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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Blow up limits for SDE

Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE $$dX_t = \sigma(X_t) \, dW_t \, , \, X_0 = 0$$ with $\sigma: \mathbb R \to \mathbb R$ Lipschitz continuous....
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Convert a discrete stochastic process with non-normal noise to continuous stochastic process

Suppose I have a discrete stochastic process, in the form of $$x_{t+1} = x_t + \varepsilon_t$$ where $\varepsilon_t$ is the random noise. The caveat is by examining the existing data, $\varepsilon_t$ ...
DiveIntoML's user avatar
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195 views

Distribution of "occupation times" of Brownian Motion

Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal{B}(\mathbb R)$ be a Borel set. I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
Andrea Aveni's user avatar
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1 answer
169 views

Stochastic integral with non-anticipating integrand

Let $B$ be a Brownian motion. We want to define $$ \int_{0}^{t} B_{s} dB_{s} : = \lim_{n \to \infty } \sum_{k = 1}^{2^{n}t} B_{\frac{k-1}{2^{n}}}[ B_{\frac{k}{2^{n}}} - B_{\frac{k-1}{2^{n}}}]. $$ To ...
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Exponential of supremum of Brownian bridge on short time frame

For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$. Question: Is it true that $\mathbb E[|\text{exp}\, (\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $...
Nate River's user avatar
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Distribution of zeros and angles of a function with additive coloured noise

Let us consider some real-variable function $$ f(t) = f_0(t) + \xi(t), $$ where $f_0$ - some "regular" (a continuously differentiable function without any noise [one can consider $f_0 = \...
MightyPower's user avatar
1 vote
1 answer
316 views

Expectation of stochastic integral

Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$. Is the following expectation $$ \left\langle \int^...
can't stop me now's user avatar
1 vote
0 answers
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Lipschitzness of conditional law of a stochastic filtering problem wrt the Wasserstein distance

Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be a pair of stochastic processes taking values in $\mathbb{R}^n$ and in $\mathbb{R}^m$; defined on a filtered probability spaces $(\Omega,\mathcal{F},(\...
Justin_other_PhD's user avatar
5 votes
1 answer
311 views

Joint distribution of drawdown time and value of geometric Brownian motion

Let $X$ be a geometric Brownian motion, satisfying the SDE $$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$ for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant. Define the ...
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Upper left Dini derivative of Brownian motion at a hitting time

Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by $$D^-W_t := \limsup_{h \to 0^-} \frac{W_{t+h} - W_t}{h}.$$ Fix $a > 0$, and define the stopping time $\tau$ by ...
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1 vote
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Martingale representation theorem up to a stopping time

Let $W$ be a standard one dimensional Brownian motion, and let $\mathcal F_t$ be its completed natural filtration. Let $\tau$ be an $\mathcal F_t$ stopping time with $\tau < T$ almost surely for ...
Nate River's user avatar
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2 votes
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Full version of Cameron Martin theorem for Brownian motion

I’m looking for a version of the Cameron Martin theorem for the Brownian motion under random shifts. Here is the precise statement: Let $\mathbb P$ be Wiener measure on $\Omega := C[0, 1]$. Given a $C[...
Nate River's user avatar
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A double sum with complex numbers having stochastic variables

I am very confused by a sum I have been trying to solve analytically/ numerically for a long time. It comes from the idea of a physical problem where the observation is made that has a combined ...
CfourPiO's user avatar
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Converse of Itô's formula

Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$ Prove that $f$ is ...
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Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?

In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
Martin Geller's user avatar
1 vote
1 answer
126 views

Is it possible to sum this analytically in any way?

The sum I am looking for is the following sum as $M \to \infty$: $$ L(\omega) = \sum_{m = 1}^{M} \frac{\sin\left( N \frac{\omega_m - \omega}{2} \right)}{\sin\left( \frac{\omega_m - \omega}{2} \...
CfourPiO's user avatar
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1 answer
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Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion

We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...
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Second moment of stochastic integral wrt Levy Processes

I have a question about the second moment of the integral wrt Levy Processes. Let Z a Levy processe. We know that: And a few page later is written that by differentiation of the characteristic ...
Ginger 17's user avatar
0 votes
1 answer
227 views

Integrated square difference of Brownian bridges

I am doing some work with measuring the distance between distributions, and someone pointed out to me that I should look into calculating the integrated squared difference of two brownian bridge ...
John Smith's user avatar
5 votes
1 answer
467 views

Riemannian metric induced by a stochastic differential equation

Following this paper, a diffusion process in $\mathcal{R}^d$ $$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$ with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
can't stop me now's user avatar
1 vote
1 answer
372 views

Is there an inverse Lamperti transformation for diffusions?

The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion. For multidimensional processes there are some conditions on the ...
can't stop me now's user avatar
2 votes
1 answer
190 views

Comparing diffusion processes in different metrics

I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$. Is there a way to apply ...
can't stop me now's user avatar
1 vote
0 answers
111 views

Stratonovich version of Girsanov

One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density $$\frac{d\mu}{d\mu_0}:=\exp\left(\...
user479223's user avatar
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2 votes
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What is the quadratic variation of $W(B(t))$?

Let $W$ be a two sided real valued Brownian motion. Let $B$ be a one sided Brownian motion independent of $W$. Consider the process $X(t)=W(B(t))$. Is the quadratic variation finite and if it is, what ...
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1 answer
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Does a continuous martingale converge almost surely on the event that its quadratic variation is finite?

Let $M$ be a continuous martingale. Denote by $E$ the event that its total quadratic variation is finite, i.e. $$E := \{\langle M, M \rangle_\infty < \infty\}.$$ Question: Is it true that as $t \to ...
Nate River's user avatar
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Patching together weak solutions of SDE's at random time points

Suppose we are given a sequence of drift coefficients $b^n : \mathbb R \to \mathbb R$ and we know that the following SDE has a weak solution, unique in law on $[0,\infty)$ $$dX_t^n(\mu) = b^n(X_t^n(\...
Stefan Perko's user avatar
2 votes
1 answer
142 views

Does the time of maximum of a diffusion process admit a continuous density?

Let $W$ be a standard one dimensional Brownian motion, and consider the solution $X$ to the SDE $$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$ with $X_0 = 0$ a.s., and where $\mu, \sigma: \mathbb R \...
Nate River's user avatar
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Regularity of solutions to forward-backward stochastic differential equations

Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy $$ \left\{ \begin{aligned} d X_t &= aP_t dt +bdB_t;\\ X_0 &= x_0;\\ d P_t &=cP_t dt + c^*Z_t dB_t; \\ P_T &...
mnmn1993's user avatar
1 vote
1 answer
145 views

For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$

Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that \begin{align*} 0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
nate's user avatar
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0 votes
1 answer
166 views

Abstract Wiener spaces for pinned processes (e.g., Brownian Bridge)

In introductions to abstract Wiener spaces, the sample paths usually form a Banach space; so, in particular, the sum of two sample paths is a valid sample path and also an element of the Banach space. ...
MrArsGravis's user avatar
4 votes
1 answer
311 views

Convergence of a continuous time stochastic gradient descent algorithm

Let $f: \mathbb R \to \mathbb R$ be a $C^1$ convex function, satisfying the growth conditions $$\lim_{x \to -\infty} \nabla f(x) = -\infty, \lim_{x \to \infty} \nabla f(x) = \infty.$$ and let $\...
Nate River's user avatar
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1 vote
0 answers
129 views

Martingale representation with respect to weak solution of driftless SDE

Let $X$ be a weak solution to the one dimensional SDE $$dX_t = \sigma(t, X_t) \, dW_t \, , \, X_0 = x_0$$ on $[0, T]$, where $W$ is a Brownian motion, $\sigma: \mathbb R \times \mathbb R \to \mathbb R$...
Nate River's user avatar
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2 votes
1 answer
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On a certain deterministic integral related to Tanaka’s formula

Note: We define the signum function, $\text{sgn}$ by $\text{sgn}(x) = 1$ if $x \geq 0$, and $-1$ otherwise. Suppose $f: [0, \infty) \to \mathbb R$ is continuous and of locally bounded variation, with $...
Nate River's user avatar
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2 votes
1 answer
403 views

Is a martingale constant on the event that its quadratic variation is zero?

Let $M_t$ be a continuous time martingale, and assume its quadratic variation is identically zero with some positive probability less than $1$. To be more precise, assume there exists some event $E$ ...
Nate River's user avatar
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3 votes
0 answers
90 views

Martingale problem for regime switching SDE

Let $\mu$ be a continuous time Markov process switching between two possible values $a, b \in \mathbb R$ with time homogeneous transition rates. Consider the one dimensional SDE $$dX_t = \mu \, dt + ...
Nate River's user avatar
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1 vote
0 answers
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Showing that the natural scale function is a martingale under specific conditions

My question is related to this post How to find the "natural scale function" for more general stochastic processes?. Indeed, I am trying to solve an exercise in which I have to show that if ...
vfsh's user avatar
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4 votes
1 answer
398 views

What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$?

Consider the $d$-dimensional SDE, $d > 1$, $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$ where $W$ is a standard $d$-dimensional Brownian motion. I am interested in the case where $\sigma: \mathbb ...
Nate River's user avatar
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1 vote
1 answer
183 views

Lower bound on ratio of extreme order statistics

This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...
oyy's user avatar
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0 votes
1 answer
144 views

Change of measure formula for the Föllmer process

While reading a preprint Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process I came across the following SDE in Section 3: $$d X_t=d B_t+\nabla \log P_{...
Student's user avatar
  • 601
7 votes
1 answer
218 views

Onsager-Machlup functional when drift is time-dependent

Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by \begin{align} \mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}, \end{align} where $b_i(x) \in \mathcal{C}_b^2(...
Enforce's user avatar
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0 answers
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Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term

Consider the Kolmogorov forward equation for a Langevin dynamic: $$\DeclareMathOperator{\Div}{div} \begin{cases} \dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\ \\ \displaystyle\int_{\...
Junlong's user avatar
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1 vote
0 answers
68 views

Markov property of jump type diffusions

Consider the following jump-type SDE with two random Poisson measures $N_1$, $N_2$ and a Brownian motion $B_t$: $dX_t= b(X_t)dt + \sigma(X_t)dB_t + \int{}F_1(X_t,u)N_1(dt,du) + \int{}F_2(X_t,u)N_2(dt,...
cogitoergoboom's user avatar
4 votes
1 answer
310 views

Lebesgue differentiation theorem at a stopping time

Let $W$ be a standard Brownian motion, and $\mathcal F_t$ it’s natural filtration. Let $H$ be a continuous process, adapted to $\mathcal F_t$ and integrable with respect to $W$. Question: Is it true ...
Nate River's user avatar
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1 vote
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Fokker-Planck equation for a 3D Bessel bridge

Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by $$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$ where $B_t$ is a ...
AD Le's user avatar
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2 votes
1 answer
436 views

A question related to Girsanov’s theorem

I’ve recently realised there is a subtlety in Girsanov’s theorem that I don’t really understand. Consider a standard one dimensional Brownian motion $W$, and consider the SDE $$dZ_t = \mu(t, Z_t) \, ...
Nate River's user avatar
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2 votes
2 answers
471 views

Divergence of Riemann sums in the Itô integral

Some motivation: Let $W$ be a standard Brownian motion, and $X$ an integrable process with respect to $W$, i.e. progressively measurable with respect to the natural filtration of $W$ and square ...
Nate River's user avatar
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2 votes
2 answers
383 views

Short time limits for SDE

Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE $$dX_t = \sigma(X_t) \, dW_t \;, \quad X_0 = x_0\;.$$ where $\sigma:\mathbb R \to \mathbb R$ is a ...
Nate River's user avatar
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7 votes
0 answers
178 views

Why do people who work in stochastic analysis and geometry tend to work in sub Riemannian geometry?

There is a rich theory of diffusions on manifolds. Every time I see someone who studies diffusions on manifolds, it seems like they study the sub Riemannian setting. I get that this is more general ...
user479223's user avatar
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2 votes
1 answer
280 views

Large noise limit for SDE with general volatility coefficients

Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE $$dX_t = \sigma(X_t) \, dW_t \;, \quad X_0 = 1 \;.$$ where $\sigma:\mathbb R \to \mathbb R$ is a ...
Nate River's user avatar
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1 vote
1 answer
350 views

Is the solution to this SDE always positive?

Let $W$ be a standard one dimensional Brownian motion, and consider the SDE $$dX_t = \sigma(X_t) \, dW_t, \, \, \, X_0 = 1 \, \text {a.s.}$$ Assume $\sigma$ is regular enough that the above SDE admits ...
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