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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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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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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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1answer
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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1answer
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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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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1answer
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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1answer
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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0answers
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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2answers
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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0answers
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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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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1answer
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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1answer
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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0answers
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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3answers
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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0answers
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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0answers
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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1answer
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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1answer
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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1answer
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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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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,\...