Tagged Questions

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

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2
votes
0answers
19 views

Black Scholes PDE

How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + ...
0
votes
0answers
6 views

Characterise expression involving white noise

I would like to characterise an expression, for example by finding its spectral density. The function is $\int_{-t_0}^{t}\mathbf{C}_s^t(\tau)\mathbf{q}_{\omega}(\tau)d\tau\cdot\mathbf{q}_{a}(t)$ ...
1
vote
1answer
20 views

Prerequisite before learning Stochastic Analysis

I'm engineering major student with some experiences on basic mathematical analysis courses (2 semester courses with using baby Rudin) and also have some basic background on undergrad probability and ...
0
votes
0answers
12 views

BDT model and the $\theta(t)$ function

Question about the answer in this question: Black Derman & Toy Model Where $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t)}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ is rewritten as $$dr=A\,dt+B\, ...
2
votes
1answer
77 views

Ito's Lemma for negative exponential

I'd been reading on Hull-White model, when I encountered the bond-pricing formula, that is if $$ dr(t) = (\alpha(t)-\beta(t)r(t))dt + \sigma(t)dW(t)$$ for some deterministic function $\alpha, \beta, ...
-1
votes
0answers
26 views

Expectation and variance of a stochastic process

How to calculate the expectation and variance of $$ U_t = e^{-\gamma t}\,U_0 + \int_0^t e^{\gamma (s-t)}\sigma\, dX_s? $$
2
votes
1answer
26 views

Black Scholes Differential Form

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that $$ S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu\, dt + ...
4
votes
0answers
53 views

Converge to Brownian Motion problem

Consider the following sequence of SDEs: $dX^n_t = \sin(nX^n_t)dt + dW_t, X^n_0 = 0\,\,\,$ Show that the solutions $X^n$ converge in finite dimensional distribution to Brownian Motion. I have been ...
2
votes
1answer
31 views

Proof of the Début theorem

I was reading the stochastic calculus notes on this website and I read the following in the proof of the Début theorem but I could not understand what does it mean. Can someone explain it to me ...
1
vote
1answer
35 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: ...
0
votes
1answer
28 views

Can someone help me understand the following?

I was reading George Lowthers notes on Stochastic Calculus and , he says the following but I cannot figure out what it exactly means? ...
0
votes
1answer
10 views

question on a stopping time problem.

I borrowed some lecture notes on stochastic calculus, which contained the following exercise: Let $(X_n)_{n>0}$ be a sequence of random variables with $X_n: \Omega \to [0,\infty)$. We set $S_n= ...
0
votes
0answers
13 views

Random starting point for Brownian motion

The hitting probability for balls centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$ where $|x|>r$. Now consider hitting time $T_{A}$ of sphere A disjoint from ...
0
votes
0answers
26 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
0
votes
1answer
12 views

Transition density of Brownian bridge using generators

Let $X_{t}:=(1-t)\int_{0}^{t}\frac{1}{1-s}dB_{s}$. This satisfies SDE: $$dX_{t}=-\frac{X_{t}}{(1-t)}+dB_{t}$$ So the generator will be $A(f)=\frac{-x}{1-t}f'+\frac{1}{2}f''$ and so I think the pde ...
0
votes
0answers
21 views

Estimate on the Positive probability of not hitting finite measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $\lim_{t\to \infty}|B_t|=\infty$. But does this imply $1-P_x(T_A<\infty)>0$ for Borel sets $A\subset \mathbb{R}^d$ with ...
0
votes
0answers
17 views

Computational rules for expectations of functions of wiener processes.

What are some general rules that are helpful for computation/calculation of expectations such as $$ E(X_t | \mathcal{F_s} ), $$ where $X $ is a function of Brownian motions $W_t$ and $\mathcal{F}$ is ...
1
vote
1answer
25 views

Joint convergence in distribution

I've one question concerning convergence in distribution of random variables: Let $X_n \rightarrow X$ and $Y_n \rightarrow Y$ for $n \to \infty$ where $\rightarrow$ denotes convergence in ...
1
vote
0answers
15 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
2
votes
1answer
56 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
1answer
46 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define $X_t=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ ...
0
votes
0answers
27 views

Measure Preserving tranformation of the space of brownian paths

Let $O$ be an orthogonal transformation of $L2_{[0,\infty)}$. Let $1_{[0,1]}$ be the indicator function for $0 \leq s \leq t$. Also let $B(t)$ be a standard brownian motion. Define $W(t) = ...
0
votes
1answer
14 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
0
votes
1answer
27 views

Integrated Brownian motion: independent stationary increments?

Let $B_t$, $t\in [0,T]$ be a $d$-dimensional standard Brownian motion. Let $\sigma:[0,T] \rightarrow \mathbb R^{d\times d}$ be a deterministic function such that $$\sigma(u) = diag( \sigma_1(u), \dots ...
0
votes
1answer
43 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
0
votes
0answers
12 views

Determining Bounds of a Generating Function of a Stopping Time [duplicate]

Consider the diffusion process $$DX_t=b(X_t)dt+\sigma(X_t)dW_t$$ where $\sigma\sigma*$ is positive definite and $b, \sigma$ are smooth and bounded. Given a one-dimensional domain bounded from 1 side ...
1
vote
1answer
36 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
0
votes
1answer
16 views

Mean of stochastic exponential

Suppose $X_t$ solves an SDE. Is it true to say that the identity, $$ \mathbb{E}\left[e^{X_t}\right] = e^{\mathbb{E}[X_t]+\frac{1}{2}\text{Var}[X_t]} $$ holds only when the drift and volatility of ...
0
votes
2answers
36 views

simple stochastic differentiate

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
2
votes
1answer
22 views

About modifications of right-continuous stochastic processes

Lemma : Let $X$ and $X'$ be two right continuous(or left continuous) processes defined on the same probability space $(\Omega,F,P)$ be a modification then the two processes are indistinguishable. ...
2
votes
1answer
22 views

Kunita Watanabe Identity

I am looking for a proof of the following version of Kunita Watanabe Identity: "Let $M,N \in M_{c,loc}$ and $H$ be a locally bounded previsible process. Then $[H \cdot M, N ] = H \cdot [M,N]$" I ...
1
vote
2answers
58 views

Stratonovich integral

I'm having some troubles to calculate the Stratonovich integral $I(sin)(t)=\int_{0}^{t}\sin{B_{s}}dB_{s}$. I've tried with the limit of ...
0
votes
1answer
56 views

Show independence of stochastic integral and stochastic process

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s ...
1
vote
1answer
38 views

Why can $\int_0^t f''(X_s) \, d\langle X \rangle_s$ not be a local martingale?

We know from Itos formula, if $X$ is a continuous local martingale and $f$ has two continuous derivatives, we can write $f(X_t)$ as $$ f(X_t) = \int_0^t f'(X_s) dX_s + \frac{1}{2} \int_0^t ...
0
votes
0answers
26 views

basic Stochastic differential equation

I'm sorry but I'm having some troubles to find a solution of this simple stochastic differential equation, $dX_{t}=2\sqrt{X_{t}}dB_{t}+2dt$ where $B_{t}$ is a Brownian motion, please can you help ...
0
votes
0answers
11 views

Asymptotic results for functions of order statistics

There are $n$ ($n \ge 3$) iid random variables $\{ {c_i}\} _{i = 1}^n$ on the interval $[\underline c,\bar c]$ ($\underline c>0$). The cdf $F(\cdot)$ and pdf $f(\cdot)$ are unkown to us, but we ...
1
vote
0answers
19 views

Infinitesiman generator of Time dipendent process

I'm trying to find the infinitesiman generator of this process $dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$ where $B_{t}$ is a brownian motion; and I've found the solution: ...
0
votes
0answers
12 views

What does it mean for a stochastic process to be measurable?

In my first class of Stochastic calculus the professor said that a process X is measurable if the map $(t,\omega) \mapsto X_t(\omega)$ is measurable from $(\mathbb{R^+ \times ...
6
votes
1answer
128 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
0
votes
0answers
19 views

Numerical solution of SDEs with fractional Brownian motion

I am trying to numerically solve some SDEs representing a nonlinear circuit (possibly chaotic) driven by noise: $$ dX = f(X) dT + \sqrt{P_{w}} dW + \sqrt{P_{f}} dC $$ where $X$ is my circuit state, ...
0
votes
0answers
14 views

Second Moment of Intensity Function of Stochastic Process

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
0
votes
1answer
26 views

What is the Stochastic Differential Equation representation of a Poisson Process

What is the Stochastic Differential Equation representation of a Poisson Process? Can it be expressed as a function of $dW$ where $W$ is a Wiener process?
1
vote
1answer
26 views

Is exit probability monotonic in drift and diffusion coefficient?

Let $W$ be Brownian motion. Let $b_t$ and $\sigma_t$ be adapted to $\mathcal{F}_t^W$. Consider the SDE $$dx_t=b_tdt+\sigma_tdW_t.$$ Assume that $b$, $\sigma$ are such that $x$ stays non-negative. Fix ...
0
votes
1answer
35 views

Deriving Black Scholes using CAPM

I am referring to http://www.frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Section 3, which is a bit more detailed version of the original derivation from ...
0
votes
1answer
23 views

An equality in SDE.

I read an example in Shreve: How to get the equality in the last line?
1
vote
1answer
39 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
0
votes
0answers
14 views

Second Moment of Intensity Function of Stochastic Process

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
0
votes
1answer
60 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
0
votes
0answers
15 views

Empirical Intensity Function

I would like to ask for help determining what other ways are there to compute the "empirical intensity function" of a process. In essence, given that I observe the occurrences of an event in time ...
8
votes
1answer
314 views

Final Step in calculating option prices under the Heston Stochastic Volatility Model

Let: $$ \alpha = -\frac{u^2}{2}-\frac{iu}{2}+jiu\\ \beta = \lambda-\rho \eta i u - j \rho \eta\\ \gamma = \frac{\eta ^2}{2}\\ $$ where $j \in \{0,1\}$ and $i^2=-1$, $g=\frac{r_-}{r_+}$ and ...