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

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2
votes
0answers
42 views

Cross Variation of two stochastic processes

I am currently working on a stochastic calculus exercise at the moment and I am slightly confused when it comes to finding cross variation. We are given that the process $X_t = W_t^3$ ($W_t$ is ...
3
votes
3answers
64 views

Showing time changed brownian motion is martingale.

Let $W$ be a one dimensional Brownian motion and define, $$ X_t=W_{(\text{exp}(\beta t)-1)}\\ \hat{W}_t=\frac{1}{\sqrt{\beta}}\int_0^te^{-\frac{\beta s}{2}}dX_s $$ Show that $\hat{W}_t$ is a local ...
1
vote
1answer
28 views

Ito's Isometry using Brownian Motion

Let $B_t$ be standard Brownian Motion. Could someone please help me to show that $$E[(\int_{0}^{t}B_sdB_s)^2] = \int_{0}^{t}E[B_s^2]ds$$ I am sure that it has something to do with Ito's formula but ...
0
votes
1answer
42 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
1
vote
1answer
40 views

Construction on Ito Integral with Brownian Motion

I have just started learning stochastic calculus and my professor posed the following as exercises to help understand how we construct the Ito Integral. Let $B$ be a standard Brownian motion. Fix ...
0
votes
1answer
51 views

$e^{X_t - \frac{t^3}{6}}$ is a martingale - show it [closed]

I am trying to use Ito's integral properties to prove it is a martingale, but am getting stuck in the preliminaries. More so, I wanted to confirm, do I use this formula?
0
votes
2answers
34 views

What is a valid range of applicability of Ito Lemma?

If I have e.g. such process $$ Z_{t}=t^{5}B_{t}+10\int_{0}^{t}sB_{s}ds $$ can I take $$ f(t,x):=t^{5}x+10\int_{0}^{t}sB_{s}ds $$ as a function to which I apply Ito formula? I'm concerned about ...
1
vote
0answers
21 views

Covariance between random variables in a stochastic differential equation

Suppose I have a SDE of the form: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda ...
2
votes
1answer
26 views

how to derive the stochastic differential equation of this process

How can I derive the SDE for the vasicek model : $$r_t = 0.1 + 0.1 e^{-t} + e^{-t}\int_0 ^t e^s dB_s$$ From observation, the SDE vasicek's model is such that: $$dr_t = b(a-r_t)dt + \sigma dB_t$$ ...
2
votes
0answers
33 views

continuous time super martingale

I am trying to prove that if I have super-martingale $(S_t,F_t)_{t\geq0}$ right continuous, and $\tau <\infty$ stopping time that $(S_{\tau \wedge t},F_{\tau \wedge t})$ also super martingale. I ...
5
votes
1answer
35 views

$x_t := a_t -b_t c_t $ , with $dx_t = \theta (\mu-x_t) dt+ \sigma dW_t$

I would like to solve the following equation explicitly using Ito's lemma: $$ x_t := a_t -b_t c_t , $$ where $x_t$ is an Ornstein-Uhlenbeck process (see here) $$ dx_t = \theta (\mu-x_t) dt+ \sigma ...
2
votes
0answers
25 views

Variance of Riemann integral of Stochastic integral

Let $f: \mathbb{R} \to \mathbb{R}$ be deterministic and let $W$ be a standard Brownian motion. Then by Ito's isometry we know $$ Var\left( \int_0^u f(s) dW(s) \right) = \int_0^u f^2(s) ds. $$ Now, ...
3
votes
0answers
30 views

Does Ito's Isometry hold if the integrand has a brownian motion in it?

I am wondering what is the distribution of: $$ \int_0^tW_sdW_s $$ Solution: (Thanks to @muaddib) Applying Ito's Formula to $W_t^2$ gives $d(W_t^2) = 2W_tdW_t +dt$, and so: $$ \int_0^tW_sdW_s= W_t^2 ...
3
votes
0answers
62 views

Finite Moments of complicated Stochastic Differential Equation

Suppose I have a SDE of the form: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda ...
1
vote
0answers
25 views

Stopping time, trace, filtration

I have a question about probability theory. Let $(\Omega,\mathcal{F},P)$ be a probability space. The completion of $\mathcal{F}$ w.r.t $P$ is denoted by $\mathcal{F}^{P}$ Given a ...
0
votes
1answer
18 views

SDE solution using Itô formula

I'd like to solve the Langevin SDE $$dX(t)=-bX(t)dt+\sigma dW(t),\\X(0)=X_0,$$ $W(t)$ being a standard Brownian motion, using the Itô formula $$du(t,X(t)) = \frac{\partial u}{\partial t}dt + ...
2
votes
1answer
85 views

Stochastic calculus book recommendation

I'm a quantitative researcher at a financial company. I have a PhD in math, but I'm an algebraist, so I only took the two required analysis courses in grad school (measure theory for the first, and I ...
2
votes
0answers
14 views

How to solve SDE that looks like OU process

I'm trying to figure out how to solve the following SDE, $$ dZ_t = -\kappa(Z_t-\mu)dt + Z_tdW_t $$ It looks really similar to the OU process but applying the integrating factor approach which ...
1
vote
1answer
30 views

Itô's formula yields an Itô process

In our course on stochastic analysis, we proved the following version of the one-dimensional Itô formula: Let $\{W_t\}_{t\ge 0}$ be a one-dimensional Brownian motion w.r.t. some (right-continuous and ...
2
votes
0answers
18 views

Can we integrate brownian motion with respect to a deterministic function

Let $B_t$ be brownian motion at time $t$, and $x$ be some random variable. For instance, I know that $$\int_0^T 1 dB_t = 1(B_T-B_0)$$ And that $$\int_0^T \cos(B_t) dB_t$$ cannot be directly ...
0
votes
0answers
21 views

Partial differential equation involving a random process (literature advice)

In articles like this one (end of page one and page two), physicists often tend to treat a random process with discrete time and countable space set as a differentiable function (whose domains are ...
2
votes
0answers
28 views

Construction of Brownian motion - differentiability

I'm working on the the construction of BM given by Lévy-Ciesielski. The author begin to prove another result and for this reason he assume that BM exists and that it is also differentiable. For this ...
2
votes
1answer
20 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
0
votes
0answers
29 views

Expectation of an ito process

I came across this sub-question as a part of a bigger question, the question itself seems very simple but I'm having hard time figuring out a solution. Just to give a little background, this comes in ...
1
vote
0answers
33 views

Modified Stochastic Integral

Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$ Suppose ...
1
vote
1answer
47 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...
1
vote
0answers
48 views

Superposition of renewal processes: Variance of lifetimes

I've a question concerning the superposition of renewal processes. Assume we have $n$ independent renewal processes with the same lifetime distribution (especially mean $\mu$ and variance $\sigma^2$). ...
3
votes
1answer
28 views

Stochastic calculus rules $d(B_t^2) = 2B_t\,dB_t + dt$ - why?

Let $B_t$ = Brownian motion at time $t$ I know that $(dB_t)^2 = dt$ and $d(f(x)) = f'(x)\,dx$ for some differentiable function. Now, I have that $$M_t = B_t^2 - t$$ $$dM_t = d(B_t^2) - d(t)$$ ...
1
vote
1answer
38 views

How to check if integral wrt Brownian motion is a martingale

As in title, I have a process $$X_{t}=\int_{0}^{t}s^{2}dB_{s}$$ I found here a sufficient condition for such integral to be a martingale on the interval. But I am asked if it is a martingale, not ...
4
votes
1answer
44 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, ... , ...
0
votes
1answer
48 views

Prove this expectation of Brownian motion?

Prove $E[(\Delta B_j)^4]=3(\Delta t_j)^2$ where the Delta stands for the change of something i.e $B_j-B_{j-1}=\Delta B_j$ and the $B_j$ stand for the standard Brownian motion I won't show my step ...
1
vote
0answers
14 views

What is the Euler Lagrange condition for SDEs?

Does the Euler Lagrange condition... $$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0$$ ...have a meaningful extension to Stochastic Differential ...
5
votes
1answer
77 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 ...
1
vote
0answers
63 views

Can these random variables be seen as products of indicator functions?

Spin-off from here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ How about using indicator functions? I was thinking that $X_n = 2^n 1_{A_1}$, but I guess ...
0
votes
1answer
43 views

Reasoning in “Prove X is a martingale” [duplicate]

From here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ Why exactly? In retrospect, I'm not sure I really got it. I'm trying to think about it in terms of ...
1
vote
1answer
55 views

$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$

Under what conditions does it hold that $$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$$ if we are given a stochastic process $X = (X_n)_{n \geq 0}$ on a filtered probability space $(\Omega, ...
3
votes
1answer
93 views

Reversible Ito Diffusions

I have given a diffusion equation $$ dX_t = -\nabla V(X_t) \, dt + \sigma dB_t.$$ I found here(1) a characterization when $X_t$ is reversible, aslong as $\sigma=1$. Is this also true for $\sigma ...
1
vote
1answer
27 views

Expectation of a stochastic process

Ok, I'm new to stochastic calculus and I'm having some troubles with a simple exercise that I don't seem to get. Here it is: Recalling that $\mathbb{E}[e^{W_t}]=e^{\frac{t}{2}}$ compute ...
0
votes
1answer
42 views

Proving existence of Itō Integral

Here's an extract from some Continuous Martingales notes I can see how K-W implies the blue box inequality but how does that inequality give continuity? Also what is the functional theorem that ...
0
votes
0answers
18 views

Expectation of Modified Stochastic Integral

Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$ Suppose ...
-2
votes
1answer
43 views

Calculate a differenciation [closed]

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
3
votes
1answer
38 views

Stochastic calculus - Ito confusion

We have $W(t) = f(t)X(t)$. My textbook says that $dW = fdX + X\dfrac{df}{dt} dt$. I don't get how they arrived at this conclusion. I get the first part, because $\dfrac{dW}{dX}dX = fdX$. But for the ...
1
vote
0answers
53 views

Ito's lemma - mistake in text book?

Ito gives us $$dW = \dfrac{dW}{dX} dX + \left(\frac{dW}{dt} + \frac{1}{2} \frac{d^2W}{dX^2}\right) \, dt$$ We have a function $W(t) = 1 + t + E^{X(t)}$. My text book says that $$dW = e^{X(t)} \, dX + ...
0
votes
0answers
14 views

zero drift brownian motions and barriers problem [duplicate]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > ...
1
vote
1answer
33 views

Does Brownian Motion return to the origin infinitely soon? [closed]

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
0
votes
0answers
25 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
2
votes
2answers
60 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
2
votes
1answer
43 views

What is meant by a linear SDE?

I am sure this is a ridiculous question, but I can't seem to find a definition. I know the definition of linear ODE or PDE just by saying that the differential operator should be linear, but how does ...
0
votes
0answers
23 views

Solving an integral to solve a statistical problem

In solving a statistical problem, I want to know $$\mathbb{P}( (Y_2 \leq q_\alpha | Y_1 \geq q_\alpha) $$ where $Y_1 = X_1 + Z$ and $Y_2 = X_2 + Z$ and $Z \sim N(0,\sigma_z^2), X_i \sim N(0, ...
0
votes
0answers
29 views

Stochastic Differential Equation for Time Integral of Stochastic Process

Let $X(t)$ denote standard Brownian motion $dX(t) = a X dt + X dW(t)$ with solution $X(t) = e^{a t + W(t)}$. I want to consider the time-integrated process \begin{equation} Y(t) := \int_0^t d\tau~ ...