# 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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### Where does the extra term in the density of a diffusion with respect to $c B(t)$ come from?

It is well known that for the diffusions \begin{align*} dX&=f(X)dt+&cdB\\ dY&=&cdB \end{align*} the density of the law of $X$ with respect to the law of $Y$ is \begin{align*} \frac{d\...
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### Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$

Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse. It is evident that $$\mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p,$$ where $X\sim N(0,1)$. Is ...
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### Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$

Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...
1 vote
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### What is the Cameron-Martin norm associated to $X(t)=\int_0^t B(s) ds+B(t)$?

The process $X(t)=\int_0^t B(s) ds+B(t)$ is a centered continuous Gaussian process. Therefore it defines a Gaussian measure on $C[0,T]$. Therefore there is a Cameron-Martin space with Cameron-Martin ...
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### A large noise limit

Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion. Denote $Y := \int_0^1 f(t) \, dW_t$. For each $\varepsilon > 0$, consider the conditioned random ...
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### Solution to a fully nonlinear SDE

Let $W$ be a standard one dimensional Brownian motion. Does the following (fully nonlinear) SDE admit a strong/weak solution? $$dX_t = X_{t + W_t} \, dt \, ,\, X_0 = 1 \text{ a.s.}$$ Explictly, we ...
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### On the Lipschitz constant of $\Gamma$

Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
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### Does higher volatility of SDE imply lower probability of staying positive?

Given two SDEs $X^1$, $X^2$ : $$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$ where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
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1 vote
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### Where to submit a new proof of the continuous martingale convergence theorem?

There were various proofs of the discrete martingale convergence theorem, but as far as I know there is only one proof of the continuous version of this theorem using the up-crossing lemma. I wrote a ...
Good evening, I was thinking about the following situation: Let $I \subset \mathbb{R}^2$ be a bounded subset and $X$ be a stochastic process such that $$dX_t = b(X_t) dt + \sigma(X_t)dW_t,$$ where $W$ ...
Let $\sigma, \mu: \mathbb R_+ \times \mathbb R \to \mathbb R$ be Lipschitz continuous functions, with $\mu > C > 0$ for some constant $C$. Let $W$ be a standard Brownian motion, and let $X$ be ...