Questions tagged [stochastic-calculus]
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.
5,866 questions
1
vote
0
answers
40
views
Good references for Kolmogorov Equations
I’m looking for a solid reference book that goes deep into the analytical side of the Forward and Backward Kolmogorov equations for diffusions.
Specifically, I'm looking for a more rigorous and ...
4
votes
1
answer
149
views
+50
Why Euler-Maruyama doesn't seem to work for Stratonovitch SDE
For the Stratonovitch SDE on $R$ (the real line),
$$ dX_t = b(X_t) dt + \sigma(X_t) \circ dw_t, X_0 = x \in R $$
with $w=(w_t)$ 1D-Brownian motion and $\sigma,b$ globally Lipschitz, consider its ...
- 1,366
2
votes
0
answers
32
views
Derive a Stochastic Differential Equation for the Geometric Average of a Geometric Brownian Motion
I working on a problem in "Problems and Solutions in Mathematical Finance (Ch 3 pg. 123-147 Question 19)". The problem is stated as follows:
$dS_u/S_u = \mu dt + \sigma dW_t$ (i.e. a GBM)
...
- 566
4
votes
2
answers
148
views
Covariation of $X_t = tB^1_t$ with $Y_t = B^1_t B^2_t$ where $B^1$ and $B^2$ are two independents Brownian motions
I consider a filtered probability space, $B^1$ and $B^2$ two independents brownian motions on the filtration and I would like to compute the covariation of $X_t = tB^1_t$ with $Y_t = B^1_t B^2_t$.
We ...
- 1,709
0
votes
0
answers
35
views
Study recommendation to get into McKean Vlasov processes
I'd like to gain some knowledge on McKean Vlasov processes but I wouldn't know where to start reading about them. I have a good knowledge of the general theory of stochastic processes and standard ...
- 157
1
vote
0
answers
35
views
Deriving the Poisson kernel on the ball from its stochastic interpretation
Let $\mathcal{A} = \sum_{i,j=1}^d c_{i,j}(x)\partial_{x_i x_j} + \sum_{i=1}^d b_i(x)\partial_{x_i}$ be an operator whose coefficients are nice enough that the existence of a solution to the Dirichlet ...
- 469
3
votes
1
answer
85
views
Stochastic differential of exponential process
Trying to build some familiarity with Ito's formula with the following problem: let $\lambda$ be a $d$-dimensional process in $\mathbb{L}^2$ and $W$ a Brownian motion. Consider the process $M^{\lambda}...
- 157
0
votes
0
answers
31
views
Convergence of Ito integral happens almost surely taking a subsequence of any starting partition; which subsequence?
Piggybacking off of Understand better stochastic integral through a.s. convergence.
Let's take the "simplest example" of
$$\int_0^T B_t d B_t$$
It is known that for any sequence of ...
- 11.3k
3
votes
2
answers
57
views
What does $X_{\tau}$ mean w.r.t non-finite stopping times and strong Markov property
Let $(X_t)_t$ be a measurable process (i.e $X(t,\omega)$ jointly-measurable). Let $\tau$ be a random variable with values in $[0,+\infty]$.
In Karatzas and Shreve (definition 1.15) we define on the ...
- 51
0
votes
0
answers
63
views
Brownian bridge's upper and lower level reaching probability
I am reading Emmanuel Gobet's paper "Advanced Monte Carlo Method for Barrier and Related Exotic Options". On Page 5, it states that the probability of going above the upper barrier $U$ or ...
- 261
0
votes
0
answers
103
views
The Girsanov theorem proof
I am trying to understand The Girsanov theorem and one of its proofs that is given in Oksendal, Stochastic Differential Equations.
So, I have come across some problem and the problem is that in the ...
4
votes
1
answer
97
views
Escaping time of a modified CIR process
I have a process in $\mathbb{R}$ satisfying $X_0 = \xi$,
$$dX_t = (aX_t + b)dt + c\sqrt{|X_t|}\sigma_t dB_t,$$
where $a, b$ and $c$ are all positive constants, $B_t$ is a Brownian motion in $\mathbb{R}...
- 189
1
vote
0
answers
77
views
Continuity of Volterra processes
I am currently working on a problem related to path-dependent stochastic Volterra equations, i.e. continuous adapted solutions to $$X_t=\xi+\int_0^t k_b(t,s)b(s,X^s)\,ds+\int_0^t k_\sigma(t,s)\sigma(s,...
- 67
1
vote
1
answer
44
views
Pathwise vs Strong Solution to SDEs
Given a real-valued Brownian motion $(B_t)$ on some probability space $(E,F,P)$ with filtration $(F_t)$, a strong solution with initial condition $x$ to $dX_t = b(t,X_t) dt + a(t,Z_t) dB_t$ is usually ...
- 1,366
3
votes
0
answers
46
views
How to deal with Ito processes product with time delay?
I have the weight process $\omega_t$, and the asset price process $S_t$, both of them are Ito processes:
$$\frac{d\omega_t}{\omega_t}=\mu^1_t\,dt+\sigma^1_tdW^1_t,\qquad
\frac{dS_t}{S_t}=\mu^2_t\,dt+\...
- 153