Questions on the calculus of stochastic processes, or processes that have a random component.

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32 views

Ratio distribution of independent exponentially distributed variables

first things first: I am not a studied mathematician and therefore lack thorough knowledge of the topic - please consider this, even though I will of course try to express myself as accurately as ...
2
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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 ...
2
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1answer
16 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, \{\...
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0answers
13 views

Partial Integration for Semimartingales

Let $X,Y$ be 2 continuous semimartingales. It could be shown that for every $t>0$, \begin{align} X_tY_t = X_0Y_0 + \int_0^t X_s dY_s + \int_0^t Y_s dX_s + \langle X, Y \rangle _t. \end{align} Let ...
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0answers
22 views

Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate ...
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0answers
28 views

Ito formula proof

Is there a simple way to prove $$x=f(t,x_t)\\df(t,x_{t})=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dB_t+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dx_t)^2$$? can we prove it by ...
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0answers
36 views

Options on Futures Black-Sholes

I am taking the Financial Risk Management course, and the topic now is "Variations on the Black-Scholes Model". I am following Paul Wilmott's "The Mathematics of Financial Derivatives: A Student ...
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0answers
85 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\...
4
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1answer
52 views

Ito's formula and Taylor expansions for jumps processes.

Consider some model $$ dX_t = \mu d t + \sigma dW_t $$ where $\mu, \sigma$ are some constants. Now let $f \in C^{1,2}$ and consider $$ Y_t = f(t,X_t). $$ Say we (informally) consider a second order ...
3
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0answers
30 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
23 views

Probability of exit from compact set

I have a continuous real valued diffusion $\{ X_t \}_{t\ge0 }$ that is contained in a compact set $[a,b] $of $\mathbb{R}$, where $a > 0$ and. Define the stopping times \begin{equation} \tau_c=\inf \...
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1answer
29 views

the exact integrand space for stochastic integral?

I found it in Schilling, Partzsch's textbook "Brownian Motion": only consider in $[0,T]$, they define the Dolean's measure $\mathbb P\times\mu$, and the corresponding $L^2$ norm on $L^2(\Omega\times [...
2
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0answers
39 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's "Probability and Measure" (and other books) show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say the joint measurability ...
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0answers
30 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}^...
7
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1answer
48 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 ...
4
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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
19 views

Why does Euler-Maruyama method use a square root of the time step

Euler-Maruyama method is supposed to be an extension of the Euler method for ODE, but applied to SDE. This means that if we have an equation: $$ dY_t = Y_t dW_t $$ where $W_t$ is the Wiener process, ...
1
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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 ...
2
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0answers
30 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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0answers
14 views

Comparing two hitting times of Bessel process

Suppose $X$ is a Bessel process of dimension $1 < d \le 3$ with $X_0 = 0$. Then $X$ satisfies the SDE $ dX_t = \frac{d - 1}{2X_t} dt + d W_t$ for some Brownian motion $W_t$. Let $a > 0$. Let $\...
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0answers
9 views

Show that $d\hat{B}_t = B_t - \frac{c}{t}\int_0^t B_s ds$ is a BM given BM $B$.

I need to show that the solution to the SDE $$\hat{B}_t = B_t - \frac{c}{t}\int_0^t B_s ds$$ is a BM in its natural own filtration. From the fact that $\int_0^t B_s ds$ is measurable w.r.t. the ...
2
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0answers
55 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $(exp(\lambda X_t-\frac{\lambda ^2}{2}t))_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process X ...
1
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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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0answers
11 views

Describe the law of a Bessel process conditioned on hitting $b$ before $0$

We are given the Bessel process SDE $$dX_t=\frac{\delta -1}{2X_t}dt+ dB_t, X_0>0.$$ Where $B_t$ is a standard Brownian motion, at least until $X_0=0$. We need to solve four problems: Show that $...
0
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1answer
33 views

What is the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$?

I am trying to find the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$, but am unable to use Ito's Formula because of the nasty integral. Is there another solution I am missing? Thanks!
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0answers
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 ...
5
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1answer
103 views

Regularity, Dirichlet form

I have a question about Dirichlet form. Let $\Omega$ be an Euclidean domain of $\mathbb{R}^{N}$ and $X=\bar{\Omega}$. The measure $m$ on the Borel $\sigma$ algebra $\mathcal{B}(X)$ is given by $m(A)...
1
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0answers
25 views

How to find the mean of $\int_0^t W_s ds$, where $W_s$ is a Wiener process?

am trying to find the expectation of $\int_0^t W_s ds$, with $W_s$ being the Standard Wiener process. I am trying to use Ito's formula, by decomposing as: $$ \frac{W_t^3}{6} = \frac{1}{2}\int_0^t B_s^...
2
votes
1answer
48 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + \int_0^...
1
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0answers
11 views

In stochastic calculus, what is the importance behind quadratic variation?

I am learning stochastic calculus right now and I came across several mentions of the computation of the quadratic variation of a Wiener process random variable. However, most of the resources I have ...
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0answers
13 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 \...
2
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1answer
26 views

Chain rule for derivatives in SDE

I'm having trouble understanding applying chain rule to SDEs or actually chain rules in general. It has been a while since I took rudimentary calculus classes, so I might be slipping on the basics. ...
2
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2answers
67 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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0answers
13 views

Gaussian Hilbert spaces indexed by a Hilbert space

Let $H$ a real Hilbert space. Then, there is a real Gaussian Hilbert space $G$ indexed by $H$. I know this result is a consequence of Kolmogorov Extension Theorem, but I have not idea of how begin....
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17 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R (\cdot,\...
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0answers
27 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 statement:...
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0answers
12 views

How to formulate and analyze systems of stochastic differential equations?

I'm having trouble finding reference material on how to deal with systems of stochastic differential equations. Specifically, I'm interested in ecological models. For example, consider the standard ...
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0answers
48 views

Martingal-property of stochastic Integral w.r.t. Brownian Motion

To Show that $(e^{B_t^1}cos(B_t^2))_{t \in \mathbb{R_+}}$ (where: $B=(B_s^1,B_s^2)$ is a 2-dimensional Brownian Motion) is a Martingal I used Ito's Lemma and showed that this is equal to: $ 1+ \int_0^...
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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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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 ...
2
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1answer
47 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 ...
1
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1answer
37 views

Lebesgue measure of union of semi-open interval

Given $\mathbf{A} = \bigcup_{n\geq0}[n,n+ \frac{1}{2^n}[$ and the Lebesgue measure $\lambda$, find $\lambda(\mathbf{A})$. My solution: \begin{align} &\lambda\left(\bigcup_{n\geq0}[n, n+\frac{1}{...
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0answers
19 views

Markov property of ito diffusion [duplicate]

Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$. In this ...
1
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0answers
27 views

Itō isometry in Hilbert spaces

Let $U$ and $H$ be separable Hilbert spaces $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $\mathfrak L:=\mathfrak L(U,H)$ be ...
2
votes
1answer
94 views

Ito integral of average of the square of a Wiener signal?

How do we evaluate the average of the square of a Wiener signal? Standard case: Typically, the signal average is $S(t)=\frac{1}{T}\int_{0}^{T}s(t)dt$, where we can write the integral in Ito form $S(...
2
votes
1answer
18 views

Product of deterministic function and Ito process

In a case such as the Cox-Ingersoll-Ross where $$ \mathrm{d}{R\left(t\right)}=\left(\alpha-\beta R\left(t\right)\right)\mathrm{d}{t}+\sigma\sqrt{R\left(t\right)}\mathrm{d}{W\left(t\right)}, $$ is it ...
0
votes
0answers
38 views

Ito's formula application

Let $ \alpha, \beta \in R$ and define $$ N(t)=e^{\beta t} \cos(\alpha W (t)) $$ I need to use Ito formula to compute $dN(t)$ Suppose $\alpha$ is fixed. What should $\beta$ be so that $N$ is a ...
0
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0answers
35 views

Yet another application of Ito's formula

Question : Let $dW^4(t) $ be the sum of an ordinary integral with respect to time and an Ito integral. Where $W^4(t)$ are standard Brownian motion. I am trying to apply Ito's formula to this, say ...
1
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1answer
47 views

SDE Solution: Hull-White extension of Vasicek model

I am trying to figure out the particular ansatz (if that's all there is) for the solution to the SDE: $ dr_t = [v_t - ar_t]dt + \sigma dW_t, $ where $a$ is constant and $v,t$ are, potentially, time-...
2
votes
0answers
23 views

Use of Itô isometry for correlation calculation

When calculating the covariance of the Ornstein-Uhlenbeck process, the Wikipedia article applies implicitly the Itô isometry with the fact of non-overlapping independent increments of the Wiener ...