# Questions tagged [stochastic-calculus]

Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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### Strong blow up limits for SDE

Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications. Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution ...
1 vote
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### Reference request: Gaussian estimates for SDE with discontinuous diffusion coefficient

Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R_+ \times \mathbb R^d \to \mathcal M_{d \times d}^{\text{sym}} (\mathbb R)$ be bounded measurable where $\sigma$ is ...
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### Calculation of the difference of two Brownian bridges

I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e., $$B_{1t} - B_{2t} = \sqrt{2}B_t$$ where $B_{1t}$ and $B_{2t}$ are ...
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### A notion of SDE via the martingale representation theorem

$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
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### On a real smooth version of white noise distribution theory

In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
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### Local martingale with increasing process

Here is a problem in stochastic calculus: If $M_t$ is a continuous process and $A$ an increasing process, then $M$ is a local martingale with increasing process $A$ if and only if, for every $f\in C^2$...
1 vote
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### Modulus of "set"-continuity for Wiener Field

My question concerns some "set-wise" continuity properties of Gaussian random fields, more specifically of Wiener fields (see definition here: https://encyclopediaofmath.org/wiki/...
1 vote
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### When is the probability measure on the "direct product" via the Kolmogorov extension theorem supported on the "direct sum"?

Let me restrict to the case of Hilbert spaces, which seem simplest. Let $\{H_n\}$ be a sequence of (possibly infinite dimensional) Hilbert spaces and $\{ \mu_n \}$ be a sequence of Borel probability ...
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1 vote
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### How to obtain this differential relation about moments of a stochastic process?

$\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of Proposition 3.8. in the paper Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos. ...
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1 vote
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### The space of linear operators between Hilbert spaces has martingale type 2

I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded ...
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### Reference request for a Riemannian Fokker-Planck equation

The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...