# Tagged Questions

The tag has no usage guidance.

0answers
40 views

### Regularity requirement for convergence of Euler scheme for stochastic integral?

Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process $\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$ which is like a naive hedge of a European put, which does not work in ...
0answers
39 views

### How to solve this system of ODEs?

Im trying to replicate the procedure of the Hackbarth et al. 2006 paper. Im trying to solve the ODEs (12) and (13) on page 525 in the paper, following the solution by the authors given in appendix A. ...
1answer
99 views

### Second Moment of Stock Process

I have a stock process which I have decided to model as $$S_T=S_t\exp((r-q-\frac{1}{2}\sigma^2)(T-t)+\sigma(W_T-Wt))-D_T$$ where $D_T$ is a cash dividend at time $T$. This dividend is known. I then ...
3answers
131 views

### How to understand nonrandom/random process in Shreve book?

I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, $\Delta(t)$, defined on Page 126, which is just a constant inside a given ...
0answers
56 views

### What is the maximum of a brownian motion with drift over the interval [t_1,t_2]

I am having a problem deriving the equation: $$P(max_{(t_1 \leq t \leq t_2)} S(t) > B | S(t_1),S(t_2))= e^{-\frac{2}{T}ln\bigg{(}\frac{B}{S(t_1)}\bigg{)} ln\bigg{(}\frac{B}{S(t_2)}\bigg{)}}$$ ...
2answers
182 views

### Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
0answers
50 views

### For a square-root process (CIR), how to verify the characteristic function of the transition density?

I am trying to solve a financial mathematical question. I derived PDE (a) for the characteristic function as follows. But, I don't know how to verify the following characteristic function of the ...
2answers
231 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 ...
2answers
105 views

### Question about the martingale property of stochastic integral

Let $W_{t}$ be a Wiener process, and let $$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$ Is $X_{t}$ a martingale? We can rewrite in differential form as $$dX_{t} = W_{t}dt$$ ,which means $X_{t}$ is a diffusion ...
0answers
25 views

### self financing property vs. unlimited borrowing

How the self financing property of a portfolio should be understood in the problems where the unlimited access to the borrowing is assumed?
2answers
240 views

### Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1$ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
2answers
79 views

### Problem with deriving the dynamics of a process

I'm trying to solve the following problem. Given a process $X_t$ and a process $Z_t$, with the dynamics of $X_t$ as $$dX_t = (\alpha + \beta X_t)dt + (\gamma + \sigma X_t)dW_t$$ and $Z_t$ defined ...
0answers
87 views

1answer
46 views

### Please help me with this problem of double exponential distribution

please help me with this problem of double exponential distribution
0answers
63 views

### Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
2answers
77 views

1answer
86 views

Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $... 1answer 93 views ### How do one solve$ \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u $? Double integral with general deterministic function$\delta(t)$How do one solve$ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $?$\delta(t)$is a general deterministic function.$r$is constant. 1answer 45 views ### Motivation: Stochastic Interest rate model what is a reason that someone might be interested in a stochastic-interest model such as the Chen model? Also can you provide me with a link to an easy to read motivational paper/part of a paper on ... 1answer 55 views ### How to change to risk neutral measure in a mean reversion process? For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case? 2answers 106 views ### Stochastic process theory question *S follows a process$dS= mSdt + oSdz$where m and o are constant. What is the probability followed by$ Y=(Se)^{(r-t)} $. If S follows a process$ dS= k (b-S) dt + oSdz $where k, b, o are ... 2answers 86 views ###$ \mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace $? Exp. of an indicator funct and a diffusion with non-proportional vol How to compute$ \mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace $? where$ dS_t = S_t r dt + \sigma dW_t $and$ 1_{S_T > K} $is the indicator function being one when ... 1answer 78 views ### Brownian motion. Solve stoc. integral by using Ito's lemma I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^... 1answer 85 views ### How to prove$\int_0^t W_s^2dWs = \frac{1}{3}W_s^3 - \int_0^t W_s ds$using Ito's formula? [closed] Please help me with this problem. 1answer 147 views ### How to apply the Feynman-Kac formula? I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ... 1answer 175 views ### Extended Hull White Interest Rate Model for Zero Coupon Bond Please taking the following SDE dr = u (r; t) dt + w (r; t) dX: u (r; t) = a(t)-br; w (r; t) = c; b&c are constants and a(t) arbitrary function of time. If Zero Coupon Bond Z (r; T; T) = 1 ... 0answers 146 views ### How to compute the stochastic integral of log-normal process? How do you compute the following integral: $$\int_0^t e^{\mu s + \sigma W_s} ds$$ or $$\int_0^t e^{\mu s + \sigma W_s} dW_s$$ ? Are those integrals stochastic processes of some well-know type (... 1answer 79 views ### Obtaining the drift of a Wiener process formed from a random walk I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ... 3answers 65 views ### Is a bond expiring at$T$clean or dirty price a martingale under the$T$-Forward measure? When we say Bond prices are martingale under T-Forward measure, do we mean their Clean Price is a martingale or is it their dirty price. I guess it should be dirty price, as clean price is just a ... 1answer 100 views ### Proof that the stopping time for a Brownian Motion is finite for given target levels Given a standard brownian motion$W_t$and defining$\tau$as:$\tau :=inf\{t\geq0:W_t=1$or$W_t=-2\}$The proof below shows that the stopping time is finite:$P(\tau < t) \geq (|W_t|>2)\\$... 1answer 123 views ### What is the correlation between these two functions of GBMs? Let's say that I have two correlated GBMs: $$dA_t = A_t \sigma^A dW^A_t$$ $$dR_t = R_t \sigma^R dW^R_t$$ $$dW^R_t dW^A_t = \rho dt$$ I am trying to price a derivative which payoff at time$T$is: $$... 1answer 164 views ### Why is GARCH more often applied in risk analysis than stochastics? I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ... 0answers 41 views ### Is there anyone tried to use simultaneous stochastic differential equations? I am looking for some examples or attempts of using simultaneous stochastic differential equations for financial analysis but there has been none so far. Is it just so nasty to apply such thing in ... 2answers 215 views ### Is the average of independent Brownian Motions still a Brownian Motion? If W and B are independent Brownian Motions (BM thereafter), then the average of W and B is X_t=\frac{1}{2}(W_t+B_t). Where do I begin to show that indeed it is still a BM? Also, if both ... 1answer 105 views ### Difference between stochastic calculus and newton calculus As I am not a student of hard core mathematics,I just want to know how stochastic calculus is different from newton calculus. What make stochastic calculus different from simple newton calculus ? 1answer 111 views ### Probability of Stock breaching barrier If a stock has a process: dS(t) = sigma*dB(t), where B(t) is a standard Brownian motion, and current stock price is S(0). There is a barrier H>S(0). What is the probability that the stock ... 0answers 124 views ### stochastic calculus and multidimentional itos lemma I am considering a number of assets (N) in a portfolio. each asset follows a geometric Brownian motion process therefore the stochastic differential equation is dS(i) = S(i)μdt + S(i)σdX(i). The ... 1answer 161 views ### Derivation using Ito's Lemma of price process Define q(t) as the log price minus a linear trend$$ q(t) = \ln P(t) - \mu t $$Assume the log price process = Equation 1:$$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$Can you show that the ... 1answer 93 views ### About the boundary conditions of the Black-Scholes-Merton PDE I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let c(t,x) be the value of the ... 1answer 137 views ### Computation of Expectation This question has so long preoccupied my mind.Please help me to solve it. Question: Assume X_t described by the following stochastic differential equation$$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ... 2answers 275 views ### how we can derive$PIDE$of double exponential Jump-diffusion model (we know as kou model)? I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure$P$: ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ‎... 1answer 258 views ### Closed form solution of PDE of Option Price Let$V=V(S_t,t)be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ... 2answers 114 views ### Distribution of stochastic integral Suppose thatf(t)$is a deterministic square integrable function. I want to show $$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$. I want to know if the following approach is ... 2answers 185 views ### Stochastic Calculus Rescale Exercise I have the following system of SDE's$ dA_t = \kappa_A(\bar{A}-A_t)dt + \sigma_A \sqrt{B_t}dW^A_t \\ dB_t = \kappa_B(\bar{B} - B_t)dt + \sigma_B \sqrt{B_t}dW^B_t $If$\sigma_B > \sigma_A$I ... 1answer 92 views ### Why$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}$is not a martingale by definition? 1answer 183 views ### How can I calculate$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.
1answer
196 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 ...