Questions tagged [wienerprocess]
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48
questions
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Sample Wiener process constrained to open (initial), high (max), low (min), close (final)
With a Brownian bridge, one can sample a Wiener process constrained to a specified initial value and a final value.
Can the same be done when the process is constrained also to have a specified ...
0
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0
answers
60
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How to simulate a conditional expectation given a filtration
I had a question regarding how to simulate a certain conditional expectation. I am given two processes $X_1(t), X_2(t)$ which both follow their own SDE, but both are of the form
\begin{equation*}
dX_i(...
1
vote
1
answer
68
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Moments of the integral of the exponential of Brownian motion/Normal random variable
I'm studying arithmetic Asian options and there is integral of the following form: $$X_T=\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is a Brownian motion/Wiener process....
2
votes
1
answer
354
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Integrated Brownian motion
I occasionally see a post here: Integral of brownian motion wrt. time over [t;T].
This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$.
However, here is my derivation which is ...
3
votes
0
answers
73
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Feymann Kac pde with correlated process
I have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
2
votes
2
answers
281
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Calculate value of Integral of Wiener process $\int_{0}^t e^{\lambda u } dZ_u$
I am not quite sure how to solve this integral to be able to do numerical calculations with it. $\lambda$ is a constant, $u$ is time, and $Z_u$ is a wiener process. Can anyone provide some direction ...
1
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0
answers
88
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Is this the right way to accelerate my Monte-Carlo Simulation
I am trying to develop a pricer for Call VS Call and I'm using MonteCarlo method to do so because my stocks are correlated between each others.
Basically my inputs are ...
1
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1
answer
522
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Integral of brownian motion wrt. time over [t;T]
From the post Integral of Brownian motion w.r.t. time we have an argument for
$$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$
However, how does this generalise for the interval $[t;T]$? I.e. ...
1
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0
answers
131
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Value of trading strategy
A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process.
The problem is to find the probability of the ...
1
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1
answer
220
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Why the Esscher transform is the right transform for pricing formula?
A Wiener process has infinitely many states of the world at any time step. Does that not mean that there are infinitely many EMM's for any model that uses the Wiener process?
But then if there is only ...
1
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1
answer
103
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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
1
answer
177
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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
$$dS_{1}...
2
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0
answers
130
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The distribution of the jump diffusion process
In the Merton jump diffusion model the process of the share price can be expressed as $$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}.$$
Here $...
10
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2
answers
1k
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conditional expectation of stochastic integral
let $M_t$ be the following stochastic integral
$$
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 $...
0
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1
answer
189
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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
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0
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686
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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
1
answer
557
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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
0
answers
45
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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}}$...
1
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1
answer
285
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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
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1
answer
931
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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
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1
answer
302
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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
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2
answers
1k
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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:
$$E[Z(t+s)] = E\left[\exp\left(aW(t+s)-0.5a^2(t+s)\...
2
votes
1
answer
118
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Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Let $X_t$ be a stochastic process such that
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
$Y_t-Y_s$ is independent of $Y_s$ where $t>s$.
But is this also true ...
3
votes
1
answer
873
views
Probability Density Function of a Wiener Process Minimum
Let $W_t$ be a standard Wiener process. Find the probability density function of $m_T =
min_{t\in [0,T ]}W_t$.
I know that it is based of the concept of the reflection principle, but I wasn't too ...
1
vote
1
answer
219
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Can anyone explain to how Hull get from stock returns to continuously compounded stock returns?
I'm reading Chapter 13 of Hull's book and am stuck on how he got from stock returns to continuously compounded stock returns. As a recap, he built the generalized Wiener Process, which describes a ...
3
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1
answer
237
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Interpretation of IV and its use in stock movement prediction
I would like to validate my understanding of IV as a prediction tool.
Black-Scholes model is based on the assumption that rate of return of a stock is a Wiener process:
$$ \frac{dS_t}{S_t} =\mu \,...
7
votes
1
answer
9k
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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 ...
8
votes
1
answer
397
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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 ...
0
votes
1
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200
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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
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2
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560
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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
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0
answers
67
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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
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0
answers
161
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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
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1
answer
900
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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
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1
answer
1k
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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{...
5
votes
2
answers
523
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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 $\...
4
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1
answer
319
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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},\...
3
votes
2
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143
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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 ...
4
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1
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762
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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) =...
1
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2
answers
135
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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 ...
2
votes
1
answer
112
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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
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1
answer
534
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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
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1
answer
271
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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
1
answer
229
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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
1
answer
389
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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
1
answer
189
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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
3
answers
571
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 $B_t = W_t^3 - 3tW_t$. ...
4
votes
3
answers
1k
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
2
answers
6k
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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 ...