All Questions
Tagged with stochastic-processes stochastic-calculus
30
questions
3
votes
1
answer
302
views
The most general conditions under which Ito lemma holds
Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $f(x)=(x-K)^{+}$, I would be interested in knowing what are the least ...
- 576
29
votes
4
answers
10k
views
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?
- 2,667
8
votes
3
answers
8k
views
Variance of time integral of squared Brownian motion
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
$$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
- 95
8
votes
2
answers
855
views
close form for stochastic integral
I am new to stochastic calculus. Can I know how to compute the close-form solution for
$$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$
and
$$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$
I encounter ...
17
votes
9
answers
10k
views
Why the expected return rate of a stock has nothing to do with its option price?
OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
- 271
12
votes
3
answers
7k
views
What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
- 121
11
votes
2
answers
631
views
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
- 295
6
votes
1
answer
5k
views
Can I always use quadratic variation to calculate variance?
Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as
$X_t=X_0 + \mu ds + \sigma dW_t$
The quadratic variation of $X_t$ can be calculated as
$dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$
...
- 417
2
votes
3
answers
1k
views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
- 5,513
14
votes
3
answers
8k
views
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 ...
- 353
10
votes
1
answer
1k
views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
- 901
9
votes
2
answers
2k
views
Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
- 141
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
6
votes
1
answer
277
views
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 ...
- 445
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 ...
- 2,511
5
votes
2
answers
4k
views
Geometric brownian motion vs. Ornstein Uhlenbeck
I'm looking at the SDE of Geometric brownian motion(*):
$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$
(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)
and the SDE of ...
- 767
4
votes
3
answers
544
views
Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
- 901
4
votes
1
answer
806
views
Bond-price dynamics in the Vasicek model
Hello I am studying about interest rate modeling
There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...
- 41
4
votes
1
answer
695
views
Discretization of Wiener process
The Wiener process $(W_t)$ is a continuous stochastic process that satisfies the following there conditions:
$W_0 = 0$,
the increments $\mathrm{d}W_t = W_{t + \mathrm{d}t} - W_t$ are normally ...
- 111
4
votes
2
answers
763
views
Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
- 301
4
votes
2
answers
2k
views
Why does the short rate in the Hull White model follow a normal distribution?
Consider Hull White model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$
when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha t}\...
- 267
3
votes
1
answer
1k
views
How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?
Having two correlated Ito processes
($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$)
$dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $
$dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $
...
- 701
3
votes
0
answers
291
views
Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
- 576
3
votes
1
answer
2k
views
Why is the value of an adaptive stochastic process known at time t?
I am having a hard time to understand the concept of an adapted stochastic process. Using an analogy to finance, I have been told we can think of adaptiveness of a stock price process as having an ...
- 33
2
votes
2
answers
274
views
Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price
(My Question)
Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M.
$$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
- 179
2
votes
1
answer
235
views
Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$
which is ...
- 221
2
votes
1
answer
333
views
Discounted price process - martingale
I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
- 423
2
votes
1
answer
762
views
Differential of integral of a stochastic process
Let $Y_{t}$ be
\begin{equation}
Y_{t}=\int_{\Omega} g(X_{u}) du
\end{equation}
where $g(.)$ is a deterministic function and $\Omega=[t_{0},t]$ continuos partition of $\mathbb{R}$.
Furthermore let $...
- 23
1
vote
1
answer
577
views
Vector of differences of Brownian motion integrals is multivariate normal
Given a 2-dimensional Wiener process $(W_{1},W_{2})$ with correlation $\rho$.
Let \begin{equation*} X(t):= F(t) + \int_{0}^{t} f(s) dW_{1}(s) + \int_{0}^{t} g(s) dW_{2}(s)\end{equation*}
for some ...
- 237
1
vote
1
answer
763
views
How to change to risk neutral measure in a mean reversion process?
For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case?
- 221