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.
1,023 questions
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Explosion of the moment of double stochastic exponential
We consider the stochastic system
$$\frac{dS_t}{S_t}=-R_t\,dW_t,$$
with
$$dR_t=-R_t\,dt-R_t\,dW_t, \quad R_0>0.$$
We conjecture, and would like to show that
$$\mathbb{E}[S_t^2]
=
S_0^2\,\mathbb{E}\...
- 696
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Transfer of stochastic integrals under identical distributions
Let $(\Omega^i, \mathcal{F}^i, \mathbb{P}^i)$ $(i = 1, 2)$ be two probability spaces. On each space we have a predictable process $X^i$ and a continuous semimartingale $Y^i$. Assume that the pairs $(X^...
- 155
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Can we utilize SDE to study differential equations with multiple solutions?
Could you please confirm if the following results have been proven?
Suppose there is a differential equation with multiple solutions. If we add random terms, it becomes a stochastic differential ...
- 105
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Which processes admit good conditional growth rates bounds?
Fix $T>0$ and let $X_{\cdot}:=(X_t)_{t\ge 0}$ be a semi-martingal adapted to a filtered probability space $(\Omega,\mathcal{F},\mathbb{P},\mathcal{F}_{\cdot})$ which is the unique strong solution ...
- 4,902
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Speed measure for sticky drifted Brownian motion
Consider the following SDE:
$dD_t=1_{\{D_t>0\}}d(B_t+\sqrt{2}t)+dL_t\\dL_t=\sqrt{2}1_{\{D_t=0\}}dt$
under the constrain that $D_t\geq 0$, where $L_t$ is the local time of $D_t$ defined by Tanaka's ...
- 41
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Solving decoupleable families of FBSDEs
Let $d_1,d_2\in \mathbb{N}_+$, a stopping time $\tau$, and consider a system of BSDEs
\begin{align}
Y_{\tau}^1 & = \xi^1+\int_{t\wedge \tau}^{\tau}\, f_1(t,Y_t^1,Z_t^1,Y_t^2,Z_t^2)dt - \int_{t\...
- 4,902
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Is $\mathcal C(\mathbb R_+, \mathbb R)$ measurable in the cylindrical $\sigma$-algebra $B(\mathbb R)^{\otimes \mathbb R_+}$?
Let
$$
\mathcal A(\mathbb R_+, \mathbb R) = \{ f : \mathbb R_+ \to \mathbb R \text{ a map}\}.
$$
For each $t \ge 0$, denote by
$$
\operatorname{ev}_t : \mathcal A(\mathbb R_+, \mathbb R) \to \mathbb R,...
- 141
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Can Atlas-model minimize the distance between two stochastic processes?
Consider a filtered probability space in which there exist two independent Brownian motions $W$ and $B$. For every progressively measurable process $u=(u_t)_{t\ge 0}$ taking values in $[0,1]$, define ...
- 189
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Lifting of non-reversible Markov chains for convergence acceleration
Background and motivation
Let $(\Omega, \pi)$ be a finite state space with a stationary distribution $\pi$. Consider an ergodic Markov chain on $\Omega$ with transition matrix $P$ that is irreducible ...
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Approximation by Staircase Processes and $L^2$ Convergence of Malliavin Derivatives
Let $(x_s)_{s\in[0,T]}$ be a stochastic process such that, for every $s\in[0,T]$,
$$
x_s \in \mathbb{D}^{1,2}(\mathbb{R}),
$$
and assume
$$
\mathbb{E}\left[\int_0^T |x_s|^2\,ds\right] < \infty,
\...
- 696
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Generalized Malliavin Chain Rule
Let $ E $ be the Banach space of continuous functions on $[0, T]$, equipped with the supremum norm. Let $ \sigma : E \to \mathbb{R} $ be a function of class $ C^1 $ (Fréchet differentiable). Let $ X = ...
- 696
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Is this also known as BDG inequality?
Let $W=(W_t)$ be a real-valued Brownian motion on some filtered probaiblity space. Let $H=(H_t)$ be a progressively measurable process taking values in some Hilbert space $\mathcal H$, endowed with ...
- 111
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Pointwise estimate in Ito isometry
Let $W$ be a classical Wiener process on $[0,1]$ and let
$$
\mathcal{I}\colon a\mapsto \int_0^1a(t) dW(t)
$$
be the stochastic integral with respect to $W$. Ito isometry states that $\mathcal{I}$ is ...
- 637
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Localized Yamada–Watanabe
I wonder if a Localized Yamada–Watanabe theorem up to a stopping time exists. Here is more details:
Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space ...
- 696
6
votes
1
answer
551
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What curve maximizes the Levy area?
The Levy area of a $C^1$ curve $f:[0,\infty)\to \mathbb R^2$ is defined to be $$L_f(t):=\int_0^t (f_1(s)f_2'(s)-f_2(s)f_1'(s))ds. $$ It is called Levy area because by Green's theorem, it is twice the ...
- 2,377