New answers tagged

9 votes

Are differentials on their own in stochastic calculus just an abuse of notation?

Differentials in stochastic calculus have very precise interpretations. The stochastic differential equation in differential form $$dX_t = \mu (t, X_t) dt + \sigma (t, X_t) dB_t$$ translates precisely ...
Jose Avilez's user avatar
  • 12.9k
1 vote

Exercise on Girsanov's theorem

There is no difference with the case your drift is non-deterministic. Your calculation shows that $X$ is a $Q$-Brownian motion; since $Q$ and $P$ are equivalent and $Q(\{X_t > M\}) > 0$ it ...
Jose Avilez's user avatar
  • 12.9k
0 votes

Help with Integration by Parts for a Markov Chain

OP here, I think I was able to figure out the problem. Part 1: Let's start with the Q Matrix: $$ Q = \begin{bmatrix} B & A=-BI \\ 0 & 0 \end{bmatrix} $$ Within this matrix, we are interested ...
stats_noob's user avatar
  • 3,182
0 votes

Finding mean first passage time for reflecting Brownian motion

Define $\tau_x=\inf \{t\ge 0: X_t\in \partial B_{\epsilon}(0)|X_0=x\}$ and $G(x,t)=P(\tau_x>t)$. Using the backward equation involving the transition kernel for a time-homogenous process, i.e., $p(...
Fellow InstituteOfMathophile's user avatar
1 vote
Accepted

How to prove an adapted Feynman Kac Formula for $v_t + \frac{1}2 \sigma ^2 (t,y) v_{yy} + b(t,y) v_y - \delta(t,y) v + h(y) = 0$ using SDE techniques?

We see $$\begin{aligned}dv(t,Y_t)-\delta(t,Y_t)v(t,Y_t)dt&=-h(Y_t)dt+v_y(t,Y_t)\sigma(t,Y_t)dW_t\\ \implies d(v(t,Y_t)e^{-\int_0^t\delta(s,Y_s)ds})&=-h(Y_t)e^{-\int_0^t\delta(s,Y_s)ds}dt+v_y(t,...
Snoop's user avatar
  • 15.2k
1 vote

Solve SDE $dX_t = X_tW_tdt + dW_t,$

To solve problems like these, rather than trying to guess the correct process to introduce, it may be easier to just let $Y_t := e^{Z_t}X_t$ where $dZ_t = \alpha_t dt + \beta_t dW_t$ for some ...
user6247850's user avatar
  • 13.5k
4 votes

Proof of a martingale condition regarding martingale transform

$$ \begin{align} C_n(\mathbb E[X_n\vert\mathcal F_{n-1}]-X_{n-1})&=\mathbb E[C_n(X_n-X_{n-1})\vert\mathcal F_{n-1}]\\ &=\mathbb E[(C\bullet X)_n-(C\bullet X)_{n-1}\vert\mathcal F_{n-1}]\\ &...
Will's user avatar
  • 6,927
1 vote
Accepted

Converse of a martingale transform theorem

Under the stated conditions, it is not true as you can easily construct a counterexample since you basically left the process $X$ to be unrestricted. For example, assume for simplicity that $C_{n}(\...
minginator's user avatar
2 votes
Accepted

Understanding stochastic integration with semimartingale as integrator: The locally bounded variation part.

Indeed, for each $\omega$ you get a different measure $\mu_g$. So the integral can be defined path-by-path: i.e. for a fixed $\omega$ $$\left( \int_a^b f(s) dA_s \right)(\omega) = \int_a^b f(\omega, s)...
Jose Avilez's user avatar
  • 12.9k
2 votes
Accepted

Is the quadratic variation process also a martingale?

The quadratic variation process is non-decreasing, so it is not a martingale (unless it is constant at $0$). The flaw in your argument is that $M_t - \langle M\rangle_t$ is not a martingale, rather, $...
user6247850's user avatar
  • 13.5k
3 votes
Accepted

What is the solution to the SDE $X^x_t = B_t + \int_0^t \frac{x − X^x_s}{1-s} ds$

Setting $Z_t$ as what you did in your work, and then writing (2) in Differential form gives us that $$dX^x_t = xdt + dB_t - \frac{B_t}{(1-t)}dt + Z_tdt \tag{4}\label{eq4} $$ And (2) also gives us that ...
vinayak's user avatar
  • 48
1 vote
Accepted

Stochastic integration: Computing $\mathbb{E}[\exp(- \lambda \int_0^{t∧T_\epsilon}\frac{ds}{B_s^2}) ]$

The trick to being able to use the Optional Stopping Theorem here is to realize that, in the equation $\alpha (\alpha - 1) = 2\lambda$, we can always choose $\alpha < 0$. This implies $Y_t = (B_{t ...
user6247850's user avatar
  • 13.5k
0 votes

deriving covariance of SDE from fokker-planck

It holds $$ \begin{aligned} \frac{\partial \phi}{\partial t}&=-\frac{\partial m_u}{\partial t}(t)m_v(t)-m_u(t)\frac{\partial m_v}{\partial t}(t)\\ \frac{\partial \phi}{\partial x_u}&=x_v\\ \...
user408858's user avatar
  • 2,462
1 vote
Accepted

Reference request on rough paths sequence of Brownian bridges

One article where they study the stochastic Lévy area for Brownian bridges in relation to the one for Brownian motion is "BROWNIAN BRIDGE EXPANSIONS FOR LÉVY AREA APPROXIMATIONS AND PARTICULAR ...
Thomas Kojar's user avatar
  • 3,621
0 votes

Stochastic integration and use of ito's lemma

By stochastic integration by parts, which is a simple corollary of Ito's formula, $$ X_t Y_t-X_0 Y_0=\int_0^t X_S d Y_s+\int_0^t Y_S d X_S+\int_0^t d X d Y $$ we have $$\left((X \cdot M)_{T \wedge t}-(...
random_0620's user avatar
  • 2,250
0 votes

Rate of increase of maximum process of Brownian Motion

This indeed just follows from reflection principle and Borel-Cantelli $$\sum_n P[M_{t_{n}}\frac{1}{g(t_{n})}\geq \epsilon]=\sum_n P[|B_{t_{n}}|\geq g(t_{n})\epsilon]\leq c \sum_{n}exp(-\frac{(g(t_{n}))...
Thomas Kojar's user avatar
  • 3,621
3 votes
Accepted

Characteristic function of a random variable by Fourier transform

I'll admit that I'm not familiar with most if not all stochastic finance terminology. So, I'll assume that $K$ and $S_{0}$ are constants. See that for $f(x)=\log(x)$, by Ito's Lemma, you get, \begin{...
Mr.Gandalf Sauron's user avatar
1 vote
Accepted

Question about convergence in stochastic integration

No here we only assume $f$ is bounded and in $L^{2}$, he just skipped a few steps. In particular, he skipped the proof that $$g(y):=\left(\int_{R} |f(x-y)-f(x)|^{p}dx\right)^{1/p}$$ is a continuous ...
Thomas Kojar's user avatar
  • 3,621
2 votes
Accepted

Confused by the notation in Steele's Stochastic Calculus

The superscript $d$ on $\tau$ means the standard thing; i.e. $\tau^d$ is $\tau$ to the power of $d$.
Jose Avilez's user avatar
  • 12.9k
5 votes

Generator of the joint process $(X_t,Y_t)$ where $Y_t= e^{-t}W(e^{2t})$ and $X_t = \int^t_0 Y_sds$.

To find the generator I will look for the two-dimensional SDE that is solved by $(X_t,Y_t)\,.$ It is not hard to see that $$\tag{1} B_t:=\int_0^te^{-s}\,dW_{e^{2s}} $$ is a continuous martingale with ...
Kurt G.'s user avatar
  • 14.3k
1 vote

Proof that an Ito diffusion start with 0 will be positive immediately

1. By Girsanov's theorem, the laws of $X$ and of the solution $Y$ of $dY_t=\sigma(Y_t)dW_t$ are locally absolutely continuous, so you can assume without loss of generality that $\mu=0$. 2. When $\mu=0$...
John Dawkins's user avatar
  • 25.9k
2 votes

Weak Uniqueness of solution of SDE

We will add a few more details, let me know if you need more. Let $X_t$ and $Y_t$ be the strong solutions constructed from $\tilde{B}_t$ and $\hat{B}_t$, respectively, as above. Here they simply ...
Thomas Kojar's user avatar
  • 3,621
1 vote

Ito's Lemma applied to the Solution to the General Linear Equation

With your finally correct definitions but with better notation $$ \Phi_t=\textstyle\exp\left(\int_0^t( a - \frac{1}{2}b^2)\,\mathrm{d}s+\int_0^tb\,\mathrm{d}W_s\right)\,, $$ $$ I_t=X_0 + \int_0^t\frac{...
Kurt G.'s user avatar
  • 14.3k
0 votes

Ito's lemma and Stratonovich calculus

It is not very clear from Kurt's answer to Fred's answer why $\frac{\partial}{\partial f}$ is legitimate. I write a detailed explanation here. The Ito $\Leftrightarrow$ Stratonovich conversion is only ...
Fellow InstituteOfMathophile's user avatar

Top 50 recent answers are included