Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
605
questions
5
votes
1answer
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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}
5
votes
1answer
2k views
Geometric Brownian Motion: percentage returns vs log-returns
In classical calculus, we know that the limit of percentage return (ie $dS/S$) equals that of the log return (ie. $dln(S)$ ).
With uncertainty, we rely on Ito Lemma to draw a relationship between the ...
5
votes
1answer
2k 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 ...
5
votes
2answers
1k views
Libor Market Model: numeraire change
I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
5
votes
3answers
131 views
Volatility of Exchange Option
I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process:
Question:
How would you price an exchange call option that pays $max(S_{...
5
votes
2answers
245 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....
5
votes
1answer
511 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
1answer
664 views
Square of arithmetic brownian motion process
We have an arithmetic Brownian motion process $X_t$ that follows $dX_t=\mu dt + \sigma dZ_t$ and we define the asset price $S_t=X_t^2$ and we are asked to find the stochastic differential equation ...
5
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1answer
334 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 ...
5
votes
1answer
280 views
Expected value of exponential of hitting time of GBM
We have a stopping time
$$
\tau=\inf\{t\geq 0: S_0e^{\sigma B_t+(r-\sigma^2/2)t}=S^* \}
$$
where $S_0,\sigma,r,S^*$ are constants and $S^*<S_0$, and $B_t$ is a brownian motion. I wish to compute ...
5
votes
1answer
130 views
Why $W_{t}^3$ is not a martigale?(by Definition)
If $W_t$ be a wiener process then,how can i show that $W_{t}^{3}$ is not a martingale by definition?
5
votes
2answers
186 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:
$$...
5
votes
1answer
115 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
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)$$
Now ...
5
votes
1answer
260 views
What is the probability that a OU process hits an upper barrier U before a lower barrier L?
What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?
5
votes
2answers
595 views
Bayes Theorem with change of measure
Tomas bjork- arbitrage theory in continuous time.
Appendix B, proposition B41 says:
The proof is not clear to me.
Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
5
votes
1answer
613 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
5
votes
2answers
659 views
Ito vs. Stratonovich: Why is it the exact midpoint that renders Ito-correction zero?
Perhaps I am approaching this from the wrong direction but I was just thinking about the relationship between Ito and Stratonovich integrals:
It is a well known result that to convert one into the ...
5
votes
2answers
2k views
Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
5
votes
1answer
187 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$
with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$.
It is now often claimed that this ...
5
votes
2answers
3k views
Geometric brownian motion vs. Ornstein Uhlenbeck
I'm looking at the SDE of Geometric brownian motion(*):
$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$
(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)
and the SDE of ...
5
votes
2answers
684 views
Why is the black-scholes model arbitrage free when σ>0?
I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
5
votes
1answer
235 views
Explicit solution SDE
I have the following SDE:
$$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$
where $W_{t}^{...
5
votes
1answer
238 views
What is the stochastic differential of a general semimartingale?
By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3:
$$H = B + H^c + h(x) \...
5
votes
0answers
99 views
Summary of Stochastic Derivatives, Integrals, Expectations, and Variances
I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
5
votes
0answers
125 views
Complete Financial Market: Integrability condition for Contingent Claims
Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
4
votes
3answers
1k views
Variance of Brownian Motion
Can someone point me into the right direction to calculate this one:
$E(B^4_t)=3t^2$
I had tried using the following property with no luck:
$E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\...
4
votes
3answers
2k views
For $B_t$ a Brownian motion what is the probability that $B_1>0$ and $B_2<0$?
Let $B_t$ be a Brownian Motion. What's the probability that $B_1>0$ and $B_2<0$?
4
votes
1answer
188 views
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
4
votes
3answers
471 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 $B_t = W_t^3 - 3tW_t$. ...
4
votes
2answers
727 views
Finding price of the power option
Let's assume a market with $d=1$ and $X=X^1$ satisfying
$dX_t=\sigma X_t\,dW_t,\: \: X_0=1,$
where $(W_t)$ is a standard Brownian motion. Assume that $\mathbb{F}$ is the natural filtration of $X$ ...
4
votes
1answer
1k views
What's the variance of this Ito integral?
I am reading stochastic calculus and I have understood that the process
$$X=\int_{0}^{1}\sqrt{\frac{\tan^{-1}t}{t}}dW_t$$
has normal distribution with mean zero. How can I find the variance of $X$?
4
votes
2answers
2k views
Differential equation for log-returns
I have a question that might be trivial to most of you, but somehow I'm not able to solve it by myself. I have a disagreement with my colleague on the distributional properties of a Geometric Brownian ...
4
votes
2answers
442 views
Ito lemma of Convertible Bond under Two-factor Model Interest Rate
@Behrouz Maleki has provided the PDE of two factor model in other post so
could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ...
4
votes
1answer
149 views
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 ...
4
votes
2answers
537 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 $Z_s=f(t-s,X_s)$ ...
4
votes
1answer
5k 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
4
votes
1answer
265 views
How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$
How can I calculate?
\begin{align}
Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)
\end{align}
Thank you for your attention.
4
votes
1answer
149 views
How do we calcualte $E[W_sW_t|W_s]$
$W_t$ is a Brownian motion. How do we calculate this expectation?
there are two cases:
$s < t$
$t < s$
Do we have to distinguish the two cases or there is a unified way of calculating it
4
votes
1answer
307 views
For the Brownian motion integrate
I want to calculate
$$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$
I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to ...
4
votes
2answers
3k views
Intergral of Brownian motion w.r.t. Brownian motion
I don't understand why $S$ (highlight on picture), I learned
$$\int_0^t W(s) dW(s) = \left. \frac{1}{2} (W^2(s)-s) \right \vert_0^t $$
everyone please explain for me. Thank you
4
votes
2answers
1k views
Financial Mathematics - Martingales example
Was hoping somebody could help me with the following question.
Prove that under the risk-neutral probability $\tilde{\mathsf P}$ the stock and the bank account have the same average rate of growth. ...
4
votes
1answer
687 views
negative values in geometric brownian motion
A GBM (Geometric Brownian Motion)
$ \frac{dx}{x} = \mu dt + \sigma dW $
solves to
$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$
From the solution, it is clear that $x_t$ cannot become negative. ...
4
votes
2answers
163 views
Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
4
votes
2answers
2k views
Square of Wiener process
In Ito's calculus one often comes $dW^2=dt$. How does this come about? What is it's relation to the Milstein method?
4
votes
2answers
787 views
Asymptotic behavior property of geometric Brownian Motion proof
Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:
If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \...
4
votes
2answers
482 views
question on Leif Andersen's “Interest Rate Modeling, vol 2 Term Structure Models”
I'm reading Leif Andersen's "Interest Rate Modeling, vol 2 Term Structure Models" and met a problem on Chapter 14 LM Dynamics and Measures, $\S$ 14.2.5 Stochastic Volatility, Lemma 14.2.6, on page 602....
4
votes
1answer
225 views
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be ...
4
votes
1answer
153 views
Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
4
votes
1answer
146 views
Compute distribution of a stochastic variable
$sign(x)=1$ if $x\geq0$
$sign(x)=-1$ if $x< 0$
Consider
$$
X_t = \int^t_0 sign(W_u)dW_u
$$
where $W_t$ is a wiener proces.
How can I determine the distribution of $X_t$ and compute $E[\exp(\...
4
votes
1answer
883 views
Ito's Lemma: Multiplication Rule
I have a conceptual question about Ito's lemma, in particular, the multiplication.
Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
