Abstract
Let \((W_{t})_{t\in \mathbb{R}_{+}}\) be a Wiener process on the probability space \((\varOmega,\mathcal{F},P)\), equipped with its natural filtration \((\mathcal{F}_{t})_{t\in \mathbb{R}_{+}}\), \(\mathcal{F}_{t} =\sigma (W_{s},0 \leq s \leq t)\). Furthermore, let a(t, x), b(t, x) be deterministic measurable functions in \([t_{0},T] \times \mathbb{R}\) for some \(t_{0} \in \mathbb{R}_{+}.\) Finally, consider a real-valued random variable u 0; we will denote by \(\mathcal{F}_{u^{0}}\) the σ-algebra generated by u 0, and we assume that \(\mathcal{F}_{u^{0}}\) is independent of \((\mathcal{F}_{t})\) for t ∈ (t 0, +∞). We will denote by \(\mathcal{F}_{u^{0},t}\) the σ-algebra generated by the union of \(\mathcal{F}_{u^{0}}\) and \(\mathcal{F}_{t}\) for t ∈ (t 0, +∞).
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Notes
- 1.
It suffices to make use of the following theorem for \(E\left [{\left (\int _{0}^{t}b_{N}(s,u_{N}(s))dW_{s}\right )}^{2n}\right ]\):Theorem. If \({f}^{n} \in \mathcal{C}([0,T])\) for \(n \in {\mathbb{N}}^{{\ast}}\) , then
$$\displaystyle{E\left [{\left (\int _{0}^{T}f(t)dW_{t}\right )}^{2n}\right ] \leq {[n(2n - 1)]}^{n}{T}^{n-1}E\left [\int _{ 0}^{T}{f}^{2n}(t)dt\right ].}$$Proof.
See, e.g., Friedman (1975).
- 2.
The assumption \(\vert f(z)\vert \leq K(1 +\vert z\vert ^{2n})\) implies that \(E[\vert f(z)\vert ] \leq K(1 + E[\vert z\vert ^{2n}])\) and, by Theorem 4.15, \(E[\vert u(t + h,t,x)\vert ^{2n}] < +\infty \). Therefore, f(u(t + h, t, x) − x) is integrable.
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Capasso, V., Bakstein, D. (2015). Stochastic Differential Equations. In: An Introduction to Continuous-Time Stochastic Processes. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-2757-9_4
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