# 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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### Convergence of right continuous martingale using $L^2$-completeness

I was reading on the convergence of $L^1$-bounded right continuous submartingales $(X_r)_{r \geq0}$, where in the proof (a sketch of a proof) they didn't use oscillations-upcrossing inequalities: a.s ...
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### Where does the “mixing” occur in convex combination of Girsanov measures?

In this post, Ofer says that taking the convex combination of two Girsanov measures yields a drift $BF_1+(1-B)F_2$ where $B$ is a Bernoulli random variable with parameter $\lambda$, independent of the ...
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### Schwartz regularity for the density of a stochastic process

Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables \begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*} It ...
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### Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$. It happens that the ...
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### If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?

Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
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### Is the topology generated by the convergence of finite-dimensional distributions metrizable?

Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
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### A doubt on the derivation of the Wiener Chaos expansion propagator

So I have seen the following calculation in a number of articles (for instance 1, 2 3) and I just can't get my head around it. The idea is basically as follows, let $\mathcal L$ be some differential ...
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### Expectation of first exit time of a bounded set by a time-homogeneous Ito diffusion is finite

This is a question concerning Remark(i) under Theorem 7.4.1(Dynkin's formula) on Page 124, $\textit{SDE}$, by Oksendal. It says that if $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$ is an $n$-dimensional time-...
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### Is my quadratic variation derivative bounded?

Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
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### existence and uniqueness of solution to CEV model sde

Suppose that you have the CEV model for a stock price following the sde $$dS_t = r S_t dt + \sigma S_t^{\eta} dw_t$$ where $0 \leq \eta\leq 1$, $S_0=s_0$ and $w$ is a Brownian motion. How do you ...
Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$: $$Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W;$$ ...