# 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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I want to calculate the variance of $$I = \int_0^t W_s^2 ds$$ I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get $$f(t,W_t)-f(0,0) = \int_0^t \frac{\... 8 votes 2 answers 855 views ### close form for stochastic integral I am new to stochastic calculus. Can I know how to compute the close-form solution for$$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$and$$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$I encounter ... 8 votes 2 answers 7k views ### How to compute the variance of this stochastic integral? I'm new to stochastic calculus and I did an exercise but I don't know if it is correct, so I need somebody with more experience to check if it is true. I am trying to compute the variance of the ... • 676 8 votes 1 answer 352 views ### Non-arbitrage theory and existence of a risk premium Consider a probability filtred space (\Omega, \mathcal F, \mathbb F, \mathbb P), where \mathbb F = (\mathcal F_t)_{0\leq t\leq T} satisfing the habitual conditions and isgenerated by 1 d - ... • 608 8 votes 2 answers 319 views ### Why won't Bjork ever show that the integrability condition is satisfied? A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0. This is based on a result presented ... • 81 8 votes 0 answers 286 views ### On a time integral of Brownian motion up to the hitting time Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space (\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\... • 81 7 votes 2 answers 2k views ### What is the average stock price under the Bachelier model? Let's say stock price follows following process:$$dS(t) = \sigma dW(t)$$where W(t) is Standard Brownian motion. The initial level for the stock is S(0). Define the average of stock price Z(t)... 7 votes 3 answers 718 views ### Why does the diffusion term remain the same when we change pricing measure? Consider some Itô process dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t} under the measure \mathbb P, where W^{\mathbb P} is a \mathbb P-Brownian motion In plenty of interest rate examples, I have ... 7 votes 1 answer 8k views ### 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 • 165 7 votes 1 answer 8k views ### What is an adapted process I am reading Björk, Arbitrage theory in Continous Time and I have noticed that he uses the term adapted proces a lot. I can't seem to understand what an 'adapted proces' is by the wikipedia article. ... 7 votes 2 answers 988 views ### Variance of \int_{t=o}^{T}\sqrt{|B(t)|} dB(t)% I'm new to stochastic calculus. Could someone please explain how I would calculate the variance of \int_{t=o}^{T}\sqrt{|B(t)|} dB(t)% I'm aware that I would first have to calculate the ... • 213 7 votes 1 answer 7k views ### Integral of Wiener process w.r.t. time I have a doubt with regards to the calculation of the below integral- \int_0^t W_sds where W_s is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ... • 392 7 votes 1 answer 353 views ### Show that (W_t, \int_0^t W_s ds) has a normal joint distribution I have to show that, if W_t is a 1-d Brownian motion then \biggl(W_t, \int_0^t W_s ds\biggr) has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on ... 7 votes 1 answer 2k views ### How to use the Girsanov theorem to prove \hat{W_t} is a \hat{\mathbb P}-Brownian motion? Let T > 0, and let (\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P) be a filtered probability space where \mathbb P = \tilde{\mathbb P} (risk-neutral measure) and \mathscr F_t ... • 901 7 votes 1 answer 245 views ### Proof that the stopping time for a Brownian Motion is finite for given target levels Given a standard brownian motion W_t and defining \tau as: \tau :=inf\{t\geq0:W_t=1 or W_t=-2\} The proof below shows that the stopping time is finite: P(\tau < t) \geq (|W_t|>2)\\ ... • 869 7 votes 1 answer 628 views ### How to measure a non-normal stochastic process? If I understand right, Itô's lemma tells us that for any process X that can be adapted to an underlying standard normal Wiener measure \mathrm dB_t, and any twice continuously differentiable ... • 926 7 votes 1 answer 393 views ### Option pricing with Brownian Bridge Say I have an asset following arithmetic Brownian motion$$ dX(t) = \sigma dW^\bot (t) $$with \sigma constant, and I have prices of vanilla options on X. I introduce a Brownian bridge$$ dY(t) = ... 1k views

### Expectation of exponential of 3 correlated Brownian Motion

Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
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### Can I always use quadratic variation to calculate variance?

Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as $X_t=X_0 + \mu ds + \sigma dW_t$ The quadratic variation of $X_t$ can be calculated as $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
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### Why is GARCH more often applied in risk analysis than stochastics?

I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ...
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### Application of Vibrato Montecarlo methods

Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain. In short the main idea of the method is the ...
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### Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
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### Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $\mathbb E \left \{ \bar X \right\}< \infty$ (where $\bar X =\sup _{0\leq t \leq T} X_t$) and its ...
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### Parametric Stochastic Integral

I need help. Defining the parametric stochastic integral $$F_t = \int_t^T\xi(t,s)g(s)ds$$ $\\\\$ with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
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### 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}
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 $dX_t = t\,dS_t$.