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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23 views

Convergence of right continuous martingale using $L^2$-completeness

I was reading on the convergence of $L^1$-bounded right continuous submartingales $(X_r)_{r \geq0}$, where in the proof (a sketch of a proof) they didn't use oscillations-upcrossing inequalities: a.s ...
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53 views

Where does the “mixing” occur in convex combination of Girsanov measures?

In this post, Ofer says that taking the convex combination of two Girsanov measures yields a drift $BF_1+(1-B)F_2$ where $B$ is a Bernoulli random variable with parameter $\lambda$, independent of the ...
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33 views

L2-closure of absolutely continuous stochastic processes?

Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...
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46 views

Find a SDE with a unique solution of given $Y$

Suppose that \mathbb{P}\right)$ is a filtered probability space satisfying the "usual conditions". $M$ is a continuous local martingale, and let $$ Y_{t}:=\exp \left(M_{t}-\frac{1}{2}[M, M]_{...
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17 views

Complex delay differential equation with time-dependent lag

I am trying to find a solution $g$ to the following delay differential equations (DDEs): $$ \beta(t)g^\prime(y)=g(y)-g(y-t)-t \quad (1)$$ $$ \beta(t)g^\prime(y)=g(y+t)-g(y)+t \quad (2)$$ with $g(y)=0$,...
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37 views

Decomposition of reversed processes

Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted. Is it possible to decompose $...
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44 views

Almost supermartingale and a.s convergence

After reading a paper on the convergence of almost supermartingale, the following result appeared: If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
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125 views

Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
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73 views

Log Sobolev inequality for Wiener space

I am reading https://arxiv.org/pdf/1003.1649.pdf and saw equation 10.2.3 that said that on Wiener space $$E\left[f^2\log\left[\frac{f^2}{E[f^2]}\right]\right]\leq 2 E[|\nabla f|_H^2],$$ where $\nabla$ ...
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138 views

How to find the “natural scale function” for more general stochastic processes?

In stochastic analysis, for an Ito diffusion $X_t$ such that $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$, we can exlpicitly compute a "natural scale function" $$S(x)=\int^x\exp\left(-\int^y\frac{2\mu(...
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312 views

Schwartz regularity for the density of a stochastic process

Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables $$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$ It ...
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166 views

Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$. It happens that the ...
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41 views

If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?

Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
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57 views

Laplace Equation for Brownian Motion

So, I know that there is this theorem (taken from here): For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have $$ u(x) = \...
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1answer
80 views

On a property of resolvents associated with holomorphic semigroups

This question is about semigroup theory. Let $E$ be a locally compact metric space, and $X=(X_t,t\ge 0;\,P_x,x\in E)$ be a Markov process on $E$. We assume that $X$ is symmetric with respect to $m$, ...
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21 views

What can be said about the coefficients of the expansion of a drift corresponding to a Girsanov measure?

Let $\mu_0$ be classical Wiener measure on $(C_0[0,T]), \mathcal B(\|\cdot\|_\infty))$. Let $\mu$ be another Borel measure so that $\mu\sim\mu_0$ are equivalent. Then by Girsanov there is a process $F(...
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28 views

Normalization of exponential in the context of Feynman integrals from a White noise perspective

I apologize in advance if this question is not suitable for MO (please let me know), but the fact is that since I am not familiar with the theory of Feynman integrals I don't know whether this is a ...
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1answer
79 views

For stopping times $\tau_k,\mathcal{F}_{\sup_{k \in \mathbb{N}^*}\tau_k}=\sigma(\bigcup_{k \in \mathbb{N}^*}\mathcal{F}_{\tau_k})$?

$(\tau_k)_{k \in \mathbb{N}^*}$ is a sequence of stopping times (taking values in $\overline{\mathbb{N}}$) for the filtration $(\mathcal{F}_n)_{n \in \mathbb{N}^*}.$ Let $\tau=\sup_{k \in \mathbb{N}^*}...
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176 views

English version on Dynkin's 1963 paper on stopping

I am looking for an English version of the following paper: Е. Б. Дынкин, Оптимальный выбор момента остановки марковского процесса, Dokl. Akad. Nauk SSSR 150, 238-240 (1963).
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35 views

Differentiable approximation of Brownian diffusion with unbounded volatility

Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
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1answer
41 views

Differentiable approximation of Brownian diffusion with bounded volatility

Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
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41 views

Occupation time of SDE

Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation $$ X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
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1answer
102 views

Is the topology generated by the convergence of finite-dimensional distributions metrizable?

Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
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39 views

A doubt on the derivation of the Wiener Chaos expansion propagator

So I have seen the following calculation in a number of articles (for instance 1, 2 3) and I just can't get my head around it. The idea is basically as follows, let $\mathcal L$ be some differential ...
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31 views

Expectation of first exit time of a bounded set by a time-homogeneous Ito diffusion is finite

This is a question concerning Remark(i) under Theorem 7.4.1(Dynkin's formula) on Page 124, $\textit{SDE}$, by Oksendal. It says that if $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$ is an $n$-dimensional time-...
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66 views

Is my quadratic variation derivative bounded?

Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
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1answer
72 views

Zeros of a non-degenerate bivariate Gaussian Process

Suppose $(X(t),Y(t))$ $t\in[0,1]$ is a bivariate Gaussian process. We can assume that each component is continuously differentiable, but not necessarily stationary, and that the covariance kernels of $...
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69 views

Translation of Dellacherie's Capacités et Processus Stochastiques

I have been studying the Strasbourg school's general theory of processes from Dellacherie and Meyer's Probabilities and Potential, and I really like it. I have heard very good reviews about another ...
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42 views

Convolution of Wiener measure and measure on $W_0^{1,2}$

Let $F$ by a process adapted to the filtration of a standard Brownian motion. Suppose that the Doleans Dade martingale exists and is a martingale. $F$ is a measure on $W_0^{1,2}$, call it $\nu$. Let $\...
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1answer
136 views

Associativity rule for integration against fractional Brownian motion

In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...
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1answer
95 views

Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure

I'm trying to figure out the connections between two contructions of Gaussian measure. Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
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1answer
149 views

Forwards Feynman–Kac formula

This might be a simple question, but I'm having trouble with it. Consider the Cauchy problem with final condition. \begin{equation} \begin{cases} \frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
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71 views

Continuity of the random variable defining the occupation measure of a continuous Gaussian process

Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable $$ X_\alpha = \lambda( \{t \; : \; Z(t) &...
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1answer
60 views

Local inverse bound of Cameron Martin and Banach norms

Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
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1answer
69 views

Onsager--Machlup functional as the density across a mesh of discrete points

It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint ...
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93 views

Derivative of the function of random variable

Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...
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1answer
51 views

Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?

Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...
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24 views

When is the dual infinitesimal generator of a S.D.E self-adjoint and negative definite?

Given a S.D.E and the dual of its infinitesimal generator $\cal L^*$ (as given below), are there general conditions known ("iff"?) when this $\cal L^*$ would be, self-adjoint i.e $\int f ({\...
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25 views

Regarding the definition of second order mean square derivative

Suppose we have a stochastic process $X$ on $\mathbb{R}$. Suppose at time $t$, there exists a random variable $X'(t)$ such that $$\lim_{h\to0} \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-X'(t)\right)^...
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32 views

Quadratic variation of generalized stochastic integrals

My question is based on this paper: https://pdfs.semanticscholar.org/0b5a/e41096a3b16d0756a1d36da55143d861ed7c.pdf. In summary, this talks about the generalization of stochastic integrals to a two ...
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1answer
86 views

Mean square derivatives and modifications

Suppose we have a stochastic process $X$ on $\mathbb{R}$. Suppose there exists a stochastic process $\frac{d X(t)}{d t}$ such that $$\lim_{h\to0} \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-\frac{d X(...
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2answers
122 views

Regarding sample continuity of Gaussian Processes

Suppose we have a Gaussian Process $X_t$ on $\mathbb{R}^n$ with mean function $m(t)$ and covariance function $K(t,s)$. Then is $X_t$ being sample continuous (i.e. the sample paths of $X_t$ are almost ...
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76 views

Pedestrian proof of Gaussian chaos for order-two polynomial?

Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
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53 views

Conditions for existence of a semi-martingale representing a system of probability measures

Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$. Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
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147 views

Local martingale but not martingale

For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process $Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
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58 views

Good proof of vector version of Ito Formula

does anyone have a good proof of the vector version of the Ito formula? $$f(t,B_t)=f(0,0)+\int_0^t \frac{\partial f}{\partial t}(s,B_s)ds+\int_0^t \frac{\partial f}{\partial x}(s,B_s)dB_s+\frac{1}{2}\...
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1answer
63 views

Conditions for Gaussianity of SDE

Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
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44 views

existence and uniqueness of solution to CEV model sde

Suppose that you have the CEV model for a stock price following the sde $$dS_t = r S_t dt + \sigma S_t^{\eta} dw_t$$ where $ 0 \leq \eta\leq 1$, $S_0=s_0$ and $w$ is a Brownian motion. How do you ...
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1answer
67 views

Time-Reversal of BSDE = SDE

Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$: $$ Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W; $$ ...
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49 views

Reference on infinitesimal generators for functional SDEs

I am trying to solve a problem using convergence of infinitesimal generators of functional SDEs. I havt yet found good material on this. The setting is that of Wanatbe Ikeda -88 page 167. It looks ...

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