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

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224 views

Variance of Ito Integral

I want to find the variance of the Ito integral: $X(t)=\displaystyle \int_0^t\sqrt{s}WdW$ where W is a Brownian motion and s is the variable of integration. This is what I have done so far: ...
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1answer
61 views

Determining $dX_t$ for stochastic equations, and which of these are $\mathcal{F} $ - martingales?

I want to write down an expression for $dX_t$ for both: i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and ii. $X_t=W_t^2-tW_t$ What is the process I would use for differentiating these stochastic ...
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1answer
49 views

Path solution for a SDE

I would like to get help in solving an Ito stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. Is it possible to get a path solution solution in terms ...
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1answer
69 views

Question on complex valued local martingales

So I was reading and found that the following was given as an example of a complex valued local martingale: $M_t = e^{\int_0^t f(\omega,s)dB_s - \frac 12\int_0^tf(\omega,s)^2ds}$ with $f(\omega,s) = ...
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1answer
47 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
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1answer
71 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
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1answer
65 views

Ito's Lemma for Integral

Let $S$ follow GBM with $dS=(r-q)S\,dt+\sigma S\,dW$ where $W$ is a standard Brownian motion. Define $I_t=\int_0^t qe^{r(t-u)}S_u \,du$, then how can I determine $dI_t$? The answer should be ...
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0answers
29 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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1answer
92 views

Ornstein-Uhlenbeck process written explicitly

I need to show that the Ornstein-Uhlenbeck process, $$ dX_t = -\theta X_tdt + dB(t) $$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$ X_t=B(t) - \theta ...
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1answer
103 views

Solving the SDE $dX_t=bdt+cX_t dW_t$

I want to solve the SDE $dX_t=bdt+cX_t dW_t$, $X_0=0$ for $b,c\in\mathbb R$. I start by rewriting this as $$dX_t=(\mu_1+\mu_2 X_t )dt+(\sigma_1+\sigma_2 X_t )dW_t$$ where $\mu_1=b, \mu_2=0, ...
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1answer
123 views

Solution to a stochastic differential equation

I could really do with some help on this question, have no idea where to start. Any advice would be much appreciated, thank u in advance. I am given $$\begin{align}dx(t)&=(1+x(t))dt + x(t) ...
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1answer
85 views

Ornstein-Uhlenbeck operator and divergence operator

So I'm still struggling with Malliavin calculus, and this time about the divergence operator. We are working in the classical Wiener space $(W,H,\mu)$ where $W$ is the Wiener space ...
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0answers
32 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]: $$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$ A ...
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1answer
34 views

Do stochastic processes form a Banach space?

I'm interested in solving a particular integral equation: $$g(X) = \int_0^1 K(X,p)f(p) \ dp$$ where $f(p)\in L^1([0,1])$ and $X$ is a stochastic process of finite length; i.e. a collection of random ...
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0answers
80 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
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0answers
110 views

Second Fundamental Theorem of Asset Pricing

It seems that there is a step missing in the proof of the second Fundamental Theorem of Asset Pricing in Shreve's Stochastic Calculus for Finance II: Does anyone know how to show the following: If ...
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1answer
77 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
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2answers
98 views

Density of cylindrical random variables in classical Wiener space

I'm currently working on Malliavin calculus, and a theorem in my class notes is bothering me : Denote W the Wiener space of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\mu$ the associated ...
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1answer
86 views

Deriving the PDE for basket option

The payoff for basket option is max($w_1S_1+w_2S_2 -k,0)$. Using Ito's formula, I need to derive the PDE, where $dS_1 = rS_1dt + \sigma_1 S_1dW_1$ $dS_2 = rS_2dt + \sigma_2 S_2dW_2$ I need some ...
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0answers
72 views

continuous time markov process - first passage time

Let $(X_t)_{t\ge0}$ is a continuous time-homogeneous Markov diffusion process such that $X_0=y$. Let $$p(x,t|y)=d\Pr(X_t\le x|X_0=y)/dx$$ be the respective transition probability density. Let ...
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1answer
76 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
2
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0answers
62 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
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1answer
188 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
6
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1answer
71 views

double area integrals over coherence functions on circles

I am having trouble showing the following, which shows up from coherence theory: $\frac{\pi b^2}{\alpha^2}(1-J_0^2(\alpha b)-J_1^2(\alpha b))=\int_0^{2\pi}\int_0^b\int_0^b r_1r_2\frac{J_1\left ...
3
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1answer
126 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
4
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0answers
178 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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1answer
65 views

Law of large numbers variant?

I have the following: Let $(X_n)$ be a sequence of i.i.d. random variables. (a) Assume $\frac{1}{n} S_n=\frac{1}{n} \sum_{i=1}^n X_i$ converges a.s. to a real-valued random variable $Y$. Show that ...
0
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1answer
65 views

Girsanov's theorem and simulation of bond prices

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma ...
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0answers
100 views

Ito's Lemma and Geometric Brownian Motion With Jumps

I have a price process: \begin{equation} dF_t = d\Pi_t - \mu_\pi \sigma_t F_t \gamma \, dt + \sigma_t F_t \, dz \end{equation} And wish to simulate the process $x_t = \ln(F_t)$ by Euler method, ...
2
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1answer
297 views

Product of $n$ i.i.d. random variables

Let the variable $Z$ equal $Z = XY$ where $X$ and $X$ are two i.i.d. continuous random variables which distributions are given by $f_X()$ and $f_Y$. The distribution of $Z$ is given by: $$f_Z(z) = ...
0
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1answer
111 views

Finite expectation of renewal process

Let $T_n$ be a random variable with $T_n=X_1+...+X_n$ where the $X_i$'s are iid. Further we set $N(t)=max\{ n: T_n\leq n\}$ with the property $\Pr(N(t)<\infty)=1$. I want to prove that ...
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0answers
55 views

Why does the price term in Vega disappear for a European call option?

In my course, I have been asked to prove a number of statements about "the Greeks" from the Black-Scholes model for pricing a European call option with no dividends and a strike price of $K$. One of ...
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0answers
38 views

Change of Measures for Lévy-Processes

If $X$ is a Lévy-Process on a filtered probability space $(\Omega,\mathcal{F}_t, \mathbf P)$ and $Q$ an equivalent probability measure. Under which circumstances is $X$ also a Lévy-Process under ...
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1answer
48 views

What does this mean in the context of Stochastic Calculus?

I've reading into some Stochastic Calculus books and I've been stumped by two concepts used recurringly in the book. The first is a subscripted 1 which appears in the definition of a simple process ...
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20 views

2 2-dimensional Brownian motions are close to each other

Suppose $B^1$ is a standard 2 dimensional Brownian motion and $B^2$ is a 2 dimensional Brownian motion with mean zero and covariance matrix $\Gamma = \begin{pmatrix} a & b \\ b & a \\ ...
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1answer
60 views

volume of some stochastic processes

for a continuous and differentiable curve $\vec{x}_t$ in $\mathcal{R}^n$ parameterized by a single variable $t$, there is a well defined way of computing the volume of this one-dimensional manifold ...
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0answers
340 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
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0answers
50 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
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1answer
52 views

Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
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0answers
122 views

Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
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0answers
41 views

Proving $(\int_0^t f(X_s) dW_s)_{t \in [0T]}$, $f$ a $k$-Lipschitz function, is a continuous martingale

Consider $X =(X_t)_{t \in [0T]}$ progressively measurable with $X_t \in \mathbb L^p, \forall t \in [0,T]$ for $p\geq 1$ and $f$ a $k$-Lipschitz function. I would like to show that $(\int_0^t f(X_s) ...
0
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1answer
106 views

The pure jump part of Levy process and Martingale

Assume $X_{t}$ be a Levy process with generating triplet $(\sigma, \gamma, \nu)$. Here $\nu$ is the measure on $R$ satisfying $$ \int_{R}\min( 1,y^{2})\nu(dy)<\infty $$ According to the standard ...
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1answer
51 views

Clarification about a very simple stochastic integral

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting ...
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1answer
57 views

Stopping time problem - Show that T is bounded

Let $a< 0 < b$ and $W_t$ is Brownian motion $T_a$=inf{$t\ge$0|$W_t\le a$} $T_b$=inf{$t\ge$0|$W_t\ge b$} T=min{$T_a$,$T_b$} $1)$ Show that $T$ $<$ $\infty$ My attempt : ...
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1answer
73 views

How to calculate the value of $E[X^4], E[X^6],E[X^8] $…?

I learned that when X is a normal random variable , $X$~ $N(0,1)$ , $E[X^2]=1$ $E[X^4]=1.3=3$ $E[X^6]=1.3.5=15$ $E[X^8]=1.3.5.7=105$ For the general case , when variance is s , how do you do ...
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1answer
122 views

Simple integral with stochastic Brownian motion integrand

Consider $$\int_0^t \sin(B_s) ds$$ where $B_s$ is standard Brownian motion, I was wondering can I write $$\int_0^t \sin(B_s) ds = - ( \cos(B_t) - \cos(B_0)) = - \cos(B_t) ? $$ by using the ...
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1answer
39 views

Can I make the substitution of dP when using the CDF?

Random variable $X \geq 0$ with parameter $\lambda>0$ and $X$ has the c.d.f. $$ F (a) = P{(X ≤ a)} = 1 − \exp(−λa)$$ for $a \geq 0$. Consider $Z = (λ'/λ)\exp(-(λ'-λ)X)$ Show that $E[Z]=1$ thus ...
0
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1answer
98 views

Stochastic Integration and Ito Calculus

Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way. I am having trouble ...
1
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1answer
22 views

Non-interacting particles

Suppose $X(t)$ is a Markov chain taking values in $\{0,1\}^2$. Suppose $q$ is the q matrix whose positive valued entries are $q((0,0),(1,0)) = \beta_{0}$ $q((1,0),(0,0)) = \delta_{0}$ ...
0
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1answer
20 views

Please explain $E[S_{min(n,T)} ]= E [S_{0}]=0$

If $S_{n}$ is a simple random walk i.e $X_{k}= +/- 1$ with prob = 0.5 T = inf {n > = 0 |$S_{n}$ = 1} is a stopping time. T is finite almost surely. .Explain $E[S_{min(n,T)} ]= E [S_{0}]=0$ I know ...