional stochastic differential equation (Equation (1)) has a unique t-continuous solution x(t, ) with the property that x(t, ) is adapted to the filtration  Proof (Existence) From Equation (2), the corresponding equivalent stochastic integral equation of the fractional stochastic differential equation (Equation (1)) is rewritten as where 0 ≤ t < T and 0 < < 1. For more details about this equivalence between Equation (1) and Equation (2), we refer to Jumarie (2005aJumarie ( , 2005bJumarie ( , 2006. By the method of Picard-Lindelöf successive approximations, define x 0 (t) = x 0 and x k (t) = x k (t, ) inductively as follows Applying the inequality |x + y + z| 2 ≤ 3|x| 2 + 3|y| 2 + 3|z| 2 leads to Using the Cauchy-Schwartz inequality on the first two terms, I 1 and I 2 , plus Itô's Isometry, see in Oksendal (2013), in the third term, I 3 , produces Finally, using the Lipschitz condition (Equation (14)) on all terms, J 1 , J 2 , J 3 , evaluating the first integral in the second term, J 2 , and combining the first and third terms, J 1 and J 3 , yields Thus, for locally integrable function (t), define an operator B as follows Then, iterating Equation (16) yields Since 0 < < 1 and |x 1 (t) − x 0 (t)| 2 is nonnegative and locally integrable, from the first claim, Equation (6), and the Equation (10) in the proof of the second claim in Section 2, we know that Similarly, apply the Cauchy-Schwartz inequality, the Itô's Isometry, and the linear growth condition, Equation (13), instead of Lipschitz condition, Equation (14), to compute This implies where M 0 = 3(1 + T)K 2 (1 + |x 0 | 2 ) T 2 2 + T +1 +1 is independent of k and t. Thus, for any m > n > 0, where M 1 = 3(1 + T)K 2 (1 + |x 0 | 2 ) T 3 2 + T +2 +1 is independent of k and t. This means the successive approximations (x k (t)) are mean-square convergent uniformly on [0, T]. It remains now to show that the sequence of successive approximations (x k (t)) is almost surely convergent. First, apply Chebyshev's inequality to yield By computations similar to those leading to Equation (17) and Doob's Maximal Inequality for martingales, which is finite. Then, applying the Borel-Cantelli lemma yields, So there exists a random variable x(t), which is the limit of the following sequence uniformly on [0, T]. Also x(t) is t-continuous since x k (t) is t-continuous for all k. Therefore, taking the limit on both sides of Equation (15) as k → ∞, there is a stochastic process x(t) satisfying Equation (2).
(Uniqueness) The uniqueness is due to the Itô Isometry and the Lipschitz condition, Equation (14). Let x 1 (t) = x 1 (t, ) and x 2 (t) = x 2 (t, ) be solutions of Equation (2), which have the initial values, x 1 (0) = y 1 and x 2 (0) = y 2 , respectively. Similarly, apply the Cauchy-Schwartz inequality, the Itô Isometry, and the Lipschitz condition (Equation (14)) to compute By application of the generalized Gronwall-Bellman inequality in Corollary 2.1, we have Since x 1 (t) and x 2 (t) both satisfy the stochastic integral equation (Equation (2)), the initial values y 1 and y 2 are both equal to x 0 . This means |x 1 (t) − x 2 (t)| 2 = 0 for all t > 0. Furthermore, Therefore, the uniqueness of the solution to Equation (2) is proved. ✷

Fractional Fokker-Planck-Kolmogorov equation
Based on the existence and uniqueness Theorem 3.1 developed in Section 3, we derive the fractional Fokker-Planck-Kolmogorov equation associated to the unique solution of the fractional SDE, Equation (1). Before deriving the fractional Fokker-Planck-Kolmogorov equation, we first introduce an Itô formula from Pedjeu and Ladde (2012) to the following Itô process where 0 < < 1, B t is the m-dimensional standard Brownian motion, and functions b, 1 , 2 satisfy the conditions in Theorem 3.1.
Lemma 4.1 Let X(t) satisfy the Equation (18) and furthermore, let V ∈ C[R + × R n , R m ], and assume that V t , V x , V xx exist and continuous for (t, x) ∈ R + × R n , where V x (t, x) is an m × n Jacobian matrix of V(t, x) and V xx (t, x) is an m × n Hessian matrix. Then,  (1)) whose coefficient functions b, 1 and 2 satisfy the conditions in Theorem 3.1. Then the transition probabilities P X (t, x) = P X (t, x|0, x 0 ) of X(t) satisfy the following fractional type differential equation with initial condition P X (0, x) = x 0 (x), the Dirac delta function with mass on x 0 , and A * x , B * x are spatial operators defined, respectively, by and where b = (b 1 , ⋯ , b n ) T , 1 = ( 1 1 , ⋯ , n 1 ) T , and 2 is an n × m matrix with elements [ 2 ] ij = ij 2 .