On the Accuracy of One Formula for the Case of a Stochastic Differential Equation with Jumps

Nonlinear Phenomena in Complex Systems, vol. 28, no. 3 (2025), pp. 261 - 268 On the Accuracy of One Formula for the Case of a Stochastic Differential Equation with Jumps Anatoly Zherelo∗ Belarusian State University, 4 Nezalezhnasti Ave., 220030 Minsk, BELARUS (Received 13 May, 2025) Random processes with jumps find application in various fields of research, such as economics. Generally researchers are interested in the values of mathematical expectations from processes of this kind. Previously, the author has proposed a formula for the approximate calculation of mathematical expectations from processes defined by a stochastic differential equation containing a process with jumps. In this paper an accuracy estimate is obtained for the earlier proposed formula. AMS Subject Classification: 60C30, 60H10 Keywords: stochastic process with discontinuities, moments of random process, approximate formula, weak approximation method DOI: https://doi.org/10.5281/zenodo.17235903 1. Introduction Novaday for describing various phenomena observed in the environment and social processes, random processes are increasingly used, including processes with discontinuous trajectories (see, e.g. [1, 2]). In particular, random processes with paths containing jumps are used in queuing theory. The problem of calculating mathematical expectations of various types from such processes naturally arises. One of the most widely used approaches is the approach based on simulation modeling of processes. This approach requires significant computational resources, since a significant number of trajectories must be used to calculate the value of the mathematical expectation. In a number of cases, it is possible to use an approach based not on modeling trajectories, but on an attempt to construct a weak-type approximation, i.e. an approximation that allows one to estimate some parameters of the process, for example, its moments or a distribution function. In the paper [5] such an approximation is proposed for the case of a stochastic differential equation with an Ito integral over a process whose ∗E-mail: [email protected] trajectories contain discontinuities, but the paper only describes the process of constructing an approximate formula. The purpose of this paper is to obtain an assessment of the accuracy of the proposed formula. Let us remind that the object of research in the paper [5] is a random process given in the form of a stochastic differential equation, which is rewritten below in the integral form: Xt=X0+ t Z 0 α(Xs−, s)ds + t Z 0 β(Xs−, s)d˜ Ps.(1.1) Here t∈[0,1], X0∈R,˜ Pt=Pt−λt is the compensated Poisson process, α(Xs−, s)and β(Xs−, s)satisfy the following strong solution conditions [3]: |α(y1, t)−α(y2, t)|2+|β(y1, t)−β(y2, t)|2≤ K1|y1−y2|2,(1.2) |α(y1, t)|2+|β(y1, t)|2≤K2(1 + |y1|2),(1.3) where K1, K2∈Rare constants, y1, y2∈R. 261 262 A. Zherelo The approximate formula proposed in [5] is intended for calculating a functional of the form EG≡E[G(X(·))]. The notation (·)is used to indicate, that Gcan depends on whole trajectory of a solution of the equation (1.1) on [0, t]. The formula under study has the form: EG≈J(G) = 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 G[Yj1,j2(·, u1, u2, u3)]du1du2du3,(1.4) where A1+A2= 1, a1,1=1 2 1−r−A2 A1!, a1,2=1 2 1 + r−A2 A1!, a2,1=1 2 1−r−A1 A2!, a2,2=1 2 1 + r−A1 A2!, B1=1 2π(R) 1 + 1 p1+4π(R)!, B2=1 2π(R) 1−1 p1+4π(R)!, b1=1 21−p1+4π(R), b2=1 21 + p1+4π(R), Yj1,j2(t)≡Yj1,j2(t, u1, u2, u3) = X0 +α X0+αX0+β(X0, u3)ρ(2) j2(u2−, u3), u2ρ(1) j1,2(u1−, u2) +βX0+α(X0, u2)ρ(1) j1,2(u3−, u2), u3ρ(2) j2(u1−, u3), u1!ρ(1) j1,1(t, u1) +α X0+αX0+β(X0, u3)ρ(2) j2(u1−, u3), u1ρ(1) j1,1(u2−, u1) +βX0+α(X0, u1)ρ(1) j1,1(u3−, u1), u3ρ(2) j2(u2−, u3), u2!ρ(1) j1,2(t, u2) +β X0+αX0+α(X0, u2)ρ(1) j1,2(u1−, u2), u1ρ(1) j1,1(u3−, u1) +αX0+α(X0, u1)ρ(1) j1,1(u2−, u1), u2ρ(1) j1,2(u3−, u2), u3!ρ(2) j2(t, u3), and ρ(1) j1,k(s, uk) = aj1,k1[uk,1](s), k = 1,2, ρ(2) j2(s, u3) = bj21[u3,1](s), Нелинейные явления в сложных системах Т. 28, № 3, 2025 On the Accuracy of One Formula for the Case of a Stochastic Differential Equation with Jumps 263 1[uk,1](s) = (1, s ∈[uk,1], 0,overwise. Here and below we suppose, that Ghas a Fr´echet derivative, which we denotes G0, and |G0(x)| ≤ C, (1.5) where C∈Rfor any x∈R. 2. Estimating the accuracy of approximate formula For the convenience of assessing the accuracy of formula (1.4), let us introduce the following notations: ˜ Xt=Xt−X0,˜ Yj1,j2(t) = Yj1,j2(t)−X0 and ˆ Xt= t Z 0 α(X0, s)ds + t Z 0 β(X0, s)d˜ Ps At the first let us prove some useful inequalities. Proposition 1. |E[ˆ Xt]| ≤ pK2(1 + X2 0)t1/2. Proof |E[ˆ Xt]|= t R0 α(X0, s)ds + t R0 β(X0, s)d˜ Ps = | t R0 α(X0, s)ds| ≤ t R0 α2(X0, s)ds1/2 ≤ t R0 K2(1 + X2 0)ds1/2 =pK2(1 + X2 0)t1/2.  Proposition 2. The following inequality is true E[(Xt−X0)2]≤max(t, λ)K2(1 + X2 0)t+o(t). Proof E(Xt−X0)2=E    t Z 0 α(Xs−, s)ds + t Z 0 β(Xs−, s)d˜ Ps  2   =E    t Z 0 α(Xs−, s)ds  2 +  t Z 0 β(Xs−, s)d˜ Ps  2 + 2 t Z 0 α(Xs−, s)ds t Z 0 β(Xs−, s)d˜ Ps   =E    t Z 0 α(Xs−, s)ds  2  +E    t Z 0 β(Xs−, s)d˜ Ps  2  . Using H¨older inequality and properties of stochastic Ito integral, the last expression lower or equal to the following expression: E      t Z 0 α2(Xs−, s)ds  1/2  t Z 0 ds  1/2   2  + t Z 0 E[β2(Xs−, s)]d(λs) Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 264 A. Zherelo ≤max(t, λ) t Z 0 Eα2(Xs−, s) + β2(Xs−, s)ds ≤max(t, λ) t Z 0 E[K2(1 + (Xs−)2)]ds = max(t, λ) t Z 0 K2(1 + E[(Xs−)2])ds = max(t, λ) K2t+K2 t Z 0 E[(X0+˜ Xs−)2]ds  = max(t, λ) K2(1 + X2 0)t+ t Z 0 E[2X0˜ Xs−+ ( ˜ Xs−)2]ds .(2.1) Using assumptions (1.2, 1.3) and make calculations similar to those presented above it is easy to show, that (2.1) is equal to max(t, λ)K2(1 + X2 0)t+o(t).  Proposition 3. The following inequality is true E[( ˜ Xt−ˆ Xt)2]≤1 2(max(t, λ))2K2(1+X2 0)t2+o(t2). Proof E[( ˜ Xt−ˆ Xt)2] = E    t Z 0 α(Xs−, s)ds + t Z 0 β(Xs−, s)d˜ Ps− t Z 0 α(X0, s)ds − t Z 0 β(X0, s)d˜ Ps  2   =E    t Z 0 α(Xs−, s)−α(X0, s)ds + t Z 0 β(Xs−, s)−β(X0, s)d˜ Ps  2   =E  Zα(Xs−, s)−α(X0, s)ds2 +  t Z 0 β(Xs−, s)−β(X0, s)d˜ Ps  2 +2 Zα(Xs−, s)−α(X0, s)ds t Z 0 β(Xs−, s)−β(X0, s)d˜ Ps . Нелинейные явления в сложных системах Т. 28, № 3, 2025 On the Accuracy of One Formula for the Case of a Stochastic Differential Equation with Jumps 265 Using additivity of mathematical expectation and properties of stochastic integral the last expression can be rewritten in the form: E    t Z 0 α(Xs−, s)−α(X0, s)ds  2 +  t Z 0 β(Xs−, s)−β(X0, s)d˜ Ps  2  = (∗). Here, by the H¨older inequality, we get   t Z 0 α(Xs−, s)−α(X0, s)ds  2 ≤t t Z 0 (α(Xs−, s)−α(X0, s))2ds, so (∗)≤E t t Z 0 (α(Xs−, s)−α(X0, s))2ds +E    t Z 0 β(Xs−, s)−β(X0, s)d˜ Ps  2   by the properties of stochastic Ito integral this expression is equal to: t t Z 0 E(α(Xs−, s)−α(X0, s))2ds + t Z 0 E(β(Xs−, s)−β(X0, s))2d(λs) ≤max(t, λ) t Z 0 E(α(Xs−, s)−α(X0, s))2+ (β(Xs−, s)−β(X0, s))2ds according to assumption (1.2) ≤max(t, λ) t Z 0 EK1(Xs−−X0)2ds using proposition (2) ≤max(t, λ)K1 t Z 0 max(s, λ)K2(1 + X2 0)s+o(s)ds ≤1 2(max(t, λ))2K1K2(1 + X2 0)t2+o(t2).  Theorem 1. The following estimate of the error of the formula (1.4) is valid |E[G]−J(G)|= 4 3CP2 j1,j2=1 |Aj1||Bj2|pK2(1 + X2 0)t3/2+o(t3/2). Proof Using the Taylor expansion for functional (see, e.g. [4]) we get |E[G]−J(G)|= E G(X0) + 1 Z 0 t Z 0 G0(X0+τ˜ X(·))˜ Xsdτds  Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 266 A. Zherelo − 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3 G(X0) + 1 Z 0 t Z 0 G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2dτds  . Here we need to note, that A1+A2= 1 and B1+B2= 1, then the last expression has the form:  1 Z 0 t Z 0 EhG0(X0+τ˜ X(·))˜ Xsidτds − 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3  1 Z 0 t Z 0 G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2dτds  = 1 Z 0 t Z 0 EhG0(X0+τ˜ X(·))˜ Xs−G0(X0+τ˜ X(·))ˆ Xs+G0(X0+τ˜ X(·))ˆ Xsidτds − 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3  1 Z 0 t Z 0 G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2dτds  . Here we use the properties of A1, A2, B1, B2again, then the last equality can be rewritten in the form  1 Z 0 t Z 0 dτdsEhG0(X0+τ˜ X(·))( ˜ Xs−ˆ Xs) + 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3G0(X0+τ˜ X(·))˜ Xs−G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2(s)  ≤ 1 Z 0 t Z 0 dτdsEhG0(X0+τ˜ X(·))( ˜ Xs−ˆ Xs)i + 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3G0(X0+τ˜ X(·))˜ Xs−G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2(s) ≤ 1 Z 0 t Z 0EG0(X0+τ˜ X(·))21/2Eh(˜ Xs−ˆ Xs)2i1/2dτds + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3EhG0(X0+τ˜ X(·))ˆ Xs−G0(X0+τ˜ Yj1,j2(·)) ˆ Xs Нелинейные явления в сложных системах Т. 28, № 3, 2025 On the Accuracy of One Formula for the Case of a Stochastic Differential Equation with Jumps 267 +G0(X0+τ˜ Yj1,j2(·)) ˆ Xs−G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2(s)i= (∗∗) Here we use the assumption (1.5) (∗∗)≤C t Z 0Eh(˜ Xs−ˆ Xs)2i1/2ds + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3EhG0(X0+τ˜ X(·)−G0(X0+τ˜ Yj1,j2(·)))ˆ Xsi + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 du1du2du3EhG0(X0+τ˜ X(·))( ˆ Xs−˜ Yj1,j2(s))i . Here we use the proposition 3, then the last expression is lower or equal to the expression: C t Z 0 r1 2(max(s, λ)2)K1K2(1 + X2 0)s+o(s)!ds + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 |Aj1||Bj2| 1 Z 0 1 Z 0 1 Z 0 2CE[ˆ Xs]du1du2du3 + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 G0(X0+τ˜ Yj1,j2(·)) E[ˆ Xs]−˜ Yj1,j2(s)du1du2du3 ≤C 2√2max(t, λ)qK1K2(1 + X2 0)t2+o(t2)+2C 2 X j1,j2=1 |Aj1||Bj2| t Z 0E[ˆ Xs]ds + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 G0(X0+τ˜ Yj1,j2(·))   s Z 0 α(X0, v)dv −˜ Yj1,j2(s) du1du2du3 ≤C 2√2max(t, λ)qK1K2(1 + X2 0)t2+o(t2)+2C 2 X j1,j2=1 |Aj1||Bj2|qK2(1 + X2 0)2 3t3/2+o(t3/2) + 2 X j1,j2=1 |Aj1||Bj2|C t Z 0 s Z 0 α(X0, v)dvds Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 268 A. Zherelo + 1 Z 0 t Z 0 dτds 2 X j1,j2=1 Aj1Bj2 1 Z 0 1 Z 0 1 Z 0 G0(X0+τ˜ Yj1,j2(·)) ˜ Yj1,j2(s)du1du2du3 . Using transformations similar to those given above, we get for the last expression: 4 3C 2 X j1,j2=1 |Aj1||Bj2|qK2(1 + X2 0)t3/2+o(t3/2), so the theorem was proved.  3. Conclusion This paper provides an assessment of the accuracy of the formula (1.4) proposed in [5]. It should be noted that the assessment depends on the coefficients specified by the user. The obtained estimate is quite general and can be improved if additional conditions are specified for the functional to which this formula will be applied. References [1] N. Bruti-Liberati, E. Platen. Approximations of Jump Diffusions in Finance and Economics. (Quantitative Finance Research Center, University of Technology, Sydney, 2006). [2] Y. Wu, X. Liang. Vasicek model with mixedexponential jumps and its applications in finance and insurance. Advances in Difference Equations. 2018, 138 (2018). [3] D. Applebaum. Levy processes and stochastic calculus. (Cambridge University Press, Cambridge, 2009). [4] A.D. Egorov, P.I. Sobolevsky, L.A. Yanovich. Functional Integrals; Approximate Evaluation and Applications. (Kluwer Acad. Publ., Dordreht, 1993). [5] A. Zherelo. On an Approximate Formula for Functionals with Respect to a Solution of Stochastic Differential Equation with a Drift and Random Process with Jumps. Int. J. Nonlinear Phenomena in Complex Systems. 28(2), 137 (2025). Нелинейные явления в сложных системах Т. 28, № 3, 2025