# 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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### Stochastic differential equation

$$dY_t = (-2\alpha Y_t + \sigma^2)dt + 2 \sigma\sqrt{Y_t}dB_t$$
Hint here is letting $X_t = \sqrt{Y_t},$ find that $dX_t = -\alpha X_tdt +\sigma dB_t$
Question is how do you derive the 2nd equation $...

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### What is famous mistake made by Feller?

I heard "Feller made a famous mistake in 1954 and fixed by A.D. Wentell in 1959" from one lecture. There is no further explain what is that mistake? Is there someone know it? Is it ...

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### How to use Itô's formula to show that $ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\left<\mathbf{J}x_u, x_t\right>]du+\frac{1}{N}\sum x_t^i(B_s^i-B_t^i) $?

I am reading a lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it shows that for $s\ge t$,
$$
K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\langle\mathbf{J}...

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### When is the mode of a stochastic process a better statistic than the mean?

This is a soft question.
I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.
...

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### Converse to Cameron-Martin theorem

It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....

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### Chung's law of the iterated logarithm for Brownian motion

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...

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### Probability that a geometric Brownian motion with additional determinstic drift ever hits zero

Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE
$$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$
where $\mu, \sigma, C, k > 0$ are constants, ...

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### Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...

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### How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?

Let
$E$ be a $\mathbb R$-Banach space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$-...

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### Another large noise limit

Note: Here all processes take values in $[0, 1]$.
Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Let $X$ be the solution to the SDE
$$dX_t = \sigma X_t \, dW_t$$...

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### Continuation : Does the density of a stopped drifted Brownian motion vanish at zero?

Let
$$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$
where $(b_t)_{t\ge 0}$ is a bounded adapted process and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and ...

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### What is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ ...

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### Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?

Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE.
Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...

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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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### Uniform boundedness of this SDE? And possibly a stochastic Grönwall inequality?

I have a question on Lawler – Notes on the Bessel process, on page 4. Let $X_t$ be one-dimensional Brownian motion, and we want to use $N_t$ as a measure-changing (local) martingale, defined as $$N_t=\...

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### Does this book use non-mainstream stochastic analysis constructions and is thus perhaps not a good start?

I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it.
Before I make a very ...

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### Uniform bound for the occupation time of a diffusion

Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the ...

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### On the growth of sample paths of Gaussian random fields

Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random ...

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### Does the convergence of drifted Brownian motion imply the convergence of expectation?

Let $(f_{\epsilon})_{\epsilon>0}$ be a family of non-increasing and continuous functions on $\mathbb R_+$ s.t. $f_{\epsilon}(0)=1$ and $f_{\epsilon}(\infty)=0$. Assume that $\epsilon\mapsto f_\...

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### Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$

Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...

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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, ...

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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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### Stochastic filtering with time delayed observation

Let $X_t$ be a suitably nice real valued Markov process. The primary two cases I have in mind are a finite state space Markov process, and a Ito diffusion.
Define the observation process $Y_t$ by
$$...

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### Choice of stochastic integral picking the forward point in Riemann sum approximation and reversibility?

Consider the standard Riemann sum approximation of a stochastic integral (w.r.t Brownian motion for example) which is given by
\begin{align}
\int_0^t \sigma(X_s) \circ_{\lambda}dB_s \approx \sum_{i=1}^...

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### Intersection of Brownian motion and finite variation process

Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.
Question: Denoting by $\mathcal L$ the ...

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### What are the optimal times to sample a process?

Let $X$ be a one dimensional Ito diffusion given by
$$X_t = b \,W_t$$
where $b$ is a constant, and $W$ is a standard Brownian motion.
Let $B$ be another Brownian motion independent of $W$, and define ...

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### Preservation of variance for log-normal variables under change of measure

Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...

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### Filtering the period and amplitude of a sine wave corrupted by noise

Let $W$ be a standard Brownian motion and $\mathcal F_t$ its natural filtration.
Suppose $\theta, A$ are positive $L^1$ random variables independent of $\mathcal F_t$.
Let $Y_t$ be the process
$$Y_t :=...

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### Orthogonal basis of martingale-Hardy space

Let $(X,\mathcal{B}(X),(\mathcal{A}_n)_{n=1}^N,\mu)$ be a filtered probability space with $\mathcal{A}_N=\mathcal{B}(X)$ and let $H$ be the space of $\mathcal{A}_{\cdot}$-adapted martingales $m_{\cdot}...

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### Question re Ito's Lemma in Options Pricing

Ito's Lemma states that if $x(t)$ is an Ito drift-diffusion process, i.e. if $dx(t) = \mu dt + \sigma dB(t)$, where $B(t)$ is a non-geometric Brownian motion, then, if $f()$ is a twice continuously ...

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### Ito formula for fractional BM + drift and supremum bound

Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...

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### Large deviation for empirical median

I found this exercise while reading some notes on Large Deviation Principle. This exercise is at the end of the very first chapter, including Cramer's Theorem and essentially nothing more (no Sanov ...

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### How to use Riemann sum in the equation of the multifractional Brownian motion to estimate the underlying standard Brownian motion

I'm studying the paper Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates in which the author presents the following ...

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### When enlarging a filtration makes a stochastic processes into a solution to an SDE

Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...

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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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### The different quotient of the first exit time with respect to the initial state

Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that
$$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \...

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### Is $p_t(y,x)$ the kernel of the time reversal of the diffusion $X$, for $p_t(x,y)$ the kernel of $X$?

Short version. If $X$ is a diffusion with generator $L$ and the Lebesgue measure is invariant for $X$, then $L^*$ has no term of order zero and it corresponds to another diffusion $X^*$. Denoting by $...

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### Uniqueness of the solution to some SDE

Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...

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### Dependency of first hittimg time on coefficients of SDE

Let $b: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline b,\overline b]$ and $a: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline a,\overline a]$ be Lipschitz, where $\overline b>\...

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### A convergence question in $L^2$ construction of Brownian motion

I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...

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### Does the density of a stopped drifted Brownian motion vanish at zero?

Let
$$Y_t:=1+\int_0^t b(s)ds + W_t,\quad\forall t\ge 0,$$
where $b:\mathbb R_+\to[1,2]$ is continuous and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and $X_t:=...

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### A comparison principle for the stochastic integral

Here all processes are real valued on $[0, T]$.
Let $X$ be a martingale, and $M^n$ a sequence of martingales with $\sup_{t| X_t \neq 0} \frac{|M_t^n|}{|X_t|} \to 0$ as $n \to \infty$, and $M^n = 0$ ...

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### Joint law of two stochastic integrals with respect to the same Brownian motion

Let $a:\mathbb R_+\to [1,2]$ be "smooth". Given a standard Brownian motion $W$, define for $t\ge 0$
$$X_t:=\int_0^t\frac{1}{a(s)}dW_s \quad \mbox{and}\quad Y_t:=\sup_{0\le u\le t} \int_0^u a(...

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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 ...

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### Bounded Solution for a two-dimensional SDE

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 $W$ ...

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### Occupation time of a diffusion with drift bounded away from 0

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 ...