# 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.

550 questions

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42 views

### How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...

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26 views

### p-Variation distance defines semi-martingales

Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...

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**1**answer

119 views

+50

### Time discretization in the Feynman-Kac formula with boundary conditions

I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...

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61 views

### Is there solution to a backward stochastic differential equation with $yz$ in the generator?

Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...

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62 views

### When is $f(t,W_t)$ an Ito process?

Consider a Brownian motion $(W_t)_{t\in[0;T]}$.
If $f\colon [0;T] \times \mathbb R \to \mathbb R$ is $C^{1,2}$, we know that $(f(t,W_t))_{t\in[0;T]}$ is an Ito process and we can directly write down ...

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31 views

### Reference: Stochastic Optimal Control with cost functional

There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by
$$
J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right],
$$
function $g$ and ...

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**1**answer

73 views

### Hitting probability of a transient diffusion process

I have a question about properties of transient diffusion process.
In the case of $d$-dimensional Brownian motion $B=(B_t,P_x)$ ($d \ge 3$), we can prove that
\begin{align}
(1)&\quad 0<P_{x}(\...

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45 views

### Matching Stochastic Flows

Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...

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**1**answer

72 views

### Martingale representation theorem for symmetric random walk

Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that
$$ X(t) = \int_0^t ...

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60 views

### Calculate a realization of two stochastic variables as a function of the same realization of a stochastic process?

I have the following parametric stochastic integral:
$$
I(\lambda) = \int_{t_i}^{t_{i+1}} \exp(\lambda (t_{i+1} - s)) dW_s,
$$
where $W_t$ is a zero-average totally uncorrelated stationary white-noise ...

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70 views

### Solutions to the Bond Pricing Equation

Consider a spot rate of the form:
$dr = (\eta - \gamma r) dt + \sqrt{\alpha r + \beta} dW$
where all parameters are constants.
Lets look for a solution of the form $Z(r; t) = e^{A(t;T) - r B(r; T)...

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33 views

### Supremum of a general Gaussian Process

I have a stochastic integral of the form
\begin{align*}
X(t) = \int_0^t h(v) W(v) dv
\end{align*}
where $W(v)$ is the standard Brownian motion and $h(v)$ is a positive, integrable function. While $X(t)...

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36 views

### Existence and uniqueness of the asymptotic distribution of $x(k+1) = Ax(k) + v(k)$

Consider the linear discrete-time stochastic systems:
\begin{equation}
x_{k+1} = Ax_k + v_k,
\end{equation}
with time-instants $k \in \mathbb{N}$, state $x_k \in \mathbb{R}^n$, stochastic process $v_k ...

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54 views

### Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...

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39 views

### Floquet stochastic process

Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...

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46 views

### quadratic variation on n-sphere

Is it true, and if so, is there an easy way to see that the quadratic variation of standard Brownian motion on n-sphere is $\leq$ t?
Note: I am a novice in stochastic analysis.

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20 views

### Singular direction of a particle system

Consider a system of n-sdes in $\mathbb{R}$ ( the formula is not important).
The corresponding particle system $X(t)=(X_{1}(t),X_{2}(t),...,X_{n}(t))$ lives in $\mathbb{R}^{n}$ and assume that ...

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74 views

### An Incorrect Construction of the Ito Integral

Let $B_t$ be a Brownian motion defined on the interval $[0,T]$, with underlying (filtered) probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$. Call a function $f:[0,T]\times\Omega\...

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390 views

### English translation of “Les aspects probabilistes du contrôle stochastique”

I am looking for an English translation of "Les aspects probabilistes du contrôle stochastique" written by Nicole El Karoui, or knowledge whether it exists.
Other references with similar content on ...

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26 views

### Stationary distribution of gradient dynamics

We consider the gradient dynamics $ d X_{t} = d B_{t} - \nabla(U(X_{t}))dt $ in $\mathbb{R}^{d}$.
G.Royer in the book "An initiation to logarithmic sobolev inequalities" (p29,30) says that if
(1) U ...

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99 views

### Is there a distinct Ito-Sasaki version of Riemannian stochastic development?

Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...

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**1**answer

105 views

### Multiple Wiener-Ito integral distribution

Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$
Is it possible to find the distribution of multiple Wiener-Ito ...

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122 views

### Predictability of countably valued accessible stopping times on complete and cadlag filtrations

The following question is motivated by this part of the proof of Lemma 2 on page 107 of the book Stochastic integration and differential equations of Philip Protter.
Lemma 2. Let $T$ be a totally ...

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114 views

### Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...

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329 views

### Good papers on stochastic differential equations with applications in finance

I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) ...

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93 views

### Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...

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28 views

### Can we relate Levy area to the resonant term in paracontrolled calculus in the case of a controlled differential equation?

Consider the SDE:
$$dX_t=V(X_t)dW_t^H$$
Where $W_t^H$ is fractional Brownian motion with Hurst parameter $H\in (1/3,1/2)$. This equation can be solved by rough paths theory. The idea is we have to ...

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33 views

### Differentiability of a stochastic process depending on a spatial parameter

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$T>0$
$I:=(0,T]$
$d\in\mathbb N$
$M:\Omega\times\overline I\times\mathbb R^d\to\mathbb R$ such that $M(\;\cdot\;,\;\cdot\;,x)$ is $\...

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64 views

### Question about Protter's proof of the Ito's formula

The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential ...

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68 views

### martingale representation type result

Suppose we have two (not necessarily independent) semimartingales $X$ and $Y$.
Let $\mathcal{F}_t$ be the completed filtration generated by $(X_t, Y_t)$ and let $H_t$ be a martingale with respect to $\...

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61 views

### Why is the Jain Monrad condition the right condition on general Gaussian processes?

Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...

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455 views

### Is the ito integral $\int_0^t \operatorname{sign}(W_s)\mathrm{d}W_s$ a Brownian motion?

Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$:
$$\int_0^t \operatorname{sign}(W_s)\,dW_s$$
This appears for instance in the Tanaka formula. I think this is a ...

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**1**answer

151 views

### fractional Brownian Motion driven stochastic integrals

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$
With a fractional Brownian motion $B^H_{t}$.
If $H\neq\...

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15 views

### variance of a recursive distribution

I have the following recursive equation,
$\tilde{S}(k+1) = \beta \bar{r}(k) + (1-\beta)\tilde{S}(k)$
where $\bar{r}(k) \sim \mathcal{N}(0, I)$.
How can I calculate the variance of $\tilde{S}(k)$? (...

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95 views

### Moments of the Hölder norm of Brownian process

It is well known that for a brownian process $B(t),t\geq 0$, it holds
$$
\sup_{0\leq s<t\leq T}\frac{|B(t)-B(s)|}{|t-s|^\alpha}<\infty
$$
almost surely, for any $T>0$ and $\alpha<1/2$.
...

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44 views

### The general integration by parts formula and the general change of variables formula

This question is motivated by the fact that most of the books of stochastic calculus always prove the basic results of integration by parts formula and the change of variables formula, and at the ...

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91 views

### Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...

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90 views

### Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain

Hairer in his spdes notes on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{...

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78 views

### Meaning of $. \wedge t$ (. \wedge t) in stochastic analysis

In Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I they define (on page 4) a metric :
$${\bf d}_\infty ((t,\omega),(t',\omega')) := |t-t'| + \|\omega_{.\wedge t}-{\omega'...

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79 views

### Heat equation, free boundary and dynamic programming

I have a dynamic programming problem with an underlying diffusion $$ d X_t = \mu \, dt + d b_t$$
where $b_t$ is a standard brownian motion.
The HJB equation for the value function $v(x,t)$ I get is ...

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37 views

### Jumping times on Borel sets away from zero are stopping times

The following comes from some remarks of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to prove yet.
Let $X$ a Levy process, under ...

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**1**answer

104 views

### Obtaining the distribution of the First Hitting time of the Bessel Process

Let $(X_{t})_{t\geq 0}$ be a Bessel Process starting at $x>0$ of dimension $\delta>0$. Namely
\begin{align*}
X_{t}=x+W_{t}+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{X_{s}}\, ds.
\end{align*}
where ...

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68 views

### Bounded over time versus bounded over stopping times

Consider the expectation $E(G(v,X_v)|\mathcal{F}_t)$ for $t\leq v \leq T$ for a stochastic process $X_t$. We can impose one of the two following conditions :
$E(G(v,X_v)|\mathcal{F}_t)$ has a ...

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95 views

### Continuity w.r.t time vs Continuity w.r.t. stopping times

Several places in "Optimal Stopping and Free-Boundary Problems" Peskir and Shiryaev make the assumption that a (Markov) process $X = (X_t)_{t\geq 0}$ has sample
paths which are right continuous and ...

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**1**answer

89 views

### Limit distribution of Ornstein-Uhlenbeck equation

Let
\begin{equation*}
X_t=xe^{-\lambda t}+\sigma e^{-\lambda t}\int_0^t e^{\lambda s} dB_s
\end{equation*}
be the solution of Ornstein-Uhlenbeck equation where $B$ is Brownian motion, and $x,\sigma,\...

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47 views

### Has this type of pathwise (S)DE been studied before?

I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before.
Let $(G,\ast)$ be an abelian $C^1$ Lie group....

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83 views

### Exit time of a stochastic process defined by a SDE

Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\...

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110 views

### An SDE version of a Fokker Planck Equation

Assume $\rho$ is probability density defined on $\mathbb{R}^d\times\mathbb{R}^d$. I am interested in the Wasserstein gradient flow of a functional:
\begin{equation*}
\mathcal{E}(\rho)=\iint_{\mathbb{...

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68 views

### Connection between deterministic and stochastic problems in PDEs

In the study of conservation or balance laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)):
Deterministic Cauchy problem:
$$(1) \...

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57 views

### Prove that a local martingale with spatial parameter is differentiable

Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...