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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Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to ...
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33 views

Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade. I could not find a pdf version online, and my university library does not have a printed version. Thank ...
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24 views

Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous). I'm interested in the function ...
2
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0answers
160 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), ...
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75 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the ...
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52 views

A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
3
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1answer
128 views

How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here. Say we have a stochastic process described by a ...
0
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60 views

Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let \begin{equation} I = \int_D g(\textbf{x})d\textbf{x}, \end{equation} where $D \subset ...
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27 views

Fubini's Theorem for Lévy bases

Let $M$ be an infinitely divisible independently scattered and homogeneous random measure on $\mathbb R^d$ (ie a homogeneous Lévy basis). Let $\nu$ be a sigma finite measure on $\mathbb R^k$. Let ...
2
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1answer
134 views

Question on Wiener processes not hitting 0

Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true: $$\mathbb ...
1
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1answer
96 views

Limit (Convergence) of stopping times

Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
2
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81 views

Hypergeometric function

Suppose that $V$ follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that $F(V)$ must ...
0
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38 views

regularity of the conditional expectation: from Markov to Non-Markov

Let $B=(B_t)_{0\le t\le T}$ be a standard Brownian motion and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Let $\xi=\xi(B)$ be a bounded measurable functional. Now let's ...
0
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0answers
31 views

Distribution of stopped Brownian motion in $\mathbb R^2$

Let $B=(B^1_t,B^2_t)_{t\ge 0}$ be a standard Brownian motion in $\mathbb R^2$. Let $U=(U^1,U^2)$ be an independent random variable taking values in a circle $C_1\subset\mathbb R^2$ with uniform ...
0
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0answers
53 views

limit multiple integral

I want to know if $\lim_{T-> \infty}$ of this integral $$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\ \times ...
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35 views

Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?

The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but ...
3
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125 views

Proof of Feynman Kac formula

I am trying to write a complete proof of the Feynman Kac formula in the multi-dimensional case. My starting point was the proof of the univariate form on wikipedia, at ...
1
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1answer
102 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...
2
votes
1answer
55 views

Decouple system of SDEs / handle scaling problem

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...
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32 views

Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
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1answer
63 views

Brownian motion increments

We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too. Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$ and I seek another Brownian ...
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106 views

Does the martingale property holds after changing filtration?

Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by ...
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63 views

Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$? For examples: are there: Ito ...
2
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1answer
134 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
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0answers
66 views

Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
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34 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
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74 views

Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...
0
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1answer
124 views

Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
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Does a theory of stochastic differential algebras exist?

My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
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56 views

Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours. Now, we define a space of functions $S:= ...
0
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1answer
85 views

Weak convergence of process

Background: I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist: Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) ...
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0answers
83 views

Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
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1answer
63 views

Definition: Grigelionis Process?ch [closed]

Background I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...
1
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1answer
109 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
2
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1answer
63 views

Quadratic variation and the variance of a semimartingales

I will describe an example that seemingly contradicts the following Theorem For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then ...
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53 views

Holomorphic solution to SDE

Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$. Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...
3
votes
2answers
171 views

Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
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0answers
83 views

Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable? This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion ...
3
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1answer
115 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
4
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110 views

Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold. My question is, is there any research studying this idea? ...
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0answers
99 views

Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes: [1] F. Soucaliuc, Réflexion entre deux diffusions ...
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1answer
83 views

Weak existence for modified Tanaka SDE

Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE $dX_t = dW_t + dL_t^0(X_t)$, where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at ...
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102 views

Expected value of product of Ito integrals

Assume that we have a process $F(t,T)$ that fulfills the following SDE. $$ dF(t,T) = \sigma(t,T)F(t,T)dW(t) $$ where $t$ is the running time and $T>t$ is called the delivery-time. $\sigma(t,T)$ is ...
2
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1answer
156 views

using Feynman-Kac formula

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
2
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1answer
81 views

Differentiability of value function

Suppose $X$ is a process given by - $dX_t = db_t$ where $b_t$ is a standard Brownian motion with its filtration $(\mathcal{F}_t)$. Suppose an agent earns a payoff given by $V(x) = \mathbb{E} ...
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112 views

Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} ...
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1answer
175 views

Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ...
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1answer
331 views

What is the derivative of this integral?

I have asked this question here http://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral but still has no response. Might I ask it here ? Let $\alpha(t)\in\{0,1\}: ...
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1answer
133 views

Change of variable for integration with respect to Haar measure

I know how to estimate the integral* (see the update) \begin{gather} \int f(Ub)d\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2] \end{gather} where ...
3
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90 views

European call option pricing under mean reverting stock return

Consider the stock price process satisfies the following SDE: $dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s $ and the mean return $\mu_t$ satisfies the following SDE: $d\mu_t=(a-\mu_t)dt +dB_t, ...