# 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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### Ito Process: How to calculate expected return?

On page 300 of Hull's Options, Futures and Other Derivatives It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and ...
1 vote
48 views

### modelling time series using semi-martingale process

During this week lecture my professor said that the semimartingale( brownian motion contamined by noise) is a model in reduced form because we do not specify the dynamic which leads to price ...
87 views

### Return distribution with stop loss

Assume there is a investment with payoff going like a brownian motion, i.e. $dS=\mu dt+\sigma dW$, for simplicity, setting $\mu= \sigma=1$. At $t=1$, the payoff distribution is $P=Normal(1,1)$. If we ...
70 views

### How to apply CLT on scaled symmetric random walk--Shreve unclear

"Theorem 3.2.1 (Central limit)" in the book "Stochastic Calculus for Finance II Continuous-Time Models" by Steven Shreve says: Theorem. Fix $t\geq0$. As $n\to \infty$, the ...
310 views

### 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 ...
42 views

### Pricing equation with two correlated states

Consider the following asset pricing setting for a perpetual defaultable coupon bond with price $P(V,c)$, where $V$ is the value of the underlying asset and $c$ is a poisson payment that occurs with ...
141 views

### Quadratic Variation Of Mixed Brownian Motion and Poisson Process

I am trying to solve this problem where we're asked to compute the quadratic variation of a process. I assume that it is necessary to apply Ito's formula but not sure how to get the right solution. ...
95 views

### multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
1 vote
117 views

### Confusion about the formula for gain process in a financial market

In this wikipedia page, we consider the following financial market The formulas for the stocks are given here And the gain process of a portfolio $\pi$ is defined such that From what I understand, ...
64 views

### Equivalent definition of brownian motion

I'm having a question about this characterization of Brownian Motion : Theorem : If a process : $\big( X_t \big)_{t\geq 0}$ satisfies these conditions, $\big( X_t \big)_{t\geq 0}$ is a Gaussian ...
1 vote
208 views

### If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed]

Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask ...
102 views

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### Estimating the knockout probability of a discretely observed autocall note

For simplicity, let's suppose the underlier follows a Geometric Brownian Motion $S_t\sim\text{GBM}(\mu, \sigma), t\ge 0$ with $S_0=1$. A discretely-observed binary autocall note is a derivative ...
1 vote
253 views

### On first and last zeros before t in a Brownian Motion

Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$: \begin{align*} \alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \...
1 vote
87 views

### How to derive this HJB equation?

I'm reading the paper by J.Gatheral and A.Schied (2012) - "Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework". On page 6, the authors provide a ...
81 views

### How to simulate from instantaneously correlated Brownian motions?

Say I have obtained a distribution for different forward rates F_k such that: $$dF_k (t) = \sigma (t) * F_k (t) * dW_k(t)$$ with $$dW_k(t) * dW_l(t) = \rho_{k,l} (t) dt.$$ From this I want to ...
104 views

### Standard Brownian Motion and Exponential Martingale calculation [closed]

Let $W(t)$ be a standard brownian motion and let $Z(t) = \exp (\lambda W(t) - \frac{1}{2}\lambda^2 t).$ In Xinfeng Zhou's Green Book in the section on Brownian Motion (p.130) he writes as part of the ...
167 views

### Expectation of Bt^4 given BS [closed]

What is the expectation of Bt^4 and Bt^3 given Bs? Given t>s. I understand that the expectation of Bt given Bs is Bs and that the expectation of Bt^2 given Bs is something like Bs - s + t.
1 vote
274 views

### 4th Order Brownian Motion Martingale [closed]

I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
106 views

Consider for simplicity the following brownian market: $$dS^0_t= r S^0_tdt$$ $$dS^1_t= S^1_t(r dt + dW^1_t + dW^2_t)$$ where the filtration is generated by $W^1,W^2$ Consider now $W_t:= \frac{1}{2}(W^... 5 votes 0 answers 132 views ### Covariance of (fractional) Brownian motions with different Hurst parameters I'd like to calculate the covariance function for fractional Brownian motions $$E_t \left[ dW^H(t) dW^{H'}(t) \right]$$ but where the Hurst parameters are not equal:$H \neq H'\$. My first idea would ... 