# Questions tagged [stochastic-control]

Stochastic control is widely used in finance since it is about decision taking in a random environment. Option replication, portfolio construction, intraday trading are typical applications of stochastic control.

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### Portfolio of Assets with non-constant correlation - Maximum Drawdown

How could I calculate the maximum drawdown (given a specified confidence interval, ie. 99%) of a portfolio whose assets have non-constant (deterministic or stochastic) correlation? Is there a ...
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### Interchange Expectation and Supremum in Snell Envelope/American Options

I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
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### Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
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### How to, from various hypotheses on the P&L, get known models (BS, Heston etc …)

Usually models in quantitative finance are taught by giving, let's say, stochastic differential equations, initial conditions, and then pricing, under the model, various derivatives written on the ...
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### Bounded solution for a SDE

I have this SDE $$dX(t) = [X(t)(u(t)(\delta-r)+r-\beta(t))+\theta(t)(1-\alpha(t))]dt+X(t)u(t)\sigma dW(t), t \in [0,T] \\ X(0) = X_0(1-\alpha(0))$$ I've checked some books and I find the solution ...
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### Pre-requisites for Finance Mathematics

I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
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### Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
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### Price of a stochastic game between an agent and the market

In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as \begin{align} V(x,C) = \dfrac{1}{\...
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### The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
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### How do you actually solve a stochastic HJB equation in practice?

I've read a number of recent papers on market making. Nearly all of the more recent papers focus on defining the problem in terms of a state and action space, deriving the relevant HJB equations and ...
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### Code examples of solving Stochastic Optimal Control problems

I'm currently reading a book demonstrating how Stochastic Optimal Control can solve common optimization problems encountered within quantitative finance. I haven't covered much continuous mathematics ...
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### Reference Request: Control Theory Prerequisites for Quantitative Finance

Right now, even though I have a mathematical background, I did not take up control theory in college. I'm looking for an introductory text on (stochastic?) control theory as applicable to quantitative ...
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### How to Implement an optimal Stochastic Control Optimization? [closed]

I'm currently working on an stochastic optimal control problem applied to a portfolio asset allocation. In principle, the problem is to maximize the return of a fixed income portfolio under certain ...
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### Flow Variable and Stock Variable

I am new to stochastic control and I need your help! Suppose that we are a trader and we are trading based two sources of signal. One comes from the stock's flow of dividends as well as another trader'...
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### Understanding the HJM drift condition's dimensions

In an HJM model the forward rate dynamics follow $$df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t$$ where $W_t$ is a $d$-dimensional brownian motion, $b_t$ takes values in $\mathbb{R}^{d\times d}$ and $a_t$ ...
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### Merton portfolio allocation problem proportions/weights >1 or <0?

In the classical Merton portfolio problem, lets assume: $$dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t) = \pi_t X_t (\mu dt +\sigma dW_t)$$ ie: zero interest rates for simplicity. ...
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### example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$\frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
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### Portfolio insurance strategy with path dependence

I have the following problem. Let us assume that $S_t$, the stock price follows, geometric Brownian moation with parameters $(\mu,\sigma^2)$. We are given an amount of money $M$ and at each point in ...
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### 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 ...