Girsanov Research Articles

Non-explicit formula of boundary crossing probabilities by the Girsanov theorem

Non-explicit formula of boundary crossing probabilities by the Girsanov theorem

Portfolio problem for the α−hypergeometric stochastic volatility model with consumption

This paper solves the finite horizon optimal consumption-investment problem for an investor that trades in the α−hypergeometric stochastic volatility model, with preferences based on a power utility function. To find the optimal strategy, we follow the dynamic programming approach. First, we perform a suitable change of variables in order to fully transform the non-linear Hamilton-Jacobi-Bellman (HJB) equation into a semilinear one. Next, we apply the Girsanov theorem to deduce an implicit Feynman-Kac formula depending upon the volatility process. The operator defined by the Feynman-Kac representation is shown to be a contraction operator on a designed function space. Finally, the Banach fixed point theorem yields the existence of a solution to the HJB equation in the designed space. Moreover, we apply a verification theorem to guarantee that the solution of the HJB equation coincides with the value function.

Fluctuation theorem as a special case of Girsanov theorem

Stochastic thermodynamics is an important development in the direction of finding general thermodynamic principles for non-equilibrium systems. We believe stochastic thermodynamics has the potential to benefit from the measure-theoretic framework of stochastic differential equations (SDEs). Toward this, in this work, we show that fluctuation theorem (FT) is a special case of the Girsanov theorem, which is an important result in the theory of SDEs. We report that by employing Girsanov transformation of measures between the forward and the reversed dynamics of a general class of Langevin dynamic systems, we arrive at the integral fluctuation relation. Following the same approach, we derive the FT also for the overdamped case. Our derivation is applicable to both transient and steady state conditions and can also incorporate diffusion coefficients varying as a function of state and time, e.g. in the context of multiplicative noise. We expect that the proposed method will be an easy route towards deriving the FT irrespective of the complexity and non-linearity of the system.

Quantile Hedging in the complete financial market under the mixed fractional Brownian motion model and the liquidity constraint

This article proposes a pricing framework for European option that utilizes a Quantile hedging strategy in a complete financial market. The methodology involves applying the long memory Geometric Brownian motion model, utilizing the generalized mixed fractional Girsanov theorem, and incorporating relevant findings related to quasi-conditional expectation. The first step in this framework is to derive the option price formula using the equivalent martingale measure. This approach enables investors to use the entire option price without any restrictions to hedge their contingent claim. Then, we add the initial wealth condition to hedge and determine the option price with the Quantile hedging strategy. To do this, we find the maximum probability of success in complete risk hedging for different amounts of model parameters and then use the probability values to determine the option price. Moreover, we apply the minimum variance algorithms and the error correction method to find the hedge ratio. Finally, we illustrate the hedging strategy and provide numerical results that assist investors in determining the required capital for covering the option risk under market data. This calculation takes into account both the level of risk tolerance and the anticipated return on investment.

Maximum principle for partially observed leader–follower stochastic differential game

AbstractThis paper deals with the optimal control problem for partially observed leader–follower stochastic differential game. By virtue of the classical variational method and Girsanov's theorem, the stochastic maximum principles for the follower under one type of partially observed case and for the leader under the complete information structure are derived. As applications, two partially observed cases are considered for the linear–quadratic models. Then by the stochastic filtering technique, the optimal feedback controls for the follower and the leader are represented by the new stochastic Riccati equations.

A review of Girsanov reweighting and of square root approximation for building molecular Markov state models

Dynamical reweighting methods permit to estimate kinetic observables of a stochastic process governed by a target potential Ṽ(x) from trajectories that have been generated at a different potential V(x). In this article, we present Girsanov reweighting and square root approximation: the first method reweights path probabilities exploiting the Girsanov theorem and can be applied to Markov state models to reweight transition probabilities; the second method was originally developed to discretize the Fokker–Planck operator into a transition rate matrix, but here we implement it into a reweighting scheme for transition rates. We begin by reviewing the theoretical background of the methods and then present two applications relevant to molecular dynamics, highlighting their strengths and weaknesses.

Maximum likelihood estimation for a stochastic SEIR system with a COVID-19 application

In this paper, we propose a stochastic model for epidemiology data. The proposed model is obtained as a random perturbation of a suitable parameter in a deterministic SEIR system. This perturbation allows us to obtain a set of coupled stochastic differential equations (SDEs) that still have the conservation law. Afterward, by using Girsanov's Theorem, we calculate the maximum likelihood estimation (MLE) for parameters that represent the symptomatic infection rate, asymptomatic infection rate, and the proportion of symptomatic individuals. These parameters are crucial to obtain information about the dynamic of the disease. We prove the consistency of the MLE for a fixed time observation window, in which the disease is in its growth phase. The proposed stochastic SEIR model improves the uncertainty quantification of an overestimated MCMC scheme based on its deterministic model to count reported-confirmed COVID-19 cases of Mexico City. Using a particular mechanism to manage missing data, we developed MLE for some parameters of the stochastic model that improves the description of variance of the actual data.

Feynman-Kac formula under a finite entropy condition

Motivated by entropic optimal transport, we investigate an extended notion of solution to the parabolic equation $( \partial_t + b\cdot \nabla + \Delta _{ a}/2 +V)g =0$ with a final boundary condition. It is well-known that the viscosity solution $g$ of this PDE is represented by the Feynman-Kac formula when the drift $b$, the diffusion matrix $a$ and the scalar potential $V$ are regular enough and not growing too fast. In this article, $b$ and $V$ are not assumed to be regular and their growth is controlled by a finite entropy condition, allowing for instance $V$ to belong to some Kato class. We show that the Feynman-Kac formula represents a solution, in an extended sense, to the parabolic equation. This notion of solution is trajectorial and expressed with the semimartingale extension of the Markov generator $ b\cdot \nabla + \Delta _{ a}/2.$ Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov's theorem and the Hamilton-Jacobi-Bellman equation satisfied by $\log g$.

On a class of Lévy-driven McKean-Vlasov SDEs with Hölder coefficients

In this paper, under the assumptions of Hölder continuous coefficients, the well-posedness, Euler-Maruyama approximation, stability and strong Feller property for the solution to one-dimensional spectrally one sided Lévy processes driven McKean-Vlasov SDEs are considered. Our proof are based on the tools of Yamada-Watanabe approximation technique and Girsanov theorem.

Regularization of differential equations by two fractional noises

In this paper, we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates. A key challenge in this paper is to extend and apply the Girsanov theorem for two noises given by the sum of two (dependent) fractional Brownian motions by using profound techniques of fractional operator theory.

A STOCHASTIC CONTROL APPROACH TO BID-ASK PRICE MODELLING

This paper develops a model for the bid and ask prices of a European-type asset by formulating a stochastic control problem. The state process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. A Girsanov theorem for Markov chains is implemented for the change of coefficients, including the diffusion coefficient which cannot be changed by the usual Girsanov theorem for Brownian motion. The price of a European-type asset is then determined using an Esscher transform and a system of partial differential equations. A dynamic programming principle and a maximum/minimum principle associated with the stochastic control problem are then derived to model bid and ask prices. These prices are not quotes of traders or market makers but represent estimates in our model on which reasonable quantities could be traded.

On partially observed optimal singular control of McKean–Vlasov stochastic systems: Maximum principle approach

In this paper, we study partially observed optimal stochastic singular control problems of general Mckean–Vlasov type with correlated noises between the system and the observation. The control variable has two components, the first being absolutely continuous and the second is a bounded variation, nondecreasing continuous on the right with left limits. The dynamic system is governed by Itô‐type controlled stochastic differential equation. The coefficients of the dynamic depend on the state process and of its probability law and the continuous control variable. In terms of a classical convex variational techniques, we establish a set of necessary conditions of optimal singular control in the form of maximum principle. Our main result is proved by applying Girsanov's theorem and the derivatives with respect to probability law in Lions' sense. To illustrate our theoretical result, we study partially observed linear‐quadratic singular control problem of McKean–Vlasov type.

A Wong-Zakai theorem for SDEs with singular drift

We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the formdXt=b(Xt)dt+σ(Xt)∘dWt,X0=x0∈Rd,t⩾0, with a possibly singular drift b∈Lp(Rd), p>d and p⩾2, and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem.

Girsanov theorem for multifractional Brownian processes

ABSTRACT In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of . The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].

The Gaussian structure of the singular stochastic Burgers equation

AbstractWe consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed timet, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite-dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes, which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed, as we are beyond the regime of classical solutions for much of the paper.

Stochastic maximum principle for partially observed risk‐sensitive optimal control problems of mean‐field forward‐backward stochastic differential equations

AbstractThis paper deals with a partially observed risk‐sensitive optimal control problem described by mean‐field forward‐backward stochastic differential equations and the cost functional is a mean‐field exponential of integral type. By using Girsanov's theorem as well as classical convex variational techniques, we obtain two risk‐sensitive maximum principles, which are characterized in terms of the variational inequalities. Moreover, under certain concavity assumption, we give the sufficient condition for the optimality. Through the application of results, we consider the linear‐quadratic risk‐sensitive optimal control problem under partially observed information and fully observed information, respectively.

A girsanov result for the pettis integral

A kind of Pettis integral representation for a Banach valued It\^o process is given and its drift term is modified using a Girsanov Theorem.

Об оптимальном управлении стохастическим уравнением с дробным винеровским процессом

В теории стохастических дифференциальных уравнений в последние годы возникло новое направление исследований, а именно стохастические дифференциальные с дробным винеровским процессом. Такой класс процессов позволяет достаточно адекватно описывать многие реальные явления стохастической природы в финансовой математике, гидрологии, биологии и многих других областях. Эти явления в целом описываются не стохастическими системами, удовлетворяющими условиям сильного перемешивания или слабой зависимости, а системами с сильной зависимостью, и эта сильная зависимость регулируется так называемым параметром Харста, который является характеристикой этой зависимости. В данной работе исследуются задачи существования оптимального управления для стохастического дифференциального уравнения с дробным винеровским процессом. Относительно существования оптимального управления возникают те же трудности, что при исследовании задачи существования оптимального управления для стохастических уравнений с обычным винеровским процессом. Во многих реальных задачах класс допустимых управлений достаточно широк, и для оптимальных руководств могут не выполняться условия существования сильных решений для рассматриваемых уравнений. В статье рассматривается задача существования оптимального управления для стохастического дифференциального уравнения с дробным винеровским процессом, в котором присутствует коэффициент диффузии, дающий более точные результаты моделирования. Доказана теорема существования оптимального управления процессом, которому удовлетворяет соответствующее стохастическое дифференциальное уравнение. Основной результат получен с использованием теоремы Гирсанова для таких процессов и теоремы существования слабого решения стохастических уравнений с дробным винеровским процессом.

Necessary conditions for partially observed optimal control of general McKean–Vlasov stochastic differential equations with jumps

In this paper, we establish necessary conditions of optimality for partially observed optimal control problems of Mckean–Vlasov type. The system is described by a controlled stochastic differential equation governed by Poisson random measure and an independent Brownian motion. The coefficients of the McKean–Vlasov system depend on the state of the solution process as well as of its probability law and the control variable. The proof of our result is based on Girsanov's theorem, variational equations and derivatives with respect to probability measure under convexity assumption. At the end of this paper, we apply our stochastic maximum principle to study partially observed linear quadratic control problem of McKean–Vlasov type with jumps and derive the explicit expression of the optimal control.

Risk‐neutral pricing techniques and examples

AbstractTractability and flexibility are among the two most attractive features of models in mathematical finance. For the pricing of derivative products, to avoid arbitrage opportunities, the fundamental theorem of asset pricing requires the existence of an equivalent martingale measure under which the discounted price process is a (local) martingale. In the semimartingale setting, the traditional approach of risk‐neutral valuation uses a change of measure invoking Girsanov's theorem. This approach often destroys the tractability of the process which is undesirable for quantitative finance applications. To overcome these limitations, we employ a transformation based on the concept of intertwining relationships that allows us to convert Markovian semigroups into a pricing semigroup while keeping its tractability. In order to illustrate the usefulness of this approach, we apply it to exponential Lévy, positive self‐similar, and generalized CIR processes that all fall within the class of polynomial processes introduced by Cuchiero et al. (2012). Furthermore, for the jump CIR class, relying on recent work of Patie and Savov (2019), we provide an eigenvalues expansion of the pricing semigroup and carry out some numerical experiments.