Abstract
In this chapter, we study diffusion processes at the level of paths. In particular, we study stochastic differential equations (SDEs) driven by Gaussian white noise, defined formally as the derivative of Brownian motion. In Sect. 3.1, we introduce SDEs. In Sect. 3.2, we introduce the Itô and Stratonovich stochastic integrals. In Sect. 3.3, we present the concept of a solution to an SDE. The generator, Itô’s formula, and the connection with the Fokker–Planck equation are covered in Sect. 3.4. Examples of SDEs are presented in Sect. 3.5. The Lamperti transformation and Girsanov’s theorem are discussed briefly in Sect. 3.6. Linear SDEs are studied in Sect. 3.7. Bibliographical remarks and exercises can be found in Sects. 3.8 and 3.9, respectively.
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Notes
- 1.
In fact, it is a C 1+α function of time, with α < 1∕2.
- 2.
This is the case, for example, when the SDE has smooth coefficients and the diffusion matrix Σ = σσT is strictly positive definite.
- 3.
In Sect. 4.2, we will redo this calculation using the Fokker–Planck equation.
- 4.
We can also consider the case that σ ∈ R d×m with n ≠ m, i.e., that the SDE is driven by an m-dimensional Brownian motion.
- 5.
More precisely, let \(\{\mathcal{F}_{t}\}\) be a filtration. A function τ: Ω ↦ [0, +∞] is a called a (strict) stopping time with respect to \(\{\mathcal{F}_{t}\}\) if
$$\displaystyle{\big\{\omega \,;\,\tau (\omega )\leqslant t\big\} \in \mathcal{F}_{t}\quad \mbox{ for all}\;t\geqslant 0.}$$ - 6.
Two probability measures P 1, P 2 are equivalent on a σ-field \(\mathcal{G}\) if and only if \(P_{1}(A) = 0\Longleftrightarrow P_{2}(A) = 0\) for all \(A \in \mathcal{G}\). In this case \(\frac{dP_{2}} {dP_{1}}\) and \(\frac{dP_{1}} {dP_{2}}\) exist.
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Pavliotis, G.A. (2014). Introduction to Stochastic Differential Equations. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_3
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