Questions tagged [stochastic-analysis]
For questions about stochastic analysis or stochastic calculus, for example the Itô integral.
2,349 questions
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Question in Hsu's Stochastic Analysis on Manifolds: Interpreting SDE in Frame Bundle
Let $(M,g)$ denote a Riemannian manifold of dimension $d$, and $F(M)$ denote its frame bundle. For $e \in R^d$ and $u \in F(M)$, let $H_e(u) := (ue)^*$, where $v^*$ denotes the horizontal lift of $v \...
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Showing the if $X_t$ is a Martingale and $F\in C^2(\mathbb{R})$ then $F(X_t)$ is a semimartingale
Setup
Im having a hard time understanding the following problem
If $X_t$ is a Martingale and $F\in C^2(\mathbb{R})$ then $F(X_t)$ is a semimartingale.
I know by Ito's Formula we get:
$F(X_t) = F(...
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Yamada--Watanabe theorem up to a stopping time holds
I wonder if a Localized Yamada--Watanabe theorem up to a stopping time holds.
Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space satisfying the usual ...
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Connecting two forms of the Kolmogorov Backward Equation
I am struggling to connect the standard formulation of the Kolmogorov Backward Equation (KBE) from Oksendal with the one found in a paper by Andersson (1982) on reverse stochastic differential ...
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Stochastic dominance and Expectation
Assume 2 random variables, $x$ and $y$.
The variable $x$ has first-order-stochastic dominance over $y$ ($x \succcurlyeq y$).
My question is - what can I say about $E[\frac{y}{x}]$?
My intuition is - ...
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Compare quadratic variation of predictable FV process and the special semimartingale
Question
Suppose $X = N + A$ is a special semimartingale with its canonical decomposition. Do we have $E{[X, X]_t} \ge E{[A, A]_t}$?
My attempt
If $X \in \mathscr{H}^2$, i.e. $||[N, N]_\infty^{1/2}||...
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A generalization of freezing lemma
Let $\mathscr G⊆\mathscr F$ be a sub-$\sigma$-algebra, $X\in \mathbb{L}^0(\mathscr G,\mathbb R^d)$, $\varphi : \mathbb R^d×\Omega→\mathbb R$ be bounded and $B(\mathbb R^d)×F$ -measurable. Assume for ...
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$ \omega \to \int_0^{\infty} H_s^2 \, d\langle M, M \rangle_s $ $\mathcal{F}$ is measurable?
We denote the progressive $\sigma$-field on $\Omega \times \mathbb{R}_{+}$ by $\mathcal{P}$
and, if $M \in \mathcal{H}^2$ (continuous $L_2$ bounded martingale), we let $L^2(M)$ be the set of all ...
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The Cameron Martin Space of the Wiener Measure
I am going through some notes on SPDEs and I am having some difficulties with the following problem:
The definition I am working with is
My attempt:
I know that the space $\mathring{\mathcal{H}}_\mu$...
- 1,079
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Compute the conditional expectation with respect to the filtration generated by the Brownian motion
I suspect that the following holds: if, for all $n \in \mathbb{N}$, for all $f_1, \ldots, f_n \in C_c(\mathbb{R})$, and all $0 \le t_1 < \dots < t_n \le t$,
$$
\mathbb{E} \left[ (X-Y) \prod_{i=1}...
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Proof that self-financing portfolios remain self-financing after a numéraire change
I am reading Changes of Numéraire, Changes of Probability Measure and Option Pricing by Geman, El Karoui and Rochet, and I am having some trouble with the proof of this rather elementary statement in ...
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Continuity of Stopping Times with Respect to a Parameter
Suppose $f_1 : A \to [-\infty,0)$ and $f_2: A \to (0,\infty]$ are continuous functions (one point compactify the corresponding spaces if you like), where $A$ is an arbitrary topological space, and ...
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Time version of Kolmogorov Extension Theorem and Girsanov Theorem: whether there exist a $P_\infty$ such that $P_\infty|_{ {\mathcal F}_t} = P_t$?
Consider a filtrated probablity space $(\Omega, {\mathcal F}, \{{\mathcal F}_t\},P)$ such that ${\mathcal F}=\sigma\left(\displaystyle \bigcup_{t\ge 0}{\mathcal F}_t\right)$. Assume there exist a ...
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About the definition of a jump-diffusion process
I’m studying stochastic calculus and Lévy processes, and I’ve come across two definitions of jump-diffusion processes that seem related but structurally different.
In D. Applebaum’s Lévy Processes and ...
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Computation of the geometric Levy Process
I'm a bit confused about Oksendal's computation of Itô formula in the Geometric Levy process.
In his book (Applied Stochastic Control of Jump Diffusions), Itô formula for Levy process is given as ...
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