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Results tagged with stochastic-processes
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user 477203
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
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How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $H=(H_t, t\ge 0)$ be a stochastic process with continuous trajectories. Fix $T>0$. For $n \ge 1$, we define
$$
H_{s,n} := \sum_{i=1}^{ …
- 657
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Interchange the deterministic and stochastic integrals
We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \in …
- 657
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Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = …
- 657