# Questions tagged [stochastic-processes]

stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

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### Bull/Bear and Trending/Mean reverting [closed]

The market trend can be either bull or bear depending on the direction of the price movement. Also, the price process can be either trending or mean reverting. How does Bull/Bear and Trending/Mean ...
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### Process with negative quadratic variation

Today seems to be question day for me, sorry. The complex process $$dX = i\sigma dW$$ where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
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### Covariance of logarithms of geometric Brownian motion

Suppose I have a Geometric Brownian Motion process, $$dX_t=\mu X_t dt + \sigma X_t dW_t$$ I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
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### Exercise: does Ito integral of a simple stochastic process have normally distributed increments?

I am trying to solve the following problem (exercise 4.3 from Shreve's Stochastic Calculus for Finance, Vol. 2, my adaptation): Let $W(t)$, $0\le t\le T$ be a Brownian motion, and $\mathcal{F}(t)$ ...
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### Risk Neutral Pricing and the Drift

For risk neutral pricing, why do we want to compute expectation of a martingale? why is this so important? Why do we dislike the drift so much? Avoid math heavy answers please.
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### Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
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### Stock price value as a continuous-time stochastic process

I am studying a mathematics textbook on the modelling of stochastic systems. The textbook uses the price of a stock as an example of a continuous-time stochastic process: If $X(t)$ is the value of a ...
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### Variance of a time integral with respect to a Brownian Motion function

Let process $$I_t = \int_0^t f(s) W_s \,\mathrm d s$$ where $W_s$ is standard Brownian motion. My question are the following: We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
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### OHLC prices after filtering

Assume we have minute-bars of OHLC stock prices. Then, applying Kalman filter to those prices separately, we can remove a measurement noise and obtain the estimates of the states of the price ...
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### Limit of product

Suppose $g(X, \delta_t)$ approaches a constant $J$ as $\delta_t$ approaches $0$, where $X$ is a random variable, and suppose $Y^2/\delta_t$ approaches some constant $K$ as $\delta_t$ approaches $0$, ...
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### Forward rates are martingale under the T-forward measure

Forward rates are martingale under the $T$-forward measure but this derivation is suggesting otherwise. Could anyone please point out the mistake ? Let $dW_Q$ be a Brownian Motion in the risk ...
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### Accumulation Rate of Variance in Random Walk

I am slightly confused with the terminology Shreve (2008), he states: "The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over ...
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### Justify a backward differential equation

Regards of 4.5.1, how we get 4.5.5?
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### Hedged portfolio dynamics under T-forward measure

I'm looking to find the hedging PDE for a multi-currency derivative $u(F_d, F_f, X,t, T)$ under the T-forward measure, using the delta-hedging argument (F - forward rate, X - forward FX rate). ...
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### Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
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### What the expectation of S^2 is from GBM? [closed]

I was at an interview and was asked to write down the SDE for GBM. $$dS = S\mu dt + S\sigma dX$$ Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
### Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.