Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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Partial derivative of an integral

Suppose I have a model for the short rate $r$ as ($W(t)$ is standard Brownian motion) $r(t) = c+ \int_0^t \sigma (s) ^2 (t-s) ds+ \int_0^t \sigma (s) dW(s)$ I then want to find the dynamics of $r$, ...
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How to show the equality of these two events?

For $X_t$ a brownian motion defined for $t\in[0,T]$, How to show the equality of following events: $$ \{ \displaystyle \max_{[\tau,T]}(X_t)\ge 2u-d, \tau\leq T \}=\{\displaystyle \max_{[0,T]}(X_t)\ge ...
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$P(S_T > S_u \mid S_v = s_*)$

Let $u < v < T$ and assume $S_t$ follows a lognormal $((\mu - \sigma^2/2)t, \sigma^2 t)$ process. I'm interested in computing the conditional probability $$ P(S_T > S_u \mid S_v = s_*) $$ ...
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Integral with respect of $(dW_s)^n$

I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since $ dW_s dW_s=ds$ , so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for $n\ge 3$ we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (...
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238 views

Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions

Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ...
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1answer
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What's the variance of this Ito integral?

I am reading stochastic calculus and I have understood that the process $$X=\int_{0}^{1}\sqrt{\frac{\tan^{-1}t}{t}}dW_t$$ has normal distribution with mean zero. How can I find the variance of $X$?
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Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
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Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks
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Variance of Brownian Motion

Can someone point me into the right direction to calculate this one: $E(B^4_t)=3t^2$ I had tried using the following property with no luck: $E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\...
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519 views

Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?

As stated, this is an interview question. Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?
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723 views

What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?

The probability that an arithmetic Brownian motion process $dt = \mu dt + \sigma dW$ hits an upper Barrier $U$ before it hits a lower barrier $L$ is given by $$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\...
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Is this a poorly written example, or could volatility in fact be negative?

I'm self-studying and I encountered the following example. It seems to suggest that volatility is negative in this example. I was under the impression that volatility can never be negative, both from ...
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Brownian motion simulation - scaling issue

I'm trying to simulate some BM for 500 observations. I got correlated increments as I needed and they are not exactly N(0,1), so I standardize them (x-mean(x))/sd(x). But then the resulting Brownian ...
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1answer
236 views

Probability that return exceeds a certain level before a certain time (Black-Scholes)

I am self studying for an actuarial exam on financial economics. I encountered the following problem and solution. It seems to me that the author intended to mean what is the probability that the ...
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Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
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Probability of Brownian motion particle touching barrier given path starts at $X_0$ and ends at a known $X_t$

I have been reading Su and Rieger's paper on barriers and from there have been able to work out the unconditional probability of the process $dXt = μ dt + σ dWt$ touching a down barrier $α$ to be $\...
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1answer
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Drift irrelevance on high frequency data

Let's assume that price of a certain asset follows Brownian Semimartingale process with a drift term and a Brownian-driven continuous part (no jumps for simplicity). In literature it is often stated ...
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Can trinomial trees be used to model subdiffusion?

I am modeling a sub-diffusive process where the particles follow geometric Brownian motion (GBM) with movement occurring after randomly distributed waiting times. I have set this up as a simulation ...
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Polynomial interpolation of corrected lognormal distribution

Can anyone provide a formula for a polynomial interpolation of the corrected lognormal distribution used to model returns traditionally resulting from the wrong Brownian motion generated model? ...
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2answers
751 views

Conditional probability of geometric brownian motion

I created paths using GBM to implement The stochastic mesh method. But the method requires the conditional distribution, given some S(t) the probability of S(t+1). I've searched and can't find this ...
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Modeling the Stock Market [closed]

Hi I was wondering what is the model that best describes the price movement of the stock market? A Brownian motion Process with drift? An Ornstein Uhlenbeck_process? (where the long term mean is ...
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1answer
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example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
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Correlation of Asynchronous Brownian Motion

I am trying to use the closing prices of the S&P 500 and the Nikkei Index to see how they are correlated (assuming they are exactly 12 hours apart). In order to test my method, I have generated ...
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1answer
238 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},\...
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245 views

Multivariate Ito problem $M_t=\frac{X_t}{Y_t}$

I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the $M_t$ was calculated out of the Ito formula. I ...
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1answer
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investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...
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933 views

How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $ ...
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234 views

2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?

Having two Ito processes $dX_{t} =z_{1} dt + Y_{t} dB_t $ $dX^{'}_{t} =z^{'}_{1} dt + Y^{'}_{t} dB_t $ I am analyzing a proof of the product rule $d(X_t X_t^{'})=X_t dX_t^{'}+ X_t^{'} dX_t + Y_t ...
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SDE for a portfolio of two correlated assets $ Y_{t} = 2 S^{1}_{t} S^{2}_{t}$

I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ \frac{dS^{2}_{t}}{S^{...
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On the construction of a Brownian motion from a Gaussian process

Let $X$ a Gaussian process defined by $$ X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T] $...
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What is the name of all 1-day movements, 2-day movements etc

When looking at historical data (index or stock), one can find all 1-day differences/movements, all 2-day, all 3-day etc and graph the extremes of each of these. This gives two line graphs forming a ...
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Simulate correlated Geometric Brownian Motion in the R programming language

In response to this question: How to simulate correlated Geometric brownian motion for n assets? One of the responses provides an implementation in MATLAB: http://www.goddardconsulting.ca/matlab-...
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Is it really possible to create a robust algorithmic trading strategy for intraday trading?

I'm an engineer doing academic research for my master thesis in the area of quantitative finance, basically the purpose is to study the possibility to create an intraday-trading algorithm. I've tried ...
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1answer
294 views

Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
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Asymptotic behavior property of geometric Brownian Motion proof

Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as: If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \rightarrow ...
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Two correlated brownian motions

Is it true (see here, footnote 2, p.22 / p.14, without proof) that we can obtain two discretized brownian motions $W_t^1, W_t^2$ with correlation $\rho$ by doing $$d W_t^1 \sim \mathcal N(0,\sqrt{dt}...
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Processes used in quant finance

What are the main stochastic processes (and their SDE) used in quant finance? For example to model currency prices, stock prices, etc.
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Simulations of (standard, one-dimensional) Brownian motion

Consider the following two proposed simulations of paths of standard, one-dimensional Brownian motion between time $0$ and time $1$. Normal Increments Roll out a large sequence of, say $M$, ...
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Stochastic process with non-independent increments

All stochastic process I see always have independent increments. It is true for: standard brownian motion geometric brownian motion (?) Ornstein Uhlenbeck (?) in general, Levy process etc. What are ...
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Why do we usually use normal distribution and not Laplace distribution to generate stochastic process?

When working with a stochastic process based on brownian motion, the increments have normal (gaussian) distribution. However, it seems that a Laplace distribution, with density: $$f(t) = \frac{\...
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Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?

I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the parameters $\...
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What's the name of this nearly-brownian stochastic process?

1) Does the following algorithm (my question is math, not programming-related): ...
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Geometric brownian motion vs. Ornstein Uhlenbeck

I'm looking at the SDE of Geometric brownian motion(*): $$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$ (with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$) and the SDE of ...
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Question about the martingale property of stochastic integral

Let $W_{t}$ be a Wiener process, and let $$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$ Is $X_{t}$ a martingale? We can rewrite in differential form as $$dX_{t} = W_{t}dt$$ ,which means $X_{t}$ is a diffusion ...
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Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
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511 views

Black Scholes Geometric Brownian Motion Option Pricing

I'm doing a past paper for one of my masters modules and I'm stuck on this and I have no idea how to tackle such a thing. It's worth 30% of the exam so would be great if anyone here has any ...
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Geometric Brownian Motion - increasing simulations or smaller step size

I am running Monte Carlo simulations to estimate future share prices of some stocks. For stock A, I need 1 share price exactly one year from now. For stock B, I need daily prices for each trading ...
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1answer
433 views

Geometric Brownian Motion: d(S) vs. d(ln(S))

I am quoting from "Tools for Computational Finance, 5th Edition" [Seydel]. I wonder whether the histogram of simulations of the first (yellow) SDE makes sense... especially given that Seydel (...
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1answer
557 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) =...
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1answer
245 views

Brownian motion. Solve stoc. integral by using Ito's lemma

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^...

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