Abstract
The subject of this article is an analytic approximate method for a class of stochastic differential equations with time-dependent delay, with coefficients that do not necessarily satisfy the Lipschitz condition nor the linear growth condition but they satisfy the polynomial condition. More precisely, it is assumed that equations from the observed class have unique solutions with bounded moments, their coefficients satisfy the polynomial condition and some derivatives of the specific order of the drift and diffusion coefficients are uniformly bounded. Approximate equations are defined on partitions of the time interval and their coefficients are Taylor approximations of the coefficients of the initial equation up to arbitrary derivatives. It is shown that the solutions of the approximate equations converge in the sense and almost surely toward the solution of the initial equation and the rate of the convergence is given.
Appendix
Let us assume that the Borel measurable delay function is Lipschitz continuous. In other words, let there be a positive constant L such that (41) (41)
This implies that function is also Lipschitz continuous with the Lipschitz constant
The main reason for employing more restrictive assumption on δ is to determine the points in which the derivatives of the coefficients of EquationEquation (8)(8) (8) with respect to the second argument will be defined and which will be fixed on each partitioning interval. These points are of the form where represents the greatest integer function. For simpler notation, let us introduce a new function defined as
This time, EquationEquation (13)(13) (13) have following form (42) (42)
Instead of Proposition 1 we have following assertions.
Proposition 2.
I Let , be the solutions of EquationEquation (42)(42) (42) and let the assumptions be satisfied. Then, for every where represents a positive generic constant independent of n.
II Under the assumptions from part I, assumption and Lipschitz condition (41), for every where stands for a positive generic constant independent of n.
Proof.
The proof of the part I is analogous to the proof of part I from Proposition 1. The only difference occurs in the coefficients of EquationEquation (13)(13) (13) which contain instead of
II This part is proved completely differently from the part II of Proposition 1. Let and be arbitrary numbers. Since our goal is to estimate the term we consider the difference (43) (43)
The triangle inequality, Lipschitz continuity Equation(41)(32) (32) and imply (44) (44)
Thus, (43) and (44) give (45) (45)
Consequently, there are no more than division points of the partition Equation(9)(32) (32) –Equation(10)(32) (32) between and (excluding the if it is one of those division points) and the number of such points N does not depend of n nor This is very convenient since n (and therefore ) is going to tend to
Let us first consider the case when For the following estimation it is of our interest to note the index Then, bearing in mind Equation(45)(32) (32) and the inequality Equation(2)(32) (32) , we compute (46) (46)
If in the previous sum, then provides and if the part I yields (48) (48)
Hence, Equation(47)(32) (32) and Equation(48)(32) (32) substituted into Equation(46)(32) (32) imply (49) (49) where is a constant independent of n. This estimate holds whether is the division point of Equation(9)(9) (9) –Equation(10)(10) (10) or not.
Let us note that the estimate Equation(49)(32) (32) is obtained similarly in the case when and since is arbitrary, the proof is complete.□
Analogously to Theorems 3.1 and 3.2 we state following theorems.
Theorem 5.1.
Let all of the assumptions from Proposition 2 hold and let x be the solution of EquationEquation (8)(8) (8) with initial condition Equation(7)(32) (32) and let be the solution to EquationEquation (42)(42) (42) . Then where and C is a generic constant independent of n.
Theorem 5.2.
Let the assumptions from Theorem 5.1 hold. Then the sequence of the solutions to EquationEquation (42)(42) (42) converges almost surely to the solution x of EquationEquation (8)(8) (8) .
The proofs of these theorems are omitted since they are not different from the ones in previous section.
Notes
1 This is a corollary of the Hölder inequality for r > 1, in a way that
since is an increasing function, Case is trivial and the case when follows from