Questions tagged [wienerprocess]
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39
questions
9
votes
2answers
312 views
conditional expectation of stochastic integral
let $M_t$ be the following stochastic integral
$$
M_t = \int_0^t \sigma_s dW_s
$$
where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...
1
vote
1answer
24 views
Regression of stochastic integral on Wiener process
This question is a follow-up from the following: conditional expectation of stochastic integral
so I won't repeat myself regarding assumptions and notation.
Using Brownian bridge approach, we know ...
2
votes
0answers
57 views
Arbitrage portfolio example
Can you give me a concrete example of a self financing portfolio which gives arbitrage opportunity in the two-dimensional Black-Scholes model?
By the two-dimensional Black-Scholes model I mean
$$dS_{1}...
2
votes
0answers
65 views
The distribution of the jump diffusion process
In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$
Here $...
1
vote
1answer
184 views
Can anyone explain to how Hull get from stock returns to continuously compounded stock returns?
I'm reading Chapter 13 of Hull's book and am stuck on how he got from stock returns to continuously compounded stock returns. As a recap, he built the generalized Wiener Process, which describes a ...
0
votes
1answer
62 views
Differentiability of solutions of a stochastic differential equation
I would like to clarify a confusion I have.
It is well known that a Wiener process (Brownian motion) is nowhere differentiable. I have no difficulty in understanding that. But I am wondering about the ...
1
vote
0answers
208 views
CIR model. Is there a closed-form solution or even a good proxy of analytical solution?
Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE
\begin{equation}
dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}
\end{equation}
?
Notice that $\{r_t\}$ is our ...
2
votes
1answer
240 views
Expectation on a function of Wiener Process
If $W_t$ is a standard Wiener Process, then how should I prove that $E \left[ \int\limits_{0}^{t} \frac{1}{1+W_s^2} dW_s \right] = 0$?
1
vote
0answers
30 views
How to expand lognormal approximation of Brownian motion
How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion.
If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
0
votes
0answers
25 views
Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)
For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
1
vote
1answer
126 views
Calculation of a process's drift
Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I ...
1
vote
1answer
277 views
What is the difference between standard deviation, volatility and quadratic variation?
What is the difference between standard deviation, volatility and quadratic variation?
As I know, volatility is the standard deviation of the log returns, so they are basically the same. (One of ...
1
vote
1answer
54 views
How to Evaluate Expected Value powered 4 of a Wiener Process?
Since $X(t_j) - X(t_{j-1})$ is Normally distributed with mean zero and variance $t/n$ we have
$$ \operatorname{E} [(X(t_j) - X(t_{j-1}))^2 ] = \frac{t}{n} \tag{1}$$
and
$$ \operatorname{E} [(X(t_j) - ...
3
votes
2answers
293 views
Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale
I'm trying to prove that $Z(t)=\exp(aW(t)-0.5a^2t)$ is a martingale where $W(t)$ is a Wiener process and $a$ is a constant. Here is my attempt:
$$E[Z(t+s)] = E\left[\exp\left(aW(t+s)-0.5a^2(t+s)\...
2
votes
1answer
81 views
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Let $X_t$ be a stochastic process such that
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
$Y_t-Y_s$ is independent of $Y_s$ where $t>s$.
But is this also true ...
3
votes
1answer
439 views
Probability Density Function of a Wiener Process Minimum
Let $W_t$ be a standard Wiener process. Find the probability density function of $m_T =
min_{t\in [0,T ]}W_t$.
I know that it is based of the concept of the reflection principle, but I wasn't too ...
3
votes
1answer
196 views
Interpretation of IV and its use in stock movement prediction
I would like to validate my understanding of IV as a prediction tool.
Black-Scholes model is based on the assumption that rate of return of a stock is a Wiener process:
$$ \frac{dS_t}{S_t} =\mu \,...
8
votes
1answer
189 views
Integral of the OU (Ornstein Uhlenbeck) process conditioned on hitting a threshold value for the first time
Let say I have a zero-mean OU process as follows:
$dX_t = -\alpha X_t + dW_t$
The process starts at $x_0 = 0%$ and I'm interested in the event in which the process hits the value $x_{\tau} = a$ for ...
4
votes
3answers
475 views
Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
7
votes
1answer
4k views
Integral of Wiener process w.r.t. time
I have a doubt with regards to the calculation of the below integral-
$\int_0^t W_sds$
where $W_s$ is the Wiener Process.
This has been solved very ably in the following page. It turns out to be a ...
0
votes
1answer
137 views
Two Wiener process under same martingale measure Q
Let $W_1,$ $W_2$ be to Wiener processes under the martingale measure $Q$. What can be said about $dW_1*dW_2$? I know that $$(dW_i)^2=dt$$ but what about the case with two different wiener processes?
-3
votes
2answers
386 views
Geometric Brownian Motion: Why is the Wiener process multiplied by volatility?
Below is the stochastic differential equation of the Geometric Brownian Motion:
$$dS_t = S_t \mu dt + S_t\sigma dW_t$$
My understanding of the Wiener process is that the volatility component of an ...
1
vote
0answers
61 views
Solving for roots of a stochastic pay-off function
I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = ...
2
votes
0answers
140 views
Expected value of a wiener process on an infinite time horizon with a barrier
Say I have a wiener process with $X(0) = X_0>0$ and the dynamics
\begin{equation}
dX(t) =
\begin{cases}
-\mu dt + \sigma X(t) dW(t)^{\mathbb{Q}} & \mathrm{for\ } X(t)>0\\
0 & \mathrm{...
1
vote
1answer
769 views
Matlab implementation for modelling stock price process
I am trying to model the stock's price process. Let's assume volatility and risk-free rate is given. I've come up with the code below to try and model the price process with the geometrical Brownian ...
1
vote
1answer
785 views
Is this process of Brownian motion?
Background Information:
The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if
i) $W_t$ is continuous, and $W_0 = 0$
ii) the value of $W_t$ is distributed, under $\mathbb{...
4
votes
1answer
237 views
On the reflection of a stochastic integral
Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by
$$
I_t=\int_{0}^{t}\theta_sdW_t,
$$
where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq 0},\...
4
votes
2answers
372 views
Are two stochastic processes independent if the Wiener processes inside are uncorrelated
Assume there are two stochastic processes:
$dx_t = \alpha_1(x_t,t)dt + \beta_1(x_t,t)dW^1_t$
and $dy_t = \alpha_2(y_t,t)dt + \beta_2(y_t,t)dW^2_t$.
Does $dW^1_t\times{dW^2_t} = 0$ imply that $\...
2
votes
2answers
119 views
Bounded Stochastic discrete process
I just came across this stochastic process (link):
$dY_t = (a-bY_t)dt + c \sqrt{Y_t(1-Y_t)}dW_t$, where $dW_t$ is a Wiener Process. According to the author under certain conditions this process is ...
1
vote
2answers
123 views
Stochastic process theory question
*S follows a process $dS= mSdt + oSdz$ where m and o are constant.
What is the probability followed by $ Y=(Se)^{(r-t)} $.
If S follows a process $ dS= k (b-S) dt + oSdz $ where k, b, o are ...
4
votes
1answer
554 views
Solving a backwards heat equation using stochastic calculus
Given the PDE
$$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} = 0$$
with condition $F(T,x) = x^2$, one can use the Feynman-Kac formula to arrive at
$$F(t,x) =...
2
votes
1answer
79 views
Can the differential operator be removed to get the mean/variance of an Ito process?
If $X_t$ is an Ito process, such that:
$dX_t = \mu(t, X_t)dt + \sigma(t, Xt)dW_t$ where $W_t$ is a standard brownian motion.
Then we can say that:
$E(dX_t) = \mu(t, X_t)dt$ and that $Var(dX_t) = \...
1
vote
1answer
483 views
Obtaining the drift of a Wiener process formed from a random walk
I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
4
votes
1answer
265 views
How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$
How can I calculate?
\begin{align}
Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)
\end{align}
Thank you for your attention.
2
votes
1answer
138 views
What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?
The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S:
$dS=\mu S dt + \sigma S dX$
Clearly $S$ ...
3
votes
1answer
361 views
Wiener process integral
Suppose that $W_{t}$ is a Wiener process.
Assume $W_{0}=0.$ Is it
true that $\int_{t=0}^{T}dW_{t}=W_{T}$? If so, why?
Is one preferred to the other?
5
votes
1answer
175 views
Stochastic Differential
Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because
$$
\mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta.
$$
But am I allowed to actually write $(...
4
votes
3answers
480 views
Determine $E[W_p W_q W_r]$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let 0 < p < q < r. Determine $E[W_p W_q W_r]$.
...
2
votes
2answers
4k views
How to calculate the expected value of a function of a standard brownian motion (Wiener process)
Have a problem regarding the expected value of the Wiener process inside a function, namely:
Compute $E[cos(W_t)]$.
To extend my question, what is the general method of computing these E´s when it ...
