All Questions
Tagged with stochastic-processes stochastic-calculus
312
questions
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
18
votes
8
answers
8k
views
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 ...
- 1,192
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
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
13
votes
1
answer
361
views
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 ...
12
votes
2
answers
956
views
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 ...
- 2,139
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
12
votes
2
answers
584
views
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 ...
- 612
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
11
votes
4
answers
15k
views
How to use Itô's formula to deduce that a stochastic process is a martingale?
I'm working through different books about financial mathematics and solving some problems I get stuck.
Suppose you define an arbitrary stochastic process, for example
$ X_t := W_t^8-8t $ where $ W_t ...
user1715
10
votes
2
answers
1k
views
Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.
(a) $\beta_{t}=e^{t}, S_{t}...
- 103
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
9
votes
2
answers
3k
views
Ito, Stochastic Exponential and Girsanov
This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential.
Whilst reading notes relating to Girsanov it is stated that the ...
- 227
9
votes
2
answers
531
views
Why does Black-Scholes equation hold on continuation region of American Option?
Explanation for Put Option:
$$ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0, $$
where
$\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\...
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.$$
I encounter ...
8
votes
2
answers
7k
views
How to compute the variance of this stochastic integral?
I'm new to stochastic calculus and I did an exercise but I don't know if it is correct, so I need somebody with more experience to check if it is true.
I am trying to compute the variance of the ...
- 676
8
votes
1
answer
352
views
Non-arbitrage theory and existence of a risk premium
Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
- 608
8
votes
2
answers
319
views
Why won't Bjork ever show that the integrability condition is satisfied?
A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0.
This is based on a result presented ...
- 81
8
votes
0
answers
286
views
On a time integral of Brownian motion up to the hitting time
Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
- 81
7
votes
2
answers
2k
views
What is the average stock price under the Bachelier model?
Let's say stock price follows following process:
$$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 ...
- 73
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. ...
user25295
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 ...
- 73
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) = ...
user34971
6
votes
3
answers
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 ...
- 73
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
6
votes
1
answer
200
views
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 ...
- 257
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
6
votes
2
answers
392
views
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 ...
6
votes
1
answer
323
views
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 ...
- 608
6
votes
1
answer
240
views
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 ...
- 61
6
votes
0
answers
178
views
Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)
I am doing a research on communication protocol design.
A file of $n$ blocks is transferred in several rounds and
$R_i$ denotes the number of blocks received in the $i$-th round.
The sender sends $n-...
- 161
5
votes
1
answer
186
views
Stochastic Differential
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 $(...
- 800
5
votes
1
answer
579
views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
- 53
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
1
answer
689
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 ...
- 51
5
votes
2
answers
306
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
1
answer
378
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 ...
- 463
5
votes
1
answer
219
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
301
views
Can a Process with a Stochastic Drift be a Martingale?
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:
$$...
- 5,513
5
votes
2
answers
1k
views
Bayes Theorem with change of measure
Tomas bjork- arbitrage theory in continuous time.
Appendix B, proposition B41 says:
The proof is not clear to me.
Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
- 343
5
votes
1
answer
144
views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0
$$ $$F(0,x)=g(x)$$ for some function $g$.
Let $X_t$ be a process defined by
$$dx_t=aX(t)dt+dW(t)$$
Now ...
- 301