Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [stochastic-calculus]

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.

0
votes
1answer
17 views

Constructing Correlated Wiener Processes

Construction Hello. I'm reading the attached paper about the construction of correlated processes given a correlation matrix. But I am stuck on equation (2.23) -- surely it should say $c_{ik} . c_{...
2
votes
0answers
32 views

Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
0
votes
0answers
19 views

Conditions for a process to be a martingale

Stumbled upon a problem which raised some questions in my mind, here's the deal : $ dS = (a + b.S)dt + (c + d.S) dW \\ X = \alpha(t).S + \beta(t) $ where $a,b,c,d$ are constants. No conditions were ...
0
votes
1answer
15 views

Questions about Quadratic Variation given by Brownian Motion

We know that for a submartingle $A(t)$, $A(t)-\langle A\rangle_t$ is a martingale where $\langle A\rangle_t$ is its quadratic variation. For processes like $W^3(t)$ ($W(t)$ being standard Brownian ...
1
vote
0answers
17 views

Uniform integrablility of Radon-Nikodym derivatives if measures are locally equivalent.

Before the proof of Girsanov's theorem, we were proving the following result in class:- Lemma- Let $Q$ and $P$ be mutually locally equivalent probability measures on $(C[0,\infty),\mathcal{B}(C[0,\...
0
votes
0answers
30 views

Showing the convergence in probabilty of two Ito integrals

Consider a Brownian Motion, $B(t)$, which are all $\mathcal F_t$ measurable. Let $f,f_n$ be in $L_{ad}^2\{[a,b] \times \Omega\}$, which are the set of stochastic processes that are adapted to the ...
1
vote
1answer
42 views

Prove that $\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0$, for each $t\geq0$, where $B$ standard Brownian motion.

Let $B(t)$, $t\geq0$, be a standard Brownian motion. I would like to prove that $$\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0,$$ for each $t\geq0$. In my class notes, ...
0
votes
0answers
23 views

For any continuous functions g in [0, 1], is there exist a partition sequence such that $[g, g]_{\pi}$ = 0?

Let g be a function defined on [a,b], and partition $\triangle=\{ a=t_0<t_1<\dots<t_n=b\}$ $\delta(\triangle)=\displaystyle\max_{1 \leq k \leq n}(t_{k+1}-t_k)$ and $\pi$ be a partition ...
-1
votes
0answers
25 views

Using Ito's Lemma to compute the process followed by a function

I have the following process $$dS_t = \mu S_t dt + \sigma S_t dz_t $$ and the function $$f(S) = S^2$$ where $$\frac{\partial f}{\partial t} = 0, \frac{\partial f}{\partial S} = 2S, \frac{\partial^2 ...
-2
votes
0answers
19 views

A stochastic calculus problem [closed]

enter image description here It's a stochastic calculus problem.
0
votes
2answers
38 views

Literature Request: Stochastic Calculus

Hello does can anyone tell me what is required to learn Stochastic Differential Equations? Anybody have any really good resources that make learning the subject easy? Also is it true that not many ...
1
vote
1answer
29 views

Show that $X_t=e^{B(t)}-1-\frac{1}{2}\int_0^te^{B(s)}ds$ is a martingale

The problem tells me to show that $X_t$ is a martingale but I am getting that it is not. Here the assumed filtration is ${\scr F}_t=\{\sigma(X_s):s \le t\}$. Here is what I tried. $$E(X_t|{\scr F}_s) ...
0
votes
0answers
15 views

Meaning of $H_{\mathbb{R}}(W)$

Let $W_{t}$ be Brownian motion. It is said that if $X_{t}$ is an Ornstein–Uhlenbec process of form $X_{t} = \sqrt{2\alpha} \int^{t}_{-\infty}e^{-\alpha(t-s)}dW_{s}, X_{t}$ then belongs to $H_{\mathbb{...
0
votes
0answers
15 views

Uniqueness of the integrand of a stochastic integral

Given a progressively measurable processes $\Delta_s(\omega),\Delta'_s(\omega)$ and real numbers $z',z$, there was a claim that if $$\int_0^T(\Delta_s-\Delta's)\mathrm{d}X_t=z-z'$$ for a non-trivial ...
0
votes
0answers
7 views

Under what conditions does the solution to a mean reverting sde satisfy $E[\sup_{[0,T]} r(t)^2]<\infty$ for all $T>0$

Consider the mean reverting square root SDE $dr(t)=\alpha(\mu-r(t))dt+\sigma \sqrt{r(t)}dW(t)$ Under what conditions on the coefficients does the solution to a mean reverting sde satisfy $E[\sup_{[0,...
0
votes
0answers
17 views

expected time for all balls to be put in urn 2 Ehrenfest model

I am trying to calculate the maximal hitting time of the Lazy Random Walk on the $n$-dimensional hypercube(I know it is $2^n$). I'm using the Ehrenfest urn model. Let $X_t$ be the number of balls in ...
0
votes
1answer
36 views

Introduction to Stochastic Integration book

I know this question has been asked several times (see: here, here, here, here, and here), but what are the "best" books on stochastic calculus: has anyone had experience with enough books on this ...
1
vote
0answers
23 views

Steady-State Workload for a Compound-Poisson-Process (Lévy-Driven-Queue)

Suppose that $X \in \mathbb{C}P(r, \lambda, b(\cdot))$ with the jumps being distributed exponentially with mean $\frac{1}{\vartheta}$. (Hence we are in a special case of compound Poisson input: an $M/...
2
votes
1answer
56 views

Why isn’t Brownian motion differentiable?

Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?
0
votes
1answer
35 views

Partial Derivative for Stochastic Integral

Good day, I am trying to apply Ito's lemma to find an integral but I am struggling with my choice of functions. $\int^ T _0 tdW(t) = T W(T)- \int^T_ 0W(t)dt$ Our version of Itos lemma states the ...
0
votes
1answer
45 views

Ito's Lemma for a Brownian motion

I'm attempting to prove a lemma from a paper, in the context of optimal contracts. $r,\rho,\gamma,\alpha,\sigma$ are all known constants. $dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
0
votes
0answers
24 views

What is the distribution of $dX_t = b\sigma_tdV_t$?

$$dX_t =\sigma_tX^a dW_t$$ $$d\sigma_t = b\sigma_tdV_t$$ $$ dV_tdW_t = \rho dt$$ $V,W$ are two correlated Brownian Motions. $a$ and $b$ is two real valued parameters. If $X_0$, $\sigma_0$, a, b and $...
1
vote
1answer
17 views

Sufficient condition to show density of $\mathcal{E}$ in $\Lambda^2$(Hilbert space)

I am trying to understand the proof that the space of simple processes is dense in $\Lambda^2$. The proof in my lecture notes starts by assuming that for $\phi \in \Lambda^2$ which is orthogonal to ...
0
votes
0answers
27 views

Marginal Distribution of Diffusion Process

Working on a problem that I'm having some trouble starting. I have $X_t = 2t + 3B_t$ for $t \ge 0$ where $B_t$ is a Brownian Motion. I want to find the marginal distribution of $X_t$, as well as $E(...
0
votes
0answers
17 views

Simplifying a probabilistic relation

Consider three independent random variables $A$, $B$ and $C$ and a fixed number $K$. Can we rewrite the $\text{Pr} \left(A \geq K \cap A \geq \max \left(B,C \right) \right)$ as $\text{Pr} \left(A \geq ...
0
votes
0answers
19 views

A question about measurability on the Taylor expansion of Ito's formula.

I am reading the proof of Itō's formula and self-studying stochastic integration. Essentially, the authors proved the formula for dimension $d = 1$ and they ask the learner to do for greater ...
1
vote
1answer
26 views

Optimal Number of Realizations for a Discrete Stochastic Process

I have a curiosity concerning discrete stochastic processes. Let us say we have a discrete stochastic process $X_{i} = \left(x_1,x_2,...x_i,...,x_N \right)$, hence we have N random variables with an ...
1
vote
0answers
19 views

correlation with respect to probability measure [closed]

Is correlation value dependent on the probability space? That is, if I have 2 different variables that are correlation in one probability space, after I make a change of measures, can they become ...
1
vote
0answers
15 views

SDE with stationary Log-normal distribution

Is there a stochastic differential equation whose solution follows a stationary Log-normal distribution? I was thinking in the geometric Brownian motion $$dx = (\alpha x )dt + (\sigma x )db, \quad \...
0
votes
0answers
26 views

min of two variables under different measures

I have a variable $X$ (a default time), which is a random variable in the space associated with probability measure $\mathcal{P}$. And another random variable (i.e. an another default time) associated ...
0
votes
0answers
15 views

How do we arrive at A) Zero coupon dynamics B) Forward rate dynamics under Hull white one-factor short rate model?

Below is the similar analysis from section 3.2.4 (CIR model) given in the book Interest rate models theory and Practice - Brigo
1
vote
1answer
26 views

total variation inequality involving cross variation

It's from Brownian Motion and Stochastic Calculus by I. Karatzas chapter 1, problem 5.7 property (iv) trying to show an inequality wrt total variation and cross variation of martingales. $$\check{\xi}...
0
votes
0answers
19 views

Prove limiting distribution goes to stationary distribution: $\lim_{t\to\infty} \pi_{j}(t) = \overline{\pi}_{j}$

This is a problem I'm struggling with on continuous-time Markov chains. Let $(\pi_{1}(0), \pi_{2}(0))$ be the initial distribution of the process $\pi_{i}(0) = P(X_{0} = i)$, $\pi_{i}(t) = P(X_{t}...
0
votes
0answers
23 views

A right-continuous non-decreasing process is cadlag?

Consider a measureable process $A:\mathbb{R}_+\times \Omega\rightarrow \mathbb{R}$ such that $t\mapsto A_t$ is a right-continuous non-decreasing process. Then is it true that $A$ is cadlag?
0
votes
0answers
24 views

derivative of integral of Brownian motion

\begin{eqnarray}\label{Bht} B^{H}_{t}=\int^{t}_{0}(t-s)^{H-1/2}dW_{s}\,, \end{eqnarray} where $W_{s}$ is a Brownian motion. Then, we can obtain \begin{eqnarray}\label{dBht} dB^{H}_{t}=(H-\frac{1}{2})\...
2
votes
0answers
42 views

Proving a limit of a Markov process

Let $\{X_{t}\}$ be a continuous time Markov process with a phase space consisting of only two states from the set $S = \{1, 2 \}$. Let $1 - p_{ii}(t) = q_{i}(t) + o(t)$ as $t \rightarrow 0$, where $...
2
votes
0answers
72 views

How to calculate expected value of integral?

How to calculate $E \big[(\int ^{t} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^A (\int ^{t+h} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^B\big]$, where \begin{align} \tilde{L}_\alpha (t) = \...
0
votes
0answers
13 views

Normal distribution of integral with Brownian Motion [duplicate]

I am studying stochastic calculus, and one conclusion I saw is $\int_0^t W(s)ds$ is normally distributed where $\{W(s)\}_{0\leq s\le \infty}$ is a Brownian Motion. What is the rigorous way to prove ...
0
votes
1answer
31 views

SDE Integration

Does anyone know how to get the integration of the SDE below (Assume $\sigma \to 0$)? $$\dfrac{\mathrm dS_t}{S_t}=(r_d-r_f)\mathrm dt+\sigma(t, S_t)\mathrm dW_t$$ Thank you in advance! Image Link ...
2
votes
1answer
63 views

Expectation of solution to SDE $dX_t=-\tanh(X_t) dt + dW_t$

How do I calculate the expectation of the process given by the SDE $$dX_t=-\tanh(X_t) dt + dW_t, \qquad X_0=x_0$$ and $W_t$ a Wiener process? If I start with $$ d\left(e^{t/2}\sinh(X_t)\right) = e^{...
0
votes
0answers
18 views

Choleskey Decomposition of a Complex Matrix in MATLAB

I am trying to decompose a cross spectral density matrix (A Complex Matrix) using "chol" command in MATLAB. we know that every positive definite and Hermitian matrix can be decomposed using Cholesky ...
0
votes
0answers
43 views

Moments of a Wiener process evaluated at some random time.

Assume that we have a random variable $T$ that takes values in $[0, \infty)$, and we know that for any continuous integrable function we have that for all $k \in \mathbb{N}$ the following holds $$ \...
1
vote
1answer
33 views

The average time before a person find their group

Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\...
0
votes
1answer
23 views

Solution of a second order Stochastic Differential Equation

Consider the following SDE $$dx = (-ax)dt + \sigma db\\ dy = (-by+e^{-x})dt$$ where $a,b,\sigma>0$ and $b$ is a browninan motion PROBLEM: What is the solution of this system? How can I estimate ...
0
votes
0answers
22 views

A Version of Fubini-Tonelli Theorem for Hilbert Space Valued Functions

I'm currently working on a project in which we define a new type of integral. And I'm trying to intechange the integral with expectation, something like $\mathbb{E} \left[ (\mathcal{N})\int f dW \...
1
vote
0answers
31 views

$B_t^3 - 3t B_t$ is a $L^2$ martingale ($B_t$ being a standard Brownian motion)

By Itô's formula I get that \begin{align} d(B_t^3 - 3t B_t) &= (3 B_t^2 dB_t + 3 B_t dt) - 3(B_t dt + 3 t d B_t) \\ &= (3 B_t^2 + 3t) d B_t \end{align} which seems related to martingale ...
0
votes
0answers
7 views

Exchanging Derivative and expectation

I am trying to explain why the kth derivatives of Mt(λ) = exp(λWt − λ^(2)t/2) are also martingales with respect to Gt. I was wondering that, since the derivative is with respect to λ I could exchange ...
1
vote
0answers
22 views

Second order derivatives in Ito formula for Brownian motion and local martingale

Itô's formula for a $\mathcal{C}^2$ function of two variables F reads: \begin{align} F(X_t, Y_t) &= F(X_0, Y_0) + \int_0^t \frac{\partial F}{dx}(X_s, Y_s) \, dY_s + \int_0^t \frac{\partial F}{dy}...
1
vote
1answer
44 views

Name for integrals where the integral is divided by the range of integration?

I've seen this sort of integral come up when I've been studying approximating step processes to some very simple stochastic processes, but I can't shake the feeling that they link to some more general ...
3
votes
0answers
25 views

$n \times n$ system of stochastic differential equations

Let be $n \times n$ system of SDE $$ dX_t^{(i)} = \sum_{k = 1}^n a_{ik}(t) X_t^{(k)} dt + dW_t^{(k)}$$ where $i = 1, 2, \dots, n$ and $a_{ik}(t)$ are continuous function for $t \geq 0$ and $X_0^{(i)} =...