# All Questions

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### What is a stationary process?

How do you explain what a stationary process is? In the first place, what is meant by process, and then what does the process have to be like so it can be called stationary?
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### Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
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### Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
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### How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
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### Probability density function of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
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### Filtration and measure change

I asked this question in math stackexchange but to no avail. So i'm trying the luck here. I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand ...
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### What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?

I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process. $$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$ Is anyone able to calculate the mean ...
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### Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
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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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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 $(... • 840 5 votes 1 answer 3k views ### CIR Process from Ornstein–Uhlenbeck process The wikipedia entry on the CIR Model states that "this process can be defined as a sum of squared Ornstein–Uhlenbeck process" but provides no derivation or reference. Can any one do that? I could only ... • 2,806 5 votes 1 answer 600 views ### Itos Lemma problem Can someone help me with calculus for this problem. I have these 3 equations and with Itos Lemma I have to find$dXt$. \begin{cases} dY= μYdt+σYdB \\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases} 5 votes 1 answer 760 views ### Martingale representation theorem Let$r_t, \theta_t$denote some stochastic processes driven by a$N$dimensional Brownian motion$W_t$(they are of course assumed adapted to the natural filtration$\mathcal{F}_t$of that Brownian ... 5 votes 2 answers 326 views ### Why is$Y(t)V^h(t)$a martingale? Let$\lambda$be the market price of risk:$\frac{a - r}{\sigma}$, and define$Y(t) = e^{-\lambda W(t) - (r + \frac{\lambda^2}{2})t}$. Let$V^h(t)$be the value process of any self-financing portfolio.... • 51 5 votes 2 answers 5k views ### How to apply the Feynman-Kac formula? I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ... 5 votes 1 answer 428 views ### Lipschitz condition in mathematical finance I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ... • 483 5 votes 1 answer 244 views ### Evaluating the SDE$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$. 5 votes 2 answers 539 views ### Are two stochastic processes independent if the Wiener processes inside are uncorrelated Assume there are two stochastic processes:$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$imply that$\...
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I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts? Suppose I have a process with a stochastic drift: ...