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

learn more… | top users | synonyms

0
votes
0answers
23 views

Basic Stochastic Calculus

Let $B_t$ be brownian motion. Then if I need to calculate $\mathbb{E}[2(B_2-B_0)+(B_2+B_1)(B_3-B_2)]$ is this simply $0$ as independence results in: $\mathbb{E}[2(B_2-B_0)] + \mathbb{E}[B_2+B_1] ...
1
vote
0answers
11 views

Pricing an option on a mean-reverting assets

In an universe we have two assets and a predictor: $\frac {dS_{1,t}}{S_{1,t}}=(\mu_{1,1}+\mu_{1,2}X_t)dt+\sigma_{1,1}dB_{1,t}+\sigma_{1,2}dB_{2,t} $ $\frac ...
1
vote
0answers
15 views

Functions of Brownian Motion and Time

Sorry, this will be a little long. I'm currently working on a problem where I basically have an SDE logistic equation: $$dX_t = diag(x_1,\cdots, x_n)[b+Ax-\lambda \eta(t)] dt + diag(x_1,\cdots, ...
-1
votes
1answer
62 views

How do I solve this SDE (stochastic differential equation)?

I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} Here, $b$ is a constant. I am trying to apply my usual methods for ...
2
votes
1answer
42 views

Stochastic integral where the integrator is zero in probability

We are given a continuous semimartingale $Y$ and a continuous process $B$ of finite variation. Hence, we know that $\langle B \rangle$, the quadratic covariation of $B$, is zero in probability. I now ...
1
vote
1answer
49 views

Application of Ito's formula to log and exponential

Let $X$ be a strictly positive continuous semimartingale with $X_0 = 1$ and define the process $Y$ by $$ Y_t = \int_0^t \frac{1}{X} dX - \frac12 \int_0^t \frac{1}{X^2} d \langle X \rangle. $$ Let the ...
1
vote
2answers
33 views

What is the distribution given by $\int^t_0 W_s^2ds$

Define $X_t=\int^t_0 W_s^2ds$, what will be the distribution of $X_t$? My approach is as follow: Let $f(s)=W_s^2s$, by Ito's lemma we have $X_t=W_t^2t-2\int^t_0W_ssdW_s-\frac{t^2}{2}$. Discretize ...
1
vote
1answer
35 views

Infinitesimal Random Variable

I have been very confused by the idea of infinitesimal random variables, namely letting $\{Z(\omega,t)\}_{t\in\mathbb{R}}$ be a stochastic process. What do we mean by $dZ$. Is this meant by ...
3
votes
1answer
48 views

Squared Bessel Process and Ito Lemma

$dX_t = \delta dt+ 2\sqrt{X_t} dW_t$, where $W_t$ is a standard Wiener process, Define $\tau =\frac{\sigma ^2}{2\nu(2 − \delta)}\left(1 − \exp \left(−\frac{2\nu t}{2−\delta}\right)\right)$ If ...
3
votes
0answers
31 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
1
vote
0answers
47 views

Solving an expectation related to CIR process

I encounter the following question Let $X$ satisfy the SDE $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$ for $s\geq t$ with $X_t=x$, where $k,\alpha,\sigma$ are positive constants. Find the ...
3
votes
2answers
37 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
1
vote
1answer
33 views

A distribution of a stopped Wiener process

Let $(W_s)_{s \geq 0}$ be a Wiener process and $\tau$ be a random variable with an exponential distribution with parameter $1$. Suppose that $W$ and $\tau$ are independent. Determine the distribution ...
0
votes
1answer
269 views

Does one necessarily need an MS in Math before taking a PhD in Math? [closed]

I finished bachelor's in mathematical finance and am nearly finished with master's in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but ...
1
vote
0answers
34 views

A Doob-Meyer decomposition related question

First I will state the question and then I will show my answer, which I obtained by imposing an additional condition on the processes involved. I would like to get some help on how to solve the ...
4
votes
1answer
154 views

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
2
votes
1answer
58 views

Determine for which values of some parameters a stochastic integral is a Brownian motion

Let $W_t$ be a Brownian motion on $(\Omega, F, (F_t)_t, P)$. Find all values of $a$ and $b$ such that the stochastic integral $$X_t=\int_0^t a+\frac{bu}{t} \;dW_u$$ is a Brownian motion. 1)So I need ...
0
votes
0answers
42 views

Transition density of a Geometric Brownian-motion

The solution to SDE $$dS(t)=\sigma S(t)dW_t$$ is $$S(t)=S(0)\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)$$ the transition density for this martingale is $$p(S(t),t;S(0),0)=\frac{1}{S(t)\sigma \sqrt{2\pi ...
1
vote
0answers
29 views

Convergence in finite-dimensional distributions of some integral

Let $(X^n_t)_{t \geq 0}$ be a sequence of random real-valued processes that converges in finite-dimensional distributions, i.e. for all $k \in \mathbb{N}$ and for all $0 \leq t_1 < \dots < t_k$ ...
1
vote
1answer
46 views

If $M_t$ is a martingale, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$

If $M_t$ is a martingale, for $0<t<T$, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$. I can think of $LHS=M_T \Bbb E \left[ \int_t^T h_s ...
2
votes
1answer
65 views

Proving the identity $P( X + Y = a)= \int_{-\infty}^{\infty} P( X + y = a)f_Y(y) \, \text{d}y $

Suppose $\lambda_1, \lambda_2, a \in \mathbb{R}$ and $X,Y$ are random variables. If it is needed, I can assume that $X$ and $Y$ are independent. I want to show, that the identity ...
2
votes
0answers
37 views

Differentiate probability max function

I have function as following $d(a,b):=pr(x-a>max{(y-b,0)})$ where a and b are constant and x and y are random variable. As this is a max function, it will have kink point hence, will not be ...
3
votes
1answer
57 views

Why is $\mathbb{P}(F\geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g$?

For random variables $F,G$ I have problems with understanding the equation $$\mathbb{P}(F \geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g, $$ where $D_G$ is the ...
0
votes
1answer
51 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal ...
-3
votes
1answer
45 views

Variance and expectation of the stochastic intergal [closed]

Compute the unconditional expected value and variance, and describe, as far as possible, the distribution of the random variable $Y_{t} = \int^{t}_{0} W_{s} ds $ with the hint below $\int^{t}_{0} ...
7
votes
2answers
290 views

Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, ...
1
vote
2answers
45 views

Verifying $S(t)=S(0)e^{rt} + \sigma e^{rt} \int_0^t e^{-rs} dW(s) $ satisfies $dS(t) = rS(t)dt + \sigma dW(t)$

Consider the SDE $$ dS(t) = rS(t)dt + \sigma dW(t). $$ To solve this, I let $f(t,x) = xe^{-rt}$, so $\frac{\partial f}{\partial t} = -rxe^{-rt}$, $\frac{\partial f}{\partial x} = e^{-rt}$ and ...
2
votes
1answer
97 views

Quadratic Variation of a square-integrable Lévy process

I am having a problem with the following question. I have tried using the definition of square integrable martingales and quadratic variation, but just can't seem to get anywhere. Can anybody offer me ...
1
vote
1answer
35 views

Stochastic Differential Equation Question

So I'm again working on doing something similar to this paper and could use some help. In the paper they worked with the equation $N(t)[(a(t)-b(t)N(t))dt + \alpha(t)dB(t)]$. It's a normal logistic ...
0
votes
2answers
52 views

Limit of time integral of brownian motion

Can someone help explain the following, $$ \lim \limits_{t \to 0} \frac{1}{t} \int_0^t W_u\, du=\lim \limits_{t \to 0} \frac{W_0t}{t}=W_0=0\,? $$ Thanks!
1
vote
0answers
17 views

Simulation of Brownian Motion on Borel Spaces

I am studying stochastic calculus on my own, and currently stuck to the following issue. Say my probability space is $(\Omega, \mathcal F, \mathbb P)$. Now when my $\Omega$ has sequences of finite ...
8
votes
1answer
70 views

Stochastic Calculus Question

I'm new here and was hoping someone could help me answer this question. I'm reading a paper and I'm a bit confused on how they go from 1 equation to the next. They say: Let \begin{align} x(t) = {} ...
2
votes
2answers
153 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
0
votes
0answers
12 views

Brownian Bridging Time Series Variance

Suppose I have a time series of daily levels $(X_t)_{t\geq 0}$. I want to create Brownian Bridges between these levels, such that variance is preserved. I assume that $X_t$ diffuses as, $dX_t=\mu ...
1
vote
0answers
27 views

How to show that stochastic exponent is integrable?

I need to prove that if $u: [0,T]\rightarrow \mathbb{R}$ is a deterministic square integrable function then stochastic exponential process defined : $M_{t} = exp(-\int_0^t \! u(s) \, \mathrm{d}W_{s} ...
0
votes
0answers
20 views

Integral with stochastic Brownian motion integrand

let $B_{t}$ is standard brownian motion ,then how to compute this integral $\int_0^t B_{s} ds=?$ I was working on SDE and find some integral in form of $\int_0^t f(B_{s}) ds=?$ $ f(B_{s})$ can ...
2
votes
0answers
38 views

Stcochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I wjust want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) ...
2
votes
2answers
53 views

Brownian Motion $dW_t \, dt=0$ proof!

I am facing a bit weird issue here. I am going through Shreeve book on stochastic calculus and faced the following theorem, while proving $dWdt=0$. $\sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))(t_{j+1}-t_j)$ ...
4
votes
2answers
71 views

Integral of time with respect to Brownian motion

I am trying to compute $\int_0^T t\ dB_t$ where $B$ is the standard Brownian motion. To this end I define the sequence of simple predictable functions $$ f_n = ...
0
votes
1answer
50 views

What to read after Shreve's “Stochastic calculus for finance 2”?

I am finishing the last pages of Shreve's Stochastic calculus for finance 2, and I was wondering what would be the best book to follow. I would like to go on with a book introducing more technical ...
1
vote
0answers
34 views

What is the solution to these SDP?

I am in trouble with my homework, the quesetion is to solve a pair of stochastic differential equation. $dX_t^1 = X_t^2dt + \alpha dB_t^1$ $dX_t^2 = -X_t^1dt + \beta dB_t^2$ $\alpha \ and \ \beta$ ...
0
votes
0answers
32 views

Is the variance of an Ito process strictly increasing?

Consider the Ito equation: $dX_t = f(t, X_t) dt + G(t, X_t) dW_t$ where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$, $G:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^{n\times m}$, $X_t \in ...
0
votes
0answers
31 views

Regularity of a parabolic equation

Consider the following parabolic equation on $\mathbb{R}^d$: \begin{equation} \partial_t\mu=\mathrm{div}(b\mu) + \mathrm{div}(D\nabla\mu), \end{equation} where the drift ...
0
votes
0answers
31 views

Is this simply assuming an Ito semimartingale.

I am reading a paper where they start by assuming some process follows $$ \frac{dX_t} {X_{t-}} = \alpha_t dt + \sqrt{V_t} dW_t + \int_{x > -1} x \tilde{\mu}(dt, dx) $$ with $\alpha_t$ and $V_t$ ...
3
votes
1answer
65 views

Solve a PDE with Feynman-Kac Formula

So there is the following PDE given: $\frac{\partial}{\partial t}f(t,x) + rx\frac{\partial}{\partial x}f(t,x)+\frac{\sigma^2 x^2}{2}\frac{{\partial}^2}{\partial x^2}f(t,x) = rf(t,x)$ With boundary ...
1
vote
0answers
11 views

Which filtration do we use with the bracket process of two local martingales.

I've read the following result without proof from my lecture notes: Let $X$ and $Y$ be two continuous local martingales (on the same probability space) with reducing sequences and filtrations ...
0
votes
1answer
22 views

Calculation of quadratic covariation of stopped processes

I am stuck in computing the quadratic covariation of the following two processes: Let $0< y <r$ and let $(B_t)$ be a Brownian motion started at $y$. Let $T_0 = \inf \{ t \geq 0 : B_t = 0 \}$ ...
0
votes
1answer
72 views

Derivation of Kolmogorov Forward Equation

By Ito's formula we have that for any suitable function $v(t,x)$, $$ v(T, X_T) = v(t,X_t) + \int_t^T\left( v_s(s, X_s)+ b(s, X_s)v_x(s,X_s)+\frac{1}{2}\sigma^2(s, X_s)v_{xx}(s, X_s) ...
3
votes
2answers
111 views

Good introductory book for stochastic calculus / Itō calculus?

I am looking for recommendations of a good first book to read on stochastic calculus / Itō calculus, say at the advanced undergraduate level. Does anyone have a favorite? Thanks so much!
0
votes
0answers
16 views

Quantile of the product of two random variables

Suppose two independent random variables for whom I have enough historical data to get statistical significance, but do not fit into a normal distribution. I want to get the 0,95 quantile of the ...