Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises
Abstract
:1. Introduction
- (A) , , , (models driven by a fractional Brownian motion);
- (B) , , , (time-fractional models driven by the standard Brownian motion);
- (C) , , , , (combinations of (A) and (B));
- (D) , , , , , (a multi-time scale model).
2. Preliminaries
- is the matrix norm on .
- is the indicator of the set A.
- X is the separable normed space with the norm .
- For any separable metric space S, the set of all (equivalence classes of) measurable functions (also referred to as random points in S) is again a metric space equipped with the topology of convergence in probability; note that if X is a separable normed space, then becomes a linear metric, but not locally convex, space; to simplify the notation, we always disregard the difference between equivalence classes and their representatives considering the latter as elements of the space as well.
- Bounded subsets can be described as follows: for any there is a ball such that
- is the space of all n-dimensional functions that are continuous on .
- () is the space of n-dimensional functions that are p-integrable on with respect to the Lebesgue measure.
- contains all -adapted n-dimensional stochastic processes on , whose trajectories almost surely belong to the space .
- contains all -adapted n-dimensional stochastic processes on , whose trajectories almost surely belong to the space .
- In case is replaced by another stochastic basis , we will write and instead of and , respectively, so that and
3. Basic Properties of the Integral Operators with Singular Kernels
- 1.
- The operator is compact as an operator from to .
- 2.
- If , then the operator maps to for any .
- 3.
- If , then the operator maps to if and only if .
- 1.
- h is uniformly continuous on any tight subset of .
- 2.
- h maps bounded subsets of (resp., the entire ) to tight subsets of .
- 1.
- The operator is tight as an operator from to .
- 2.
- If , then the operator maps to for any .
- 3.
- If and , then the operator maps to .
4. Main Results
4.1. Existence of Weak and Strong Solutions
- (A1) are the standard scalar Wiener processes, not necessarily independent, defined for on the stochastic basis (2).
- (A2) For each , if and if , if and if .
- (A3) , ( are random continuous Volterra operators (), which for any are -adapted stochastic processes on .
- 1.
- is defined by , where equipped with the product topology, and are all copies of either the space or the space and ν may be any natural value or ∞;
- 2.
- The marginal measure is the limit point (in the narrow topology) of a sequence of random Dirac measures , where ();
- 3.
- and are the -completions of the σ-algebras and ( (), respectively, where are copies of either the space or the space
- 1.
- If an operator is local and tight-ranged, then it has at least one weak fixed point in the sense of Definition 5.
- 2.
- Assume that an operator is local and uniformly continuous on tight subsets of . If H has at most one weak fixed point for any acceptable expansion of the stochastic basis , then any weak point of H will be equivalent to a strong fixed point of H, i.e., the one belonging to .
- 1.
- 2.
- 3.
4.2. Ordinary Fractional Stochastic Differential Equations
- (B1) The functions and are -measurable in for all and -adapted in ω for any and , continuous in for -almost all and for all
4.3. Fractional Stochastic Differential Equation with Random Delays
- (B2) The random delays , () are -measurable stopping times for all , , .
4.4. Fractional Stochastic Neutral Differential Equation
- (B3) , if and and if . ().
- (B4) The functions and are -measurable in for all , -adapted in ω for any and , continuous in for -almost all and for all and
- (B5) The random delays are -measurable stopping times a.s. satisfying , for all , , .
- (B6) are scalar Borel-measurable functions defined on and (, ) is a continuous function with the property
5. Conclusions
- 1.
- Time-fractional stochastic differential equations driven by the fractional Levy noise, e.g., the fractional Poisson-like noise.
- 2.
- The existence of strong solutions for fSDEs by applying Okinawa’s uniqueness theorem or its generalizations.
- 3.
- fSDEs with other types of fractional differentials, e.g., those with the time-dependent Hurst parameters.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
fDE | Fractional differential equations |
fSDE | Fractional stochastic differential equations |
fBm | Fractional Brownian motion |
RLfBm | Riemann–Liouville process, i.e., fractional Brownian motion of the Riemann–Liouville type |
fLm | Fractional motion based on the Lévy process |
e.g. | exempli gratia (for example) |
i.e. | id est (that is) |
Appendix A. Fixed-Point Theorem and Main Properties of Local and Tight Operators
- 1.
- A finite linear combination of local operators is local.
- 2.
- A composition of local operators is local.
- 1.
- The operator is local and uniformly continuous on tight subsets of ; in particular, is continuous.
- 2.
- The operator maps tight subsets of to tight subsets of .
- 3.
- If, in addition, is a.s. compact (resp., compact-ranged), then the operator is tight (resp., tight-ranged).
- 1.
- For any , the superposition operators are local and tight as operators acting in the space .
- 2.
- The operators are local and tight for any .
- 1.
- If is the superposition operator generated by a random Carathéodory map , and X and Y are separable metric spaces, then the unique local and continuous extension of to the space of all random points is given by .
- 2.
- If is an acceptable expansion, and , then the stochastic operators admit the unique local and continuous extension to the space given bywhere is the standard scalar Wiener process on the stochastic basis .
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Ponosov, A.; Idels, L. Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises. Mathematics 2025, 13, 204. https://doi.org/10.3390/math13020204
Ponosov A, Idels L. Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises. Mathematics. 2025; 13(2):204. https://doi.org/10.3390/math13020204
Chicago/Turabian StylePonosov, Arcady, and Lev Idels. 2025. "Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises" Mathematics 13, no. 2: 204. https://doi.org/10.3390/math13020204
APA StylePonosov, A., & Idels, L. (2025). Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises. Mathematics, 13(2), 204. https://doi.org/10.3390/math13020204