7

Of course, optimal control is at the core of math finance. Take few applications: Option Pricing: you have an exposure to a time dependent combination of market factors; you have some knowledge of their dynamics. They are partly deterministic, partly stochastic (i.e. random). At each "time step" you can adjust your portfolio at a given cost. Your goal is to ...


5

In this case it is just the notion that your payoff function should not explode at some point - made mathematically rigorous. Have a look at the following picture from wikipedia: Intuitively the Lipschitz condition (or Lipschitz continuity) ensures that your payoff function always remains entirely outside the white cone, so it cannot e.g. become infinitely ...


4

FWIW, I implemented one such control solution for my course project. See here


3

The problem of when to exercise an option with Bermudan features is an optimal stopping problem. I have a done a lot of work on how to do these things when the state space is high dimensional. There are various more complicated problems where the contract is more difficult eg swing options.


3

We assume that \begin{align*} \frac{dX_t}{X_t} &= (r+\pi Y_t)dt + \pi\sigma dW_t,\tag{1}\\ dY_t &= -\lambda Y_t + dB_t.\tag{2} \end{align*} From $(2)$, \begin{align*} Y_t = Y_0 e^{-\lambda t}+ e^{-\lambda t}\int_0^t e^{\lambda u} dB_u. \end{align*} Moreover, from $(1)$, \begin{align*} \ln X_T &= \ln X_0 + (r-\frac{1}{2}\pi^2\sigma^2)T + \pi \...


3

You can characterize local volatility (LV) models by the existence of a delta-hedging strategy which reduces the variance of P&L to zero, together with an assumption that P&L is a continuous process. You can characterize the Black-Scholes (BS) model as the unique LV model with time homogeneity and spacial homogeneity. Under BS we have expected P&...


2

Your statement should be correct, the weights into the risky asset are not bounded between $0$ and $1$. Essentially, by setting $r=0$ you omit the term which shows that your weights always sum up to one, simply by choosing the weight for the risk-free asset to be $1-\pi^*$. In other words, obtaining $\pi^*>1$ simply implies you go short in the risk-free ...


2

As you have defined the Sharpe ratio, it is independent of your position. You have $$ \mathrm{E}[rf(z_1)] = \alpha z_1 f(z_1) $$ and $$ \mathrm{Var}[rf(z_1)] = \sigma_2^2f(z_1)^2 $$ and hence $$ \frac{\mathrm{E}[rf(z_1)]}{\sqrt{\mathrm{Var}[rf(z_1)]}} = \frac{\alpha z_1}{\sigma_2} $$ independent of the function $f(z_1)$.


1

Such relationships are commonly covered in statistical mechanics, so any decent statsictal mechanics book should help. Here is an article that gives a very nice summary of the key concepts: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2716075/#!po=4.74138 I don’t have access to Rouge and Karoui article, but I think it is just trying to find an equivalent ...


1

Your issue is that you misinterpreted the NA criterion, it reads: $$ a_t(x) \triangleq \sum_{i=1}^{\infty} \left(b^i_t(x) \int_0^x (b_t^i(u))^T du\right)e_i, $$ where $b^i_t$ denotes the $i^{th}$ column of the volatility matrix $b_t$, $^T$ the transpose and $e_i$ the $i^{th}$ standard basis vector in $\ell^1$. In other words the $i^th$ coordinate of the ...


1

I assume that the problem is $$\max_{\pi} E\left(\ln Z_T^{\Pi} \right).$$ Note that $\ln Z_t^{\Pi} = \ln X_t^{\Pi} -\ln X_t^{\rho}$. Moreover, \begin{align*} d\ln Z_t^{\Pi} &= d\ln X_t^{\Pi} -d\ln X_t^{\rho}\\ &=\Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]dt + \sigma(\pi-\rho)dW_t. \end{...


1

Equations (1) to (3) are correct. Your investment strategy is then, $\forall t > 0$ $$ X_t = \theta _ t S_t $$ Provided you use this strategy as part of self-financing portfolio you can write the P&L over an infinitesimal time interval as $$ dV_t = \theta_ t dS_t $$ assuming zero safe rate, i.e. that any cash required to finance your long stock ...


1

Let $Y_t := 2 S_t^1 S_t^2 $. Applying (multivariate) Itô to the function $f(t,S_t^1,S_t^2)=2 S_t^1 S_t^2$ yields a stochastic differential equation for $Y_t$ $$ \frac{dY_t}{Y_t} = \frac{dS_t^1}{S_t^1} + \frac{dS_t^2}{S_t^2} + \rho \sigma_1 \sigma_2 dt $$ Re-applying Itô's lemma to the function $f(t,Y_t) = \ln(Y_t)$ then yields $$ d\ln Y_t = (\mu_1 + \mu_2 -...


1

Actually, a lot of finance and economics are centered around optimal control problems. Traditionally, most economies are modeled as dynamic systems. In finance, portfolio optimizations, advanced option pricing etc are all optimal control problems. You could look at the book Non Linear Option Pricing, it has a lot of optimal control problems.


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