Questions on the calculus of stochastic processes, or processes that have a random component.

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2
votes
1answer
128 views

How to solve $\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$ with condition $X(0)=1$?

I want to solve the stochastic differential equation $$\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$$ with condition $X(0)=1$.
1
vote
1answer
94 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
2
votes
0answers
51 views

Proof of the Key Renewal Theorem

I try to prove the Key renewal theorem by using the renewal theorem. In my book, it is written that it can be done by proofing the theorem first for indicator functions, then for step functions and ...
5
votes
3answers
1k views

Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
4
votes
1answer
44 views

Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
2
votes
1answer
62 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
0
votes
0answers
20 views

Expected value of solution of SDE

Is there any way to find expectation of $X_t$ defined by the following SDE? $dX_t = -[\sin(2X(t)) + \frac{1}{4}\sin(4X(t))]dt + \sqrt{2}\cos^2 x dB(t), X(0)=1, t \in [0,\tau),$ where $\mathbb{B}$ is ...
0
votes
1answer
19 views

Question on Ito Isometry and bounds of integration

I am trying to find the variance of $\int_t^T(T-s)~dW_s$ I was wondering if this approach is correct: $$ Var~(\int_t^T(T-s)~dW_s~)=\mathbb E~[~(~\int_t^T(T-s)~dW_s~)^2~]=\mathbb ...
0
votes
0answers
43 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
1
vote
0answers
31 views

Girsanov theorem conditions

If we have an adapted function $f(t)$ such that $\int_0^t f(s)ds\,<\infty$, then the Girsanov exponent can be defined: $$ Z(t):=\exp\left( \int_0^t f(s)dW(s) - \frac{1}{2} \int_0^t ...
0
votes
0answers
7 views

Stationary distribution for OU process driven by fractional brownian motion

Consider the SDE driven by a fractional brownian motion $$ dX_t = \kappa (\omega - X_t) dt + \eta dW_t^{H} $$ where $W_t^{H}$ is a fractional brownian motion with Hurst parameter H. I am interested ...
0
votes
1answer
326 views

Solving Stochastic Differential Equations

Can anyone help me with the following SDE? Solve the following stochastic differential equation: $$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$ with $Y_0=0$. Hint: Try a solution of the form $Z_tH_t$ where $Z_t = ...
1
vote
2answers
81 views

solving a stochastic differential equation

How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
0
votes
1answer
128 views

How to solve this SDE?

Suppose we have the stochastic equation $dX_t=-\frac{1}{1-t}X_tdt+dW_t$ with $X_0=0$. I have to prove that exist soma function $f=f(t)$ such that the following occurs: ...
3
votes
2answers
134 views

Simple stochastic differential equation

Solve the following stochastic differential equation: $$ dX_t=X_t\,dt+dW_t. $$ Thank you very much for help! I even don't know where to start...
3
votes
0answers
46 views

If $S_{t}$ satisfies $dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}$, will $S_{t}$ be a martingale?

If the process $S=S_{t}$ satisfies the SDE: $$dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}, \ S_{0}=1.$$ will $S_{t}$ be a martingale? It seems reasonable to say so because $S_{t}$ is clearly ...
-1
votes
1answer
66 views

Solution to a stochastic differential equation

I could really do with some help on this question, have no idea where to start. Any advice would be much appreciated, thank u in advance. I am given $$\begin{align}dx(t)&=(1+x(t))dt + x(t) ...
2
votes
5answers
314 views

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
1
vote
1answer
108 views

$dX_t=1_{X_t\not=0} dW_t$

Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the stopping time $$ ...
1
vote
1answer
64 views

Expectation of stochastic differential equation

I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito ...
1
vote
0answers
17 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
1
vote
1answer
90 views

Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion

Original Question: Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion. Attempt at an answer: Apply Ito's calculus over $f(t,b):= B^2(t)$. $$df(t,b) = \frac{\partial ...
0
votes
1answer
28 views

find the soultion $Y(t)$ of the SDE $dY(t) = \left ( \theta - \gamma Y(t) \right )dt + \sigma dw(t)$

find the soultion $Y(t)$ of the SDE $$dY(t) = \left ( \theta - \gamma Y(t) \right )dt + \sigma dw(t)$$ as a function of the inital conditon $Y(0) = y_0$ where $\theta$, $\gamma$ and $\sigma$ are ...
2
votes
0answers
108 views

How to construct the strong solution to the SDE $dX_{t}=\sqrt{X_{t}}dW_{t}$?

Given the SDE: $dX_{t}=\sqrt{X_{t}}dW_{t},$ $\ X_{0}=1$ , where $W_{t}$ is a 1-d Brownian motion. I was told that this SDE has a unique strong solution, but I don't know how to construct it. I know ...
2
votes
1answer
103 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
3
votes
1answer
450 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
1
vote
1answer
105 views

Ito vs Stratonovich SDE with irregular time-dependence in coefficients

Suppose I am interested in the Stratonovich SDE $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) \circ dB_t $$ If the coefficients are smooth enough in time and space, I can show this is equivalent to the Ito ...
0
votes
0answers
121 views

Ways to solve SDEs besides Ito's formula

I learn that Ito formula is often used to solve SDEs. I was wondering if there are other ways to solve SDEs? I searched in Oksendal's SDE (section 5.1), and Shreve's Stochastic Calculus in Finance, ...
2
votes
0answers
148 views

Solving a SDE and finding its related moments

I am attempting to answer this multi-part question, and hope you can provide any feedback on any of my workings. My apologies for the length and thank you in advance for any help! i) Let $g$ be a ...
0
votes
1answer
89 views

Solve a special non-linear Backward SDE

It is straigtforward to solve a linear Backward SDE. i.e. $dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.) How can I solve $dY_t=Z_tdW_t+ ...
2
votes
1answer
75 views

Solving a Stochastic Differential Equation (SDE)

Question: Solve the stochastic differential equation: $$ dX_t=X^3_t\,dt-X^2_t\,dW_t $$ where: $$ X_0=1 $$ My Attempt: Using Ito's with: $$ f(x)=\log(x) $$ I get that: $$ ...
0
votes
1answer
43 views

Solving the SDE $dX_t=bdt+cX_t dW_t$

I want to solve the SDE $dX_t=bdt+cX_t dW_t$, $X_0=0$ for $b,c\in\mathbb R$. I start by rewriting this as $$dX_t=(\mu_1+\mu_2 X_t )dt+(\sigma_1+\sigma_2 X_t )dW_t$$ where $\mu_1=b, \mu_2=0, ...
2
votes
0answers
126 views

Finding an SDE which satisfies $X(t)$

I am attempting the following problem, and was hoping if you guys could provide any feedback on whether my approach is valid. Thank you in advance for your time! The question is as follows: "Let ...
1
vote
1answer
250 views

What is the rationale of solving SDE by Ito's formula?

When solving a SDE by Ito's formula, we have to find a function $f(t, X_t)$ of index $t$ and the process $X$ to be solved for. I was wondering what is the criterion of choosing $f$? Is it to make ...
1
vote
2answers
79 views

Name of the formula transforming general SDE to linear

For SDE's of the general form $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t \tag{1}$$ @saz taught me that there is a formula to transform it into a linear SDE, quoting from René L. Schilling/Lothar ...
1
vote
2answers
326 views

Explicit solution of a SDE

I'd like an explicit formula as a function of $W_t$ (standard brownien motion) and $\lambda >0$ for the solution of the following SDE: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
1
vote
1answer
67 views

Solution of two (first) SDEs.

I'm about to study SDE's for the first time and I'm kinda having troubles "guessing"/"finding" solutions. Also I don't really know how and when analogies to simple ODEs are allowed (e.g. to get a ...
3
votes
1answer
65 views

Checking a solution for a SDE

I want to show that the process $Y(t) = e^t \int_0^t e^{-s}dW(s)$ satisfies the following SDE: $dX(t) = X(t)dt + dW(t), \ \ t\geq 0 , \quad X(0) = 0$ I think the right approach is to use Ito's ...
1
vote
1answer
39 views

Approach to Solving SDE

I am trying to find the solution to the SDE: $$ dX_t=aX_tdt+(b+cX_t)dW_t $$ for $t\ge0$, $X_0>0$, constants $a,b,c$ Would appreciate any hints as to how to approach this using ito's formula, I'm ...
0
votes
1answer
29 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
0
votes
1answer
25 views

SDE transformation using a primitive of a function?

Consider the following SDEs : (E) : $dX_t = (\alpha b(X_t) + {1\over2}b(X_t)b'(X_t))dt + b(X_t)dB_t$ (E') : $dY_t = \alpha dt + dB_t $ prove that E can be transformed to E' using : $ ...
1
vote
1answer
48 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
4
votes
1answer
160 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
0
votes
0answers
21 views

Visualization help for random Environment models

Hi im stuck on simple random environment models. Let $\Omega=P_{k}^{\mathbb{Z}^{d}}$ where for $k>0$ fixed. $P_{k}$ denotes the set of (2d)-vectors $(p(e))_{|e|=1,e\in \mathbb{Z}^{d}}$ with ...
1
vote
1answer
56 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
0
votes
0answers
25 views

IID implies Ergodicity

The environment space is given by $\Omega:=P^{\mathbb{Z}^{d}}$, where P contains the 2d-vectors serving as admissible transition probabilities. An Element $\omega \in \Omega$ is defined as ...
2
votes
0answers
47 views

Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
0
votes
2answers
37 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
1
vote
0answers
19 views

Product of Geometric Brownian motions

Let $S,P$ be geometric BMs: $$dS_t=S_t(\mu dt + \sigma dW_t^1)$$ $$dP_t=P_t(\tau dt + \beta (\rho dW_t^1+ \sqrt{1-\rho^2}dW_t^2)$$ Where $W^1$ and $W^2$ are independent standard BM. I want to show ...
1
vote
1answer
201 views

Integrate Brownian motion with respect to independent Brownian motion

we are wondering what can be said about the distribution of that? We are considering standard Brownian motions and the integral from 0 to 1... More precisely: What can be said about the distribution ...