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.
932
questions
3
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2
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478
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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....
0
votes
1
answer
52
views
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, $\varepsilon_t$ ...
0
votes
0
answers
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 $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
1
vote
1
answer
169
views
Stochastic integral with non-anticipating integrand
Let $B$ be a Brownian motion. We want to define $$ \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}}}]. $$
To ...
3
votes
1
answer
146
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Exponential of supremum of Brownian bridge on short time frame
For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$.
Question: Is it true that $\mathbb E[|\text{exp}\, (\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $...
0
votes
1
answer
70
views
Distribution of zeros and angles of a function with additive coloured noise
Let us consider some real-variable function
$$
f(t) = f_0(t) + \xi(t),
$$
where $f_0$ - some "regular" (a continuously differentiable function without any noise [one can consider $f_0 = \...
1
vote
1
answer
316
views
Expectation of stochastic integral
Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$.
Is the following expectation
$$ \left\langle \int^...
1
vote
0
answers
59
views
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},(\...
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 ...
2
votes
1
answer
132
views
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
...
1
vote
2
answers
213
views
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 ...
2
votes
1
answer
275
views
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[...
1
vote
2
answers
162
views
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 ...
4
votes
2
answers
411
views
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 ...
3
votes
0
answers
168
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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 ...
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 $M \to \infty$:
$$ L(\omega) = \sum_{m = 1}^{M} \frac{\sin\left( N \frac{\omega_m - \omega}{2} \right)}{\sin\left( \frac{\omega_m - \omega}{2} \...
2
votes
1
answer
391
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Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion
We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...
0
votes
1
answer
123
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Second moment of stochastic integral wrt Levy Processes
I have a question about the second moment of the integral wrt Levy Processes.
Let Z a Levy processe. We know that:
And a few page later is written that by differentiation of the characteristic ...
0
votes
1
answer
227
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Integrated square difference of Brownian bridges
I am doing some work with measuring the distance between distributions, and someone pointed out to me that I should look into calculating the integrated squared difference of two brownian bridge ...
5
votes
1
answer
467
views
Riemannian metric induced by a stochastic differential equation
Following this paper, a diffusion process in $\mathcal{R}^d$
$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
1
vote
1
answer
372
views
Is there an inverse Lamperti transformation for diffusions?
The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the ...
2
votes
1
answer
190
views
Comparing diffusion processes in different metrics
I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$.
Is there a way to apply ...
1
vote
0
answers
111
views
Stratonovich version of Girsanov
One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
2
votes
1
answer
255
views
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 ...
1
vote
1
answer
302
views
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 ...
1
vote
0
answers
58
views
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 $[0,\infty)$
$$dX_t^n(\mu) = b^n(X_t^n(\...
2
votes
1
answer
142
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Does the time of maximum of a diffusion process admit a continuous density?
Let $W$ be a standard one dimensional Brownian motion, and consider the solution $X$ to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $X_0 = 0$ a.s., and where $\mu, \sigma: \mathbb R \...
0
votes
0
answers
75
views
Regularity of solutions to forward-backward stochastic differential equations
Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy
$$
\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
&...
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}\...
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. ...
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 $\...
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, $\sigma: \mathbb R \times \mathbb R \to \mathbb R$...
2
votes
1
answer
89
views
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 $...
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 $E$ ...
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
$$dX_t = \mu \, dt + ...
1
vote
0
answers
44
views
Showing that the natural scale function is a martingale under specific conditions
My question is related to this post How to find the "natural scale function" for more general stochastic processes?. Indeed, I am trying to solve an exercise in which I have to show that if ...
4
votes
1
answer
398
views
What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$?
Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$
where $W$ is a standard $d$-dimensional Brownian motion.
I am interested in the case where $\sigma: \mathbb ...
1
vote
1
answer
183
views
Lower bound on ratio of extreme order statistics
This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...
0
votes
1
answer
144
views
Change of measure formula for the Föllmer process
While reading a preprint Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process I came across the following SDE in Section 3:
$$d X_t=d B_t+\nabla \log P_{...
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(...
1
vote
0
answers
54
views
Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term
Consider the Kolmogorov forward equation for a Langevin dynamic:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\
\\
\displaystyle\int_{\...
1
vote
0
answers
68
views
Markov property of jump type diffusions
Consider the following jump-type SDE with two random Poisson measures $N_1$, $N_2$ and a Brownian motion $B_t$:
$dX_t= b(X_t)dt + \sigma(X_t)dB_t + \int{}F_1(X_t,u)N_1(dt,du) + \int{}F_2(X_t,u)N_2(dt,...
4
votes
1
answer
310
views
Lebesgue differentiation theorem at a stopping time
Let $W$ be a standard Brownian motion, and $\mathcal F_t$ it’s natural filtration. Let $H$ be a continuous process, adapted to $\mathcal F_t$ and integrable with respect to $W$.
Question: Is it true ...
1
vote
0
answers
119
views
Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
2
votes
1
answer
436
views
A question related to Girsanov’s theorem
I’ve recently realised there is a subtlety in Girsanov’s theorem that I don’t really understand.
Consider a standard one dimensional Brownian motion $W$, and consider the SDE
$$dZ_t = \mu(t, Z_t) \, ...
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 ...
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 ...
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 ...
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 ...
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 ...