Functional Solutions of Stochastic Differential Equations
Abstract
:1. Introduction
2. Background on Ito’s Lemma and Systems of Partial Differential Equations
- 1.
- holds for all .
- 2.
- Assume (5) holds for all . Let with , and be a piece-wise continuously differentiable simple curve such that . ThenIn particular, for and with one haswhich is obtained by solving successively the ordinary differential equationsalong the horizontal path from to , andalong the vertical path from to .
3. Main Theorems: Solutions by Functionals of Brownian Motion and of Ito Processes
- 1.
- where and is any function of class . Then there exists a unique function of class such that satisfies (1). Moreover, for all , the value may be determined by solving successively the ordinary differential equationsandwith .
- 2.
4. Proofs of the Main Theorems
- 1.
- Assume (2) holds. ThenHence, according to Proposition 1(1), the system of two partial differential equationsIt follows that the function Z is at least of class . By substituting (21) and (22) into (19), we see that Z satisfies the system (17). Then Ito’s Lemma in the form of Theorem 1 implies that the process solves the stochastic differential Equation (1). By Proposition 1(2), at any point with , the value may be determined by solving successively (11) and (12).
- 2.
- 1.
- It follows from (14) thatThen the conditions for the integration of the systemHence,
- 2.
- Assume (1) has a solution of the form , such that is of class , and is given by (3), with F and G of class in time. Then (8) holds for all with . As in the proof of the uniqueness part of Theorem 4 (1), we see that also (25) holds, with f and g defined by (23). Then we differentiate as in (26). We derive that for all with , it holds that
5. Special Cases and Examples
5.1. Autonomous Case
5.2. Linear Stochastic Differential Equations
- 1.
- The solution of the associated homogeneous equationis given by
- 2.
- 1.
- If , the solution of (43) is a function of time and Brownian Motion. We have the following subcases:
- (a)
- . Then .
- (b)
- . Then , i.e., is a Geometrical Brownian Motion.
- (c)
- . Again the solution is a Geometrical Brownian Motion.
- 2.
- If , the solution may be written as a function of time and the Ito processIn fact,In particular, if , the solution is an Ornstein-Uhlenbeck process.
- 3.
- 1.
- Assume that . Then it follows from (46) that . According to Theorem 3, the Equation (43) has a solution that is a function of time and Brownian Motion.If , only the cases (1a) and (1b) are relevant, else (43) is not a stochastic differential equation. In both cases the solution formulas follow in a straightforward way from Theorem 5.If , also . Then we have indeed . Then there exists such that . By the change of variable , one obtains a homogeneous linear stochastic differential equation with constant coefficients. It follows that the solution is a Geometrical Brownian Motion.
- 2.
- 3.
- It holds thatHence, does not only depend on time. According to Theorem 4(2), the Equation (43) cannot have a solution that is a function of time and an Ito process.
5.3. On Nonlinear Stochastic Differential Equations
6. On Path-Dependent Functional Solutions
7. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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van den Berg, I. Functional Solutions of Stochastic Differential Equations. Mathematics 2024, 12, 1258. https://doi.org/10.3390/math12081258
van den Berg I. Functional Solutions of Stochastic Differential Equations. Mathematics. 2024; 12(8):1258. https://doi.org/10.3390/math12081258
Chicago/Turabian Stylevan den Berg, Imme. 2024. "Functional Solutions of Stochastic Differential Equations" Mathematics 12, no. 8: 1258. https://doi.org/10.3390/math12081258
APA Stylevan den Berg, I. (2024). Functional Solutions of Stochastic Differential Equations. Mathematics, 12(8), 1258. https://doi.org/10.3390/math12081258