All Questions
Tagged with stochastic-processes stochastic-calculus
312
questions
1
vote
1
answer
73
views
Proving an Identity between a pair of correlated Wiener processes
Suppose we have the following subordinated stochastic differential equations:
$dR(t)=\mu dt+\sigma (Y(t))dW_{1}(t)$
$dY(t)=f(Y)dt+g(Y)dW_{2}(t)$,
where $W_i$'s are standard Wiener process such that ...
4
votes
1
answer
163
views
Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
I am new to stochastic calculus but the statement below confuses me:
Beside the issue of the impossible consensus on a probability measure,
the representation of the gain from trading lacks a ...
- 73
2
votes
1
answer
227
views
Why are quadratic variation and rough paths so important in quantitative finance?
I am new to quant finance - come from a mathematics background. I am starting stochastic calculus and have been particularly interested in some papers pathwise integration and rough calculus in ...
- 73
1
vote
1
answer
93
views
Expectations in Infinite Probability Spaces with Sub Sigma-Algebras [closed]
Let $X$ be an (integrable) random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$ and let $Z=\mathbb{E}(X|\mathcal{...
- 21
1
vote
1
answer
351
views
Finding Differential and Quadratic Variation Squared Process
A question based from Springer's Stochastic Calculus for Finance II book - I've tried working this out, but keep ending up in circles.
Let $S(t)$ be given by the usual formula for an asset price ...
- 21
0
votes
0
answers
111
views
Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework
Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model
$$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\...
- 111
3
votes
1
answer
218
views
Process with negative quadratic variation
Today seems to be question day for me, sorry.
The complex process
$$
dX = i\sigma dW
$$
where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
user34971
2
votes
1
answer
207
views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
- 417
11
votes
2
answers
631
views
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
- 295
1
vote
2
answers
219
views
Heston Model and antithetic variables
I was implementing some variance reduction techniques for the heston model and came up with a question when implementing the antithetic variable technique. Namely, I was not sure if I had to implement ...
- 342
1
vote
0
answers
50
views
Realized Variance as an approximation of the Integrated Variance
Realized Variance is written as $RV_{[0,T]}^{n} = \sum_{j = 1}^{n} r_{j,n}^2$, where $r_{j,n}$ is the log return for the $j$th increment, and $n$ is the total number of sample points in the time ...
- 636
3
votes
1
answer
355
views
Boundaries for Call Spread
I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold:
Question:
What are the ...
- 587
7
votes
1
answer
393
views
Option pricing with Brownian Bridge
Say I have an asset following arithmetic Brownian motion
$$
dX(t) = \sigma dW^\bot (t)
$$
with $\sigma$ constant, and I have prices of vanilla options on $X$.
I introduce a Brownian bridge
$$
dY(t) = ...
user34971
3
votes
1
answer
267
views
Computing Itô differential of conditional expectation process (Heston SDE)
Going through this article
on Heston's model, where the variance evolves following the SDE
\begin{equation}
\label{sd1}
d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
- 33
4
votes
1
answer
695
views
Discretization of Wiener process
The Wiener process $(W_t)$ is a continuous stochastic process that satisfies the following there conditions:
$W_0 = 0$,
the increments $\mathrm{d}W_t = W_{t + \mathrm{d}t} - W_t$ are normally ...
- 111
3
votes
1
answer
1k
views
How to calculate the mean and variance of this Ito integral?
I tried to calculate this integral use Ito's lemma, $W_{t}$ is the Wiener Process.
$$I_{T}=\int_{0}^{T}\sqrt{|W_{t}|}dW_{t}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
- 83
2
votes
1
answer
110
views
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Let $X_t$ be a stochastic process such that
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
$Y_t-Y_s$ is independent of $Y_s$ where $t>s$.
But is this also true ...
- 65
4
votes
2
answers
303
views
Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$
I am wondering about the probability distribution of the stochastic process
$$X_t=\int_0^t \frac{u} {t} dW_{u}$$
I thought of using the Kolmogorov equation but after converting this into An SDE
$$...
- 65
0
votes
1
answer
85
views
Accumulation Rate of Variance in Random Walk
I am slightly confused with the terminology Shreve (2008), he states:
"The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over ...
- 313
1
vote
1
answer
102
views
4
votes
1
answer
134
views
stochastic dominance displaced diffusions
Suppose I have two processes both satisfying a displace lognormal diffusion:
$$
dX(t) = \alpha(t)[X(t) - a] dW(t)
$$
$$
dY(t) = \beta(t)[Y(t) - b] dW(t)
$$
Note that the processes are perfectly ...
user34971
1
vote
2
answers
108
views
Instantaneous change in value of portfolio
I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff).
...
4
votes
1
answer
975
views
What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
dS = S\mu dt + S\sigma dX
$$
Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
- 51
5
votes
1
answer
219
views
Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
2
votes
0
answers
126
views
Interchange Expectation and Supremum in Snell Envelope/American Options
I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options.
I know that in general,...
- 636
2
votes
1
answer
163
views
Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying
$$dX_t = \mu(X_t,t)dt + \sigma(...
- 830
4
votes
1
answer
194
views
Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
- 53
1
vote
1
answer
666
views
integration of squared brownian motion w.r.t time
How to prove $\int_0^1 B_s^2ds$ is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion.
- 53
2
votes
0
answers
50
views
Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$
Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as
$$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\...
- 21
3
votes
0
answers
220
views
Stochastic differential of a time integral
Suppose that $S$ follows a geometric brownian motion:
$$
dS(u) = r S(u)du + S(u)\sigma(u,S(u))dW(u) ,
$$
with $r$ a deterministic constant, and let the process $Z$ be defined by:
$$
Z(t) = \int_0^t ...
- 185
3
votes
1
answer
221
views
Bond Option Hedging
(My question)
Please show me how to solve from (2) to (4) with computation processes.
These are too difficult to solve.
Thank you for your help in advance.
(Cross-link)
I have posted the same ...
- 179
2
votes
2
answers
275
views
Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price
(My Question)
Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M.
$$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
- 179
2
votes
0
answers
258
views
The Ho-Lee Model (1986)'s Bond Call Option Pricing [closed]
(My Question)
I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M.
(the details in this ...
- 179
2
votes
1
answer
194
views
The Riccatti equation for The Cox-Ingerson-Ross Model
(My Question)
I went through the calculations halfway, but I cannot find out how to calculate the following Riccatti equation. Please tell me how to calculate this The Riccatti equation with its ...
- 179
2
votes
0
answers
175
views
The Ho-Lee Model (1986)
(My question)
I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M.
(Thank you for your ...
- 179
3
votes
1
answer
179
views
$\beta = 1$: Simulation of SABR and whether a solution is *exact*
Quick question regarding the conditional distributions (SABR is just an example here)
Consider
$$dS_t = \sigma_tS_tdW_t$$
$$d\sigma_t = \alpha\sigma_tdV $$
$$dW_tdV_t=\rho dt$$
Hence a SABR process ...
- 1,597
2
votes
0
answers
77
views
Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$
https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf
In the above document stochastic taylor expansions are nicely explained.
Let us now consider a typical SDE model in finance ...
- 1,597
4
votes
1
answer
807
views
Bond-price dynamics in the Vasicek model
Hello I am studying about interest rate modeling
There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...
- 41
1
vote
1
answer
53
views
Why the variance of a process is $\left( \frac{dS_T^2}{dt}\right)^2$?
Consider an Ito process $dS_t = f(t,S_t) dt + g(t,S_t)dW_t $
What is the reason that we can compute the variance as:
$\sqrt{VaR(S_t)} = \frac{(dS_t)^2}{dt}$
- 301
1
vote
0
answers
37
views
Stochastic process with determinstic frequency of regime changes
Suppose that I have an OU process. For instance, assume that I want to model the interest rates. Suppose that regime change is known ex ante, and is deterministic in terms of frequency (For instance, ...
- 1,346
5
votes
1
answer
579
views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
- 53
2
votes
1
answer
83
views
Problem finding correct SDE for Stochastic Process
I am really struggling to come up with the correct SDE for the stochastic process:
$Y(t) = a[Z(t)]^2$
where $Z(t)$ is a Brownian Motion. According to my Prof, the SDE is:
$dY(t) = adt + 2aZ(t)dZt $...
- 193
2
votes
0
answers
46
views
How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?
I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112:
I am unable to understand:
How does instability arise when we use the Euler scheme on X(t)?
What change does ...
- 31
2
votes
1
answer
272
views
Unconditional Expectation vs. Conditional Expectation at time $0$
In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
- 636
2
votes
0
answers
82
views
Novikov condition for Vasicek process
Suppose that we have a money account $S^{(0)}$ with dynamics
\begin{align}
dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt,
\end{align}
where
\begin{align}
dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}.
\...
- 121
0
votes
1
answer
153
views
When $E[f(\alpha,X)] = f(\alpha, E[X])$
When $E[f(\alpha,X)] = f(\alpha,E[X])$, where $f$ is some convex function of the first and second variables, except when the first variable takes the value $\alpha$ in which case the equality holds, ...
user34971
3
votes
2
answers
428
views
Find the brownian motion associated to a linear combination of dependant brownian motions
I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
- 73
2
votes
0
answers
115
views
Milstein discretization of the CIR process
Given the CIR process $\ dX_t = (a − bX_t ) dt +
\sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
- 342
4
votes
0
answers
120
views
Feynman-Kac to derive stochastic representation
$u_t + \frac{1}{2}\sigma^2x^2u_{xx} - \alpha + \lambda((K_d - x)^+ - u) = 0$ with terminal condition $u(T, X) = (K_m - X(T))^+$
$dX = \sigma X(t)dW_t$
$\alpha$ and $\lambda$ are constants
Ok so ...
- 273
2
votes
0
answers
600
views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
- 636