Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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10
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2answers
450 views

Area enclosed by 2-dimensional random curve

Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
9
votes
2answers
344 views

Itô's formula: Differential form

I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book: It is sometimes useful to use the following ...
8
votes
2answers
806 views

Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, ...
7
votes
1answer
111 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
7
votes
1answer
107 views

Weak convergence of stochastic integral

Consider a sequence of processes $Z_t^n$ and a procoss $Z_t$, $t\in[0,1]$ such that all $\int_0^1 Z^n dW$ and $\int_0^1 Z dW$ are martingales. Assume $$\int_0^1 Z_t^n \mathrm dW_t \xrightarrow{d} ...
7
votes
1answer
127 views

Filtrations and Sigma-Algebras and Stopping Times

In a previous post Filtrations and Sigma-Algebras I asked the question: $\textbf{Previous Question:}$ Let $\Omega=\{1,2,3\}, \mathcal{A}=\mathcal{P}(\Omega)$ and $P(\{\omega\})=\tfrac{1}{3}$ for each ...
7
votes
1answer
152 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
7
votes
1answer
812 views

Covariance of Gaussian stochastic process

Could someone help me to figure out solutions of following problems?: Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that ...
7
votes
1answer
44 views

Show uncorrelated, with Brownian motions

I have $W_t$ is a Brownian Motion and $$B_t :=W_t-\int_0^t \frac{W_u}{u}du$$ is also a Brownian Motion. I have to show that these two are uncorrelated. I know for Brownian uncorrelated is ...
7
votes
0answers
128 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
7
votes
1answer
585 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
6
votes
4answers
316 views

Probability of going into an absorbing state

If I have a random walk Markov chain whose transition probability matrix is given by $$ \mathbf{P} = \matrix{~ & 0 & 1 & 2 & 3 \\ 0 & 1 & 0 & 0 & 0 \\ ...
6
votes
0answers
92 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
5
votes
1answer
463 views

Holder continuity of Ito integral

Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder ...
5
votes
3answers
2k views

On hitting time of Brownian motion and Ito's lemma

I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion. I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...
5
votes
1answer
230 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
5
votes
1answer
145 views

Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
5
votes
1answer
54 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 ...
5
votes
0answers
130 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
4
votes
2answers
82 views

Good book that contains stochastic integration, martingales and Lévy-processes?

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes? I have tried reading: ...
4
votes
1answer
287 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
4
votes
1answer
46 views

Show local martingale

I have $\exp(\lambda X_t-\frac{\lambda ^2}{2}t)$ is a local martingale, now i have to know if $X_t$ is also a local martingale. Can anybody help me how i can show this correctly?
4
votes
2answers
164 views

Quadratic variation of the Ornstein-Uhlenbeck process

Let $(X_t)_{t\geq 0}$ be the zero-mean Ornstein-Uhlenbeck process such that $X_0 = 0$ almost surely, i.e. $$X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \quad \qquad (\triangle)$$ On the ...
4
votes
1answer
129 views

Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
4
votes
3answers
554 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
4
votes
1answer
260 views

Find the Stationary Distribution of an infinite state Markov chain

A Markov Chain on states 0,1,..... has transition probabilities $P_{ij}=1/(i+2)$ for j=0,1,....,i,i+1. I'm supposed to find the stationary distribution. So do I take the limit as n goes to ...
4
votes
1answer
115 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
4
votes
1answer
168 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
4
votes
1answer
516 views

Continuous Square integrable martingale Quadratic Variation

We know that given a continuous square integrable martingale there exists unique (up to indistinguishability) continuous, natural and increasing process which is quadratic variation process of the ...
4
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2answers
226 views

Stratonovich SDE coefficient selection

Is it possible to find a strictly positive function $\sigma:\mathbb{R}\to\mathbb{R}$, such that a solution $X_t$ to an SDE $$dX_t=-X_tdt+\sigma(X_t)\circ dB_t,$$ with $X_0$ being arbitrary, is a ...
4
votes
1answer
66 views

Deduce $\partial_tp=-\partial_x(b(x)p)+(1/2)\partial_{xx}(\sigma^2(x)p)$, for $p(x,t|y)$ of $X(t)$ and $dX=b(X)dt+\sigma(X)dW$, $X(0)=y$

I am stuck in this proof... I almost got it, but I must have made a mistake. It is part B that I am getting wrong. Thanks in advance for your help! QUESTION: Let $X$ satisfy the autonomous SDE ...
4
votes
1answer
68 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...
4
votes
1answer
92 views

A question related to reflection principle

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all ...
4
votes
1answer
105 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution ...
4
votes
1answer
221 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
4
votes
1answer
121 views

Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
4
votes
1answer
98 views

What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
4
votes
1answer
79 views

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk. My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , ...
4
votes
1answer
269 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to proof, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of ...
4
votes
1answer
613 views

What is the difference between stochastic calculus and stochastic analysis?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
4
votes
0answers
66 views

Interchangeability of the malliavin derivative with a lebesgue integral

I was curious to know the most general conditions under which a malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a lebesgue integral? I was ...
4
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0answers
367 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
3
votes
1answer
294 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
3
votes
1answer
704 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
3
votes
1answer
587 views

The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ...
3
votes
1answer
38 views

Limit Brownian Bridge Integral

As a solution of the Brownian Bridge SDE, we arrive at the solution \begin{align} X_t = (1-t) \int_0^t \frac{1}{1-s}\ dB_S \end{align} defined for $0 \leq t <1$. In order to show that for any $g ...
3
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1answer
44 views

the convergence in probability of the stochastic integral

In Jacod's Limit Theorems for Stochastic Processes:page 47,thm 4.31 (iii) $X$ is a semimartingale , $H_n$ are predictable process converge pointwise to $H$ , and $|H^n|\le K$ , where $K$ is a locally ...
3
votes
1answer
35 views

A computation using the Ito integral

I was assigned this exercise by my Stochastic Analysis Professor. Exercise. Let $B$ be a one-dimensional Brownian Motion, and consider the following processes: $X_t=\int_0^tB_sds\quad ...
3
votes
1answer
132 views

Representation theorem for local martingales

I want to prove the following local martingale representation theorem. For the statement of the theorems to come we fix a filtered probability space $(\Omega,\mathcal{A},\mathcal{F},\mathbb{P})$ where ...
3
votes
1answer
38 views

A martingale characterization

I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let $X$ be an adapted process. If ...