Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

learn more… | top users | synonyms

0
votes
0answers
38 views

How it is shown by the following integral?

Example: Ornstein-Uhlenbeck Process. Let $ dx=-\eta xdt+\sigma dz $ be an Ornstein-Uhlenbeck Process Write the moment-generating function for $x(t)$ as $$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ϕ(x_0,t_0;x,...
3
votes
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,...
0
votes
0answers
38 views

stochastic differential equation exact solution

whats (is there) exact solution of (for) this sde? $dX_{t}=\mu X_{t}dt+\sqrt{\sigma X_{t}} dW_{t}$ and what's the distribution of that? thanks
1
vote
0answers
37 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
0
votes
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 ...
5
votes
0answers
60 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
0
votes
0answers
32 views

Why an optional process could not be predictable?

We know that a predictable process is also optional (*). Why an optional process could not be predictable ? Why we cannot use the same arguments as the proof for (*) ?
0
votes
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 ...
2
votes
1answer
37 views

discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...
3
votes
0answers
26 views

A stochastic volatility model

An example of stochastic volatility model: $$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases} $$ where $Z_t$ and $W_t$ are Brownian motions and $...
0
votes
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 $\phi(z)$...
0
votes
1answer
25 views

Application of Ito's rule

I have that $\sigma$ is a piecewise continuous function on $[0,t]$, $W$ is Brownian motion, $X(t)=\int_0^t\sigma(s)dW(s)$, and $Z(t)= e^{iuX(t)},$ for some fixed $u\in\mathbb{R}$. It is then stated ...
1
vote
0answers
62 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
1
vote
0answers
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
votes
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
votes
1answer
17 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}, \{\...
0
votes
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 ...
1
vote
0answers
27 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 ...
0
votes
0answers
30 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 ...
1
vote
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 ...
7
votes
0answers
91 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
votes
1answer
85 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
votes
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 $[...
0
votes
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 \...
0
votes
1answer
32 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
votes
0answers
48 views

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

Motivation: 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 ...
0
votes
0answers
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}^...
7
votes
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 ...
4
votes
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?
0
votes
1answer
20 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
vote
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
votes
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 ...
0
votes
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 $\...
0
votes
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
votes
0answers
61 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $\left(\exp\left(\lambda X_t-\frac{\lambda ^2}{2}t\right)\right)_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{...
1
vote
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 ...
1
vote
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
votes
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!
0
votes
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
votes
1answer
104 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
vote
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
51 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
vote
0answers
13 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 ...
1
vote
0answers
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 \...
2
votes
1answer
33 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. ...
3
votes
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: $...
0
votes
0answers
14 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....
0
votes
0answers
20 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,\...
1
vote
0answers
29 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:...
1
vote
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 ...