Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
563
questions
5
votes
1answer
231 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
171 views
Computation of Expectation
This question has so long preoccupied my mind.Please help me to solve it.
Question: Assume $X_t$ described by the following stochastic differential equation
$$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ...
5
votes
1answer
122 views
Why $W_{t}^3$ is not a martigale?(by Definition)
If $W_t$ be a wiener process then,how can i show that $W_{t}^{3}$ is not a martingale by definition?
5
votes
1answer
108 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
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)$$
Now ...
5
votes
1answer
254 views
What is the probability that a OU process hits an upper barrier U before a lower barrier L?
What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?
5
votes
1answer
540 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
5
votes
2answers
595 views
Ito vs. Stratonovich: Why is it the exact midpoint that renders Ito-correction zero?
Perhaps I am approaching this from the wrong direction but I was just thinking about the relationship between Ito and Stratonovich integrals:
It is a well known result that to convert one into the ...
5
votes
2answers
1k views
Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
5
votes
1answer
183 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$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$$
(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)
and the SDE of ...
5
votes
2answers
594 views
Why is the black-scholes model arbitrage free when Ļ>0?
I want to show that: if $Ļ$ is positive then there is no arbitrage in the model, even if $r > Āµ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
5
votes
1answer
221 views
Explicit solution SDE
I have the following SDE:
$$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}$$
where $W_{t}^{...
5
votes
1answer
233 views
What is the stochastic differential of a general semimartingale?
By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3:
$$H = B + H^c + h(x) \...
5
votes
1answer
487 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
0answers
119 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
3answers
458 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
631 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
3answers
391 views
Determine $E[W_p W_q W_r]$
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 0 < p < q < r. Determine $E[W_p W_q W_r]$.
...
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
403 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
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
1answer
131 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
481 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
4k views
What is an adapted process
I am reading Bjƶrk, 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. ...
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
143 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
305 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
1answer
169 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 $(...
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
509 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= Ī¼Ydt+ĻYdB
\\ X=\frac{1}{2}cY\\ dc =-aĪ±cdt\end{cases}
4
votes
2answers
160 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
745 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
477 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
189 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
135 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
130 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
803 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. ...
4
votes
1answer
569 views
Step by Step Guide to Learn Quantitative Finance [closed]
Can some one help in creating step by step guide to learn Quantitative Finance?
The suggestions should be in the lines of
1- Which Maths topics needs to be learn 1st
2- Which Maths Books or ...
4
votes
1answer
547 views
Application of Ito's Lemma, finding the condition for the martingale
The Vasicek short rate model is
$$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$
Define the processes $x_t$ and $f(x,t)$
$$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$
$$f(x,t)=e^{a(T-t)-x_t}$$
...
4
votes
1answer
400 views
clarification to log-stock price formula
Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$
what is the log-stock price?
Using Ito formula it is ...
4
votes
1answer
338 views
Closed form solution of PDE of Option Price
Let $V=V(S_t,t)$ be the option price and
\begin{align}
V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\
V(S_T,T)=\ln (S_T)^{2}.
\end{align}
My question: How can I obtain a closed form solution of ...
4
votes
1answer
139 views
Discounted risky asset stochastic process problem
$S_t$ is the random variable representing the risky asset price at time $t$.
M_t is the riskless asset. They are governed by the equations
$\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and
$dM_t = rM_t ...
4
votes
2answers
178 views
Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$
I am wondering about the probability distribution of the stochastic process
$$X_t=\int_0^t \frac{u} {t} dW_{u}$$
I thought of using the Kolmogorov equation but after converting this into An SDE
$$...
4
votes
1answer
184 views
Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
4
votes
1answer
409 views
Pricing the Passport option
Suppose underlying asset $S$
$$dS = \mu Sdt + \sigma Sd W$$
our portfolio $\pi$ consist with $q(t)$ stock $S$ and cash $\pi - qS$...
4
votes
1answer
182 views
Notion of risk-less portfolio in derivation of Black-Scholes
EDIT: As pointed out by Gordon in the comments, the portfolio I considered in my original post is neither self-financing nor (locally) risk-free.
Though the central question is still open. Suppose ...
4
votes
1answer
997 views
How to solve this PDE using Feynman-Kac?
I have the following problem right now: solve
$$F_t(t,x) + rxF_x(t,x) + \frac{\sigma^2}{2}F_{xx}(t,x) = rF(t,x), \\ F(T,x) = (x - K)^2.$$
How do I solve this?
There exists a theorem to solve this, ...
