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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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 ...
4
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1answer
213 views

'Nonclassical' abstract Wiener space

Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...
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47 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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0answers
22 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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59 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, ...
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1answer
83 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 ...
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1answer
105 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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1answer
83 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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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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0answers
69 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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51 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:= ...
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3answers
667 views

What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?

There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point ...
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193 views

stochastic control / geometric mean

Consider the following problem: Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
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1answer
52 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 ...
3
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1answer
225 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
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1answer
1k views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
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1answer
45 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 ...
3
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3answers
207 views

Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$ dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC $$ where $W$ is a standard Brownian motion ...
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0answers
42 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
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2answers
154 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
62 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 ...
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1answer
105 views

Generalized Ito's lemma

I have the following quantity: $$ g(t)=(f(t))^{M_{t}}, $$ where $M_{t}$ is a jump process neither Markovian nor Levy and $f(t)$ is a positive, increasing but limited, right-continuous function. How ...
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1answer
404 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
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1answer
65 views

Compactness of cadlag martingales w.r.t. to the point-wise topology

Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...
3
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1answer
102 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 ...
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103 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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96 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
660 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Here is what I mean exactly. ...
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1answer
69 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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82 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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71 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 ...
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1answer
124 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
76 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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1answer
123 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
318 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
106 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 ...
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How the diffusion in the unit ball induce the boundary process on the boundary directly?

This is Example 1.2.3 from Fukushima, Masatoshi, Oshima, Yoichi and Takeda, Masayoshi's book "Dirichlet Forms and Symmetric Markov Processes". In this example, we define a Dirichlet form in the unit ...
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77 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, ...
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45 views

Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ ist standard Wiener. This solution is ...
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2answers
340 views

What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} ...
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0answers
50 views

Expectation, exponential of an additive functional of Brownian motion

I have a question about an additive functional of Brownian motion. Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a ...
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2answers
91 views

A question about Skorokhod embedding problem

The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...
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1answer
64 views

Hitting time of a stochastically continuous process [closed]

Suppose $X$ is 1-d stochastically continuous process with $X(0) = 0$, i.e. $X_s \to X_t$ in probability as $s\to t$ for all $t\ge 0$. Let $\tau = \inf\{t>0: |X_t|>1\}$. [Q.] Is $\tau>0$ ...
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1answer
578 views

Stochastic integrals as honest martingales — exponential damping

We have a given positive martingale ρt, with the dynamics: $$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$ where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...
3
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1answer
439 views

Stochastic integrals as honest martingales — comparison criterion

We have a given positive martingale $\rho_t$, with the dynamics: $$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$ where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t: ...
5
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2answers
196 views

Brownian motion in $n$ dimensions

Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in ...
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54 views

Modify exponential family representation to a semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space. We ...
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48 views

skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$? For example if $E$ is the space of ...
3
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1answer
100 views

Markov-semigroup sobolev inequality

I have a question about the following definition: A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for ...
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75 views

Maximal principle for stochastic heat equation

Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition $u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$. Then up to time $T$, the maximal value of $u$ should be ...