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

learn more… | top users | synonyms

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
0answers
10 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
votes
0answers
16 views

How is the following solution derived to solve the SDE?

Let $Y_t$ be the Ito process given by $$dY_t = \theta_t dX_t - \frac{1}{2}\theta_t^2 dt $$ By applying Ito Lemma to $f(Y_t,t) = e^{Y_t} = Z_t$, we get the following SDE $$dZ_t = \theta_tZ_t dX_t$$ ...
0
votes
1answer
69 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) ...
2
votes
1answer
55 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 ...
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
vote
1answer
33 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 ...
1
vote
1answer
45 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 ...
3
votes
1answer
577 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
1
vote
2answers
31 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 ...
3
votes
1answer
47 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 ...
7
votes
2answers
262 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, ...
3
votes
0answers
28 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, ...
6
votes
1answer
523 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
0
votes
1answer
249 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
1answer
30 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 ...
3
votes
2answers
34 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
0answers
31 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 ...
3
votes
0answers
360 views

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I ...
1
vote
1answer
45 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 ...
0
votes
0answers
29 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 ...
0
votes
1answer
49 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 ...
1
vote
0answers
27 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 $0 \leq t_1 < \dots < t_k$ ...
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
35 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 ...
0
votes
0answers
24 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
3
votes
1answer
56 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 ...
3
votes
0answers
46 views

Brownian motion with drift (stopping time and supremum)

Suppose $(B(t))_{t \geq 0}$ is a Brownian motion and $(B_{\mu}(t))_{t \geq 0}$ is a Brownian motion with drift, which is defined by $$B_{\mu}(t) := B(t) + \mu t, \ \ \ \mu <0. $$ With $T_{a} := ...
1
vote
2answers
40 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 ...
-3
votes
1answer
41 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} ...
3
votes
1answer
310 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
1
vote
2answers
108 views

How to compute the quadratic variation of a compound poisson process?

The jump diffusion model is defined as $$dS_t = \mu S_t dt + \sigma S_t dW_t + S_t d \left(\sum^{N_t}_{i=1}(V_i - 1)\right)\;\;\;\;\;\;\;(1)$$ , where ${V_i}$ is a sequence of iid non-negative random ...
2
votes
1answer
76 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
28 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
49 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 ...
2
votes
2answers
286 views

Uniform integrability of a backward submartingale

Let $\{\mathcal{F}_n\}_n$ be a decreasing sequence of sub-$\sigma$-fields of $\mathcal{F}$($\mathcal{F}_{n+1}\subset\mathcal{F}_n$) and let $\{X_n\}_n$ be a backward ...
8
votes
1answer
65 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) = {} ...
0
votes
0answers
18 views

The differential of the trace of two random matrices

I have two random matrices evolving in time, $X_{t}$ and $Y_{t}$. I know that $dX_{t} = X_{t}Adt + X_{t}dB_{t}$ and $dY_{t} = AY_{t}dt + Y_{t}dB_{t}$, where $A$ is a constant matrix and $dB_{t}$ is ...
0
votes
0answers
9 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 ...
2
votes
1answer
156 views

Quadratic variation of $X_t=\int_0^t B_s \, ds$

Let $B$ be a standard brownian motion and $$ X_t=\int_0^t B_s \, ds. $$ What is the quadratic variation $[X]_t$ of $X$? I see $dX_t$ as an sde with drift term $B_t$.
0
votes
1answer
298 views

$dt$ terms have zero quadratic variation

Why does $ds$ integral have zero quadratic variation? Even if I have a integral of the form $$\int X_s ds$$ where $X$ is a stochastic process? I know that a continuous process of finite variation ...
2
votes
0answers
126 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...
4
votes
1answer
392 views

Continuous Square integrable martingale Quadratic Variation

We know that given a continuous square integrable martingale there exists unique (up to indistinguishability) continuous, natural and increasing process which is quadratic variation process of the ...
0
votes
1answer
169 views

If quadratic variation of a local martingale is zero then it is itself zero

Let $M$ be a local martingale, if we need it, we can assume that $M$ is continuous. We know that $\langle M\rangle =0$. This implies that $M$ and $M^2$ are local martingale. Can we conclude that ...
5
votes
2answers
298 views

$L^1$ bounded martingale

If $(M_t)_{0\leq t<\infty}$ is continuous martingale and it is $L^1$ bounded, does it imply that quadratic variation $\langle M\rangle_\infty$ is finite a.s. ?
1
vote
1answer
131 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
0
votes
1answer
63 views

Quadratic variation - Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one ...
1
vote
1answer
249 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
0
votes
0answers
25 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...