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

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Probability of a right angled triangle with sides a+b=200 having hypotenuse ≥ 160

QUESTION: A $200\, cm$ long staff breaks in two at a random point. The two parts becomes the right sides of a right angled triangle. What is the probability of the hypotenuse being at least $160\,cm$? ...
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1answer
17 views

Two exponentially distributed random variables w/ different intensity. Which is more probable to take?

Let's say I have two types of light bulbs, A which has $E(A)=100$ hours of lifetime, and B which has $E(B)=200$. I have three of type A and one of type B. I randomly use one of the four, and after 200 ...
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1answer
41 views

Existence and uniqueness of SDE, is the independence requirement needed?

In Bernt Øksendals Stochastic differential equations he has this theorem in chapter 5: $\\\\\\$ However, in the proof I can not see where he uses the independence condition I marked in red. Do you ...
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2answers
126 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: http://www.cambridge.org/us/...
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0answers
53 views

Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ dX_s,...
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2answers
52 views

Probability of rectangles area being less than 0.5 w/ total length of sides = 2

Question: A random point splits the interval [0,2] in two parts. Those two parts make up a rectagle. Calculate the probability of that rectangle having an area less than 0.5. So, this is as far as I'...
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2answers
32 views

Integration by parts - Brownian motion and non-random function

Let $B$ be a standard one-dimensional Brownian motion. I want to show for a continuously differentiable non-random function $\phi$ that, \begin{align} \int_0^t \phi(s) dB_s = \phi(t) B_t - \int_0^t ...
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0answers
18 views

Is the generated semigroup by an elliptic operator be the transition semigroup?

I am considering the time homogeneous Ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where $b,\sigma$ currently are only assumed to be global Lipschitz for the existence of solution $X_t$. The ...
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1answer
53 views

Quadratic Variation Brownian motion martingale (2)

Let $B_t$ be a standard Brownian motion and $M_t = B_t^2 -t$. From here we are aware of the identity \begin{align} [M]=[B^2]. \end{align} Now, I want to apply Itô's formula to $B_t^2$ and from that ...
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1answer
32 views

What is the Difference Between a Version and a Modification of a Stochastic Process?

Under what circumstances would one say that: The stochastic process $X$ is a version of the stochastic process $Y$? Background: See here for a related but slightly different question on ...
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1answer
32 views

càdlàg adapted process of finite variation

$X$ is a semimartingale with $X_0=0$. I have to show, that $S_t:=\prod^{}_{s\le t}(1+\Delta X_s)\exp(-\Delta X_s)$ is a càdlàg adapted process of finite variation. Could you please help me?
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0answers
25 views

Distribution of a certain stochastic process

Consider on a probability space $(\Omega, \cal F, \mathbb P)$ the following stochastic process on $[0, \infty]$, where $W(t)$ is a Wiener process, all the coefficients $\lambda(t), \mu(t)$ and $\sigma(...
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0answers
15 views

Spectral Density of an ARMA process.

For an upcoming Stochastic Processes exam, we have had a sudden brief email about Spectral Density as the lecturer had forgotten to mention it in classes. He states, For an ARMA process with $\phi(z)$...
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33 views

Quadratic Variation of Brownian motion martingale

Let $B_t$ be a standard Brownian motion and let $M_t = B_t^2 -t$. I want to show that $[M]=[B^2]$. Therefore I want to use the linearity of quadratic variations \begin{align} [\alpha X + \beta Y, Z] =...
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0answers
28 views

Itô's Lemma in proof of Feynman-Kac's Formula

Let $h$ and $V$ be bounded continuous functions on $\mathbb{R}^d$. Suppose $u$ is continuous on $\mathbb{R}_+ \times \mathbb{R}^d$, bounded on $[0,T] \times \mathbb{R}^d$ for eacht $T < \infty$ and ...
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1answer
122 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 ...
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38 views

Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge 0}...
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2answers
68 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
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0answers
44 views

Feller property of Ito diffusion

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded and continuous, ...
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1answer
23 views

Equivalence between two different representations of exponential Lévy Processes

My questions are: Why do I know that $\frac{Z}{Z_-}$ looks like in the proof? Why $\int \frac{d[Z^c]}{Z_-^2}=[Y^c]$? Why does the part with the sum look like the one below? I only know that $f(x)*\...
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0answers
32 views

Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that $[...
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1answer
63 views

Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
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31 views

local martingale $\exp(\lambda X_t-\frac{\lambda^2}{2}t) $ is stochastic exponential

I have an $\mathbb{R}$ valued process $X$ which is an $\mathcal{F}^X$ Brownian motion if and only if for all $\lambda \in \mathbb{R}$ $ M_t:=\exp(\lambda X_t -\frac{\lambda^2}{2}t)$ is a $\mathcal{F}^...
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1answer
48 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?
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1answer
296 views

Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > 0$ ...
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1answer
49 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 ...
3
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30 views

How to derive the distribution of measurement noise in discrete Kalman filter which is transformed from continuous one?

With sampling time $T$, and a continuous measuring model: $$ \begin{align} y(t) &= Cx(t)+v(t) \\ v(t) & \sim \text{N}(0,R_c) \end{align} $$ we can change it into a practical discrete one, ...
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0answers
31 views

Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
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28 views

Deduce stochastic differential equation

Let $X$ be a stochastic process with $dX_t = \alpha X_t dt + \sigma X_t dW_t$ and $Y$ a stochastic process with $dY_t = \gamma Y_t dt + \delta Y_t dV_t$, where $W$ and $V$ are independent Wiener-...
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2answers
45 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle W,\phi\rangle:=\int\...
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Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle W,\phi\rangle:=\int\...
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0answers
26 views

showing a processes is martingale using ito's lemma

Let $Y(t) = t^2W_t - 2 \int_0^t sW_s \ ds$ where $W_t$ is brownian motion. I am trying to show it is a martingale by showing it is driftless. I set $Z(t,W_t) = t^2W_t$ and ito's gives $dZ = 2tW_t \ dt ...
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2answers
76 views

Showing that this is a martingale.(4.13 in Øksendals SDE)

This is an exercise from Øksendals stochastic differential equations, where I get stuck. It is exercise number 4.13.(I simplified the notation a bit.) I have that X is an Itô-process where: $...
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1answer
88 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
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1answer
33 views

Is the stochastic integral of the jumps process equal to zero for a continuous integrator?

Let $X$ be a continuous semimartingale and $H$ a progressively measurable process in $L(X)$. Assume $H$ has left limits almost surely. I claim that the jumps process of $H$, denoted by $\Delta H = H - ...
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59 views

Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
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15 views

For stochastic differential equations, why do we care if the process is $L^2$ bounded?

I have been studying Stochastic Differential Equations, and one theorem relates to the existence of a solution to the SDE: $$ dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dB_t $$ with $X_0 = x_0$ and $0 \...
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20 views

Estimate for average probability of Ito diffusion falls into an interval

Denote $E^x(X_t)$ be the solution to a Ito diffusion starting with $X_0=x$. Let $K\subset \mathbb{R}$ be a compact subset. I also assume $X^x_t$ has transition probability $p(t,y,x)$. Currently I am ...
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2answers
51 views

Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
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1answer
773 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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2answers
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Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces $(U,\langle\;\cdot\;,\;\cdot\;\...
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1answer
25 views

brownian motion and process C1 (order 1 of continuity)

Here is my problem, With probability 1 (ie: a.s) the brownian motion $(B_t)_{t\in[0,T]}$ is continuous (which is define on a classic probability space $(\Omega, \mathcal{F}, (\mathcal{F}_{t})_{t>0}...
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1answer
73 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
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28 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where     &...
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Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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15 views

Gaussian filtering

I'm reading a paper and don't get how they tackle the drift of a gaussian process. We are in the setting of isonormal Gaussian processes. Let $Z$ be a Gaussian process with covariance operator $\...
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0answers
27 views

Derive an Itō formula for $f(t,X_t)$ where $X_t=X_0+tY+W_tZ$ and $f:[0,\infty)\times H\to\mathbb R$ and $H$ is a Hilbert space

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be Fréchet ...
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0answers
29 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
2
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1answer
49 views

Is this an adapted process?(deterministic integrator in Itô-process)

Assume you have a probability space with a filtration, $(\Omega,\mathcal{F},P,\{\mathcal{F}_t\})$. Assume that the stochastic process $X_t$ is adapted to this filtration, and is jointly measurable ...
4
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1answer
115 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 $X_t$...