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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201 views

Determining Hurst exponent of a Brownian motion

I am trying to determine the Hurst exponent of a simple Brownian motion, however, I seem to get a result that differs from 0.5. I am following the instructions given on the Wikipedia-page, and here is ...
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81 views

Distribution of portfolio values with constant spending rate

If your portfolio is invested in an asset that follows a geometric Brownian motion, and you withdraw a constant dollar amount at the beginning of each year, is there an approximate analytical ...
4
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82 views

Expectation of number of hits by a brownian motion

If we denote $\tau_i$ the sequence of stopping times defined by: $\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$. If we denote N the number of stopping times below T. What ...
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72 views

mixing fractional Brownian motions

Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other. My question then: is there ...
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95 views

Reference request for research on the maximum drawdown **ratio** (NOT value)

Let's suppose the asset price process follows a Geometric Brownian motion $S_t \sim GBM(\mu, \sigma),\,t\ge 0$, and define the two process: $$ \begin{align} \text{MSF}_t &:= \max_{\tau\in[0,t]} S_\...
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320 views

Girsanov theorem and stopping time

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, equipped with a filtration $(\mathcal{F})_{0 \leq t \leq T}$ which is a natural filtration of a standard Brownian motion $(W_{t})_{0 \leq ...
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628 views

How do I artificially generate intraday ticks data from a given input (Open,High,Low,Close,Volume) using Brownian Bridge method?

How do I artificially generate intraday ticks data from a given input (Open,High,Low,Close,Volume) using Brownian Bridge method? https://en.wikipedia.org/wiki/Brownian_bridge P.S: Brownian Bridge ...
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125 views

Estimating two normal random numbers with one equation

Subtitle: Estimating the correlation of the shocks driving two commodities in two multi-factor models I am fitting two 2-factor models to electricity and gas futures, respectively. In order to ...
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167 views
+50

Geometric Brownian Motion and Energy-Efficiency Investments

Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion: $$ dX/X = \mu dt + \sigma dz\ , $$ for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
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52 views

Are the increments of a stochastic process driven by fractional Brownian motion independent?

I'm studying the following equation $$\tag1 dX_t = \mu X_t dt + \sigma X_t dB^H_t $$ where $B^H$ is the fractional Brownian motion (fBm) of Hurst parameter $H\in(0,1)$, that is a continuous ...
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370 views

Applying Ito's formula to complex functions

Within my lecture notes, the following definition is given: We say that the stochastic process $X_t$ has stochastic differential $$ dX_t = b_t dt + \sigma_t dW_t $$ if and only if $$ X_t = ...
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124 views

Prove the Markov property for the stochastic process $Y^x_t$

Prove the Markov property for the stochastic process $Y^x_t=xe^{at+bW_t}$ Given a function $u(t,x)=\mathbb{E}[f(Y^*_t)]$ with $Y^*_0=x$. For $s<t$ we have $\mathbb{E}[f(Y^*_t)]=u(t-s,Y^*_s)$ by ...
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733 views

Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)

Let $S_t$ be a process which follows a Geometric Brownian Motion: $\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$ By Ito's lemma, we have: $S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
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1k views

Standard definition of multidimensional Brownian Motion with correlations

I was wondering was the standard definition of a multi-dimensional Brownian motion is. For one-dimension, I consider the following the standard definiton. Brownian motion (or a Wiener process) is a ...
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41 views

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? ...
3
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488 views

Is Geometric Brownian Model suitable for long term price forecast?

I was thinking of using Geometric Brownian Motion to forecast future prices of timber (say one variable, the stumpage price of sawtimber). I tested the time series with Augmented Dickey-Fuller test ...
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62 views

Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown

I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2]. I do not ...
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65 views

law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
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67 views

Brownian function and Clark's formula

I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage : Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
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49 views

How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?

The integral is shown below: And how to use python to calculate pi (better if we don't need to code for each pi)?
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64 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)}. \...
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211 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 $d\ln{X}$. ...
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57 views

Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
2
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249 views

What is a stochastic processes which reasonably captures commodity price dynamics?

I ran into a stumbling block earlier when I tried to price stochastic annuities (see Asian options). This is actually technically an acturial problem, but is well adapted to the techniques of quant ...
2
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1answer
745 views

Instantaneous Volatility Estimator

Suppose a Stock follows an Itô process with instantaneous volatility $\sigma(S(t),t)$. Precisely $$dS(t)=\mu S(t)dt+\sigma(S(t),t)S(t)dW(t)$$ I have a historical data for the values of $S(t)$.How ...
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57 views

How to price a down-and-out leveraged barrier call option using Brownian motion?

I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion. My python script is below. I am not sure how to correctly model the increasing ...
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29 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 $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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44 views

Simulate correlated Brownian motions conditioned on future state(s)

Consider a model defined by 2 geometric Brownian motions $$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$ $$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$ with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
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105 views

How to solve these SDE Problems

Quuestion1. I make a solution $r(t)$ used by Ito's lemma $r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$ Is this right? and I try to make ...
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1answer
217 views

Predicting time series using Jump Diffusion model and Neural Networks

I am trying to understand the difference between using Jump diffusion model and Neural Networks or more precisely LSTM to predict time series data regardless what that data contains for example a ...
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0answers
55 views

Brownian motion from price-series, what is the time step?

If I assume a given empirical price-series is a brownian motion, I can estimate the drift and standard deviation as long as I know what the time step was when the process was 'generated'. But since ...
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138 views

The conditional expectation of a geometric brownian motion

In this question it states that $$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$ and I assume that $0 \leq s \leq t$. The accepted answer states that this step is ...
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0answers
144 views

Brownian motion for modelling future asset values

Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of $S$. I am not sure ...
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428 views

Geometric Brownian Motion with Dividends

I am working on a problem and had a quick question. I understand that for Geometric Brownian Motion we use the formula: $$X_{t_n} = X_{t_{n-1}} + \mu X_{t_{n-1}} \Delta t + \sigma X_{t_{n-1}} \...
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0answers
119 views

On quadratic covariation

I ran through an equality in a paper I was reading but couldn't check if it is correct. Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
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104 views

Correlated GBM and OU processes

I want to model two different stochastic processes, such that: $X_t , V_t$ are correlated with coefficient $\rho$. Where: $\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
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0answers
106 views

Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the ...
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0answers
103 views

Definition request - Brownian Motion Characterised by Sharpe Ratio

What is the Stochastic Differential Equation for a "Brownian Motion Characterised by Sharpe Ratio"? I saw it in a paper ("Lessons from the Mortician: volatility modulation") and the authors do not ...
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78 views

Monte Carlo Pricer for Express Certificate delivers wrong price [Mathematica]

So I wanted to price the following Express Certificate with this specific payout structure: If S1 > S0 -> 105.25 , else -> If S2 > 0.95*S0 -> 110.5 , else -> If S3 > 0.9*S0 -> 115.75 , else -> If ...
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120 views

Geometric Brownian Motion: Drawdown as a function of time

Suppose I have a strategy (model it as the usual geometric Brownian motion with a drift). Question is, how does max drawdown grow as a function of duration?
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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} = ...
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0answers
74 views

Pricing defaultable asset with finite maturity

Assume a stochastic process $X_0 = 0$ and $X_t = \nu t + \sigma W_t$ where $W_t$ is standard Brownian motion and $\nu$ is a drift (can have $\nu \leq 0$ if necessary, but prefer it to be general), ...
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0answers
130 views

Brownian bridge density and risk neutral density for derivative pricing

The book The Volatility Surface by Gatheral (2006) introduces the Brownian bridge like density $q(x_t,t;x_T,T)$ of $x_t$ conditional on $x_T = log(K)$. Can we use $q(x_t,t;x_T,T)$ as the risk neutral ...
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0answers
118 views

Smooth ornstein uhlenbeck process

I want to simulate paths for a commodity price. I use the historic data in the following way: $X_t$ is the price. $\ln\left(\frac{X_t}{X_{t-1}}\right)$ is the daily return. I calculate the slope of ...
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0answers
62 views

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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0answers
567 views

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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97 views

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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0answers
240 views

Monte Carlo simulation of Multifractional Brownian Motion in MATLAB

Code under is taken from http://en.literateprograms.org/Monte_Carlo_simulation_(Matlab) ...
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0answers
240 views

Convolution of inverse gaussian and power law distributions

I am trying to understand how the first passage time density of Brownian motion with drift is modified by the presence of waiting times that are distributed as a power law In other words, what is the ...
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1answer
77 views

Mathematical proof of $g = \mu - \frac{\sigma^2}{2}$ relationship between CAGR and average returns

I found in a paper the relation between the CAGR and the arithmetic average of returns to be $$g \sim \mu - \frac{\sigma^2}{2}$$ where g is the geometric average, $\mu$ the arithmetic average and $ ...