Girsanov’s Theorem

Enlarged Filtrations and Indistinguishable Processes

<p>Since the fractional $ G $-Brownian motion (fGBm) generalizes the concepts of the standard Brownian motion, fractional Brownian motion, and $ G $-Brownian motion, while it can exhibit long-range dependence or antipersistence and feature the volatility uncertainty simultaneously, it can be a better alternative stochastic process in the financial applications. Thus, in this paper, some empirical studies for the financial applications of the fGBm were carried out, where the recent high-frequency data for some selected assets in the financial market are from the Oxford-Man Institute of Quantitative Finance Realized Library. There are two main empirical findings. One was that the H-$ G $-normal distributions associated with the fGBm are more suitable in describing the dynamics of daily returns and increments of log-volatility for these assets than the usual distributions, since they not only characterize the properties of skewness, excess kurtosis, and long-range dependence $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{1}{2} $, at any reasonable time scale. Then a generalized stochastic model for the dynamics of the assets called rough fractional stochastic volatility model driven by fGBm (RFSV-fGBm) was developed. Finally, some parameter estimates and numerical experiments for the RFSV-fGBm model were investigated and carried out.</p>

Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by (1) where dZL is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot be priced using the standard Black–Scholes–Merton (BSM) arbitrage-free pricing criterion. Therefore, we develop a pricing theory based on a more general pricing criterion: that the expected return of all diversifiable portfolios is at the risk-free rate. We show that even though derivatives based on Lévy or tempered Lévy-driven assets cannot be hedged to make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option prices are given by (2) where the expected value uses the “real-world” probability measure, but with the asset price X(t) replaced by the alternative “pricing process” (3) which grows at the risk-free rate. We analyze these models to obtain explicit asymptotic formulas for European option prices. This analysis shows that as the time to expiry decreases, eventually all these Lévy and tempered Lévy-based models reduce to the same canonical model, and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These similarity solutions are then examined to assess the mishedging that arises from Brownian-based models in a Lévy world.

We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and infinite families of \textit{correlation measures}. Each member of this family, termed a \textit{correlation marginal}, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a $L^p$ space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting \textit{entropy statistical solutions} exist, are unique and are stable with respect to the $1$-Wasserstein metric on probability measures on $L^1$.

Let b be a Borel measurable IRd ‐ valued function, defined on some Borel subset of IRd. Consider the d‐ dimensional SDEmagnified imagewith singular drift b. A local solution (up to σ) is a tuple (X, W, Q,σ) where X is a stochastic process, W is a Brownian motion under the probability measure Q, and σ is a strictly optional time (i.e., stopping time) such that the above equation is satisfied for all t < σ. Such a local solution was constructed by the author in an earlier paper under very mild conditions on b. In this paper we give criteria for the global existence of the solution, i. e., for Q(σ = ∞) = 1.

In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637–654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Levy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.

A Visual Classification of Local Martingales

Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems.
\nOne strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation
\ntheorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained.
\nIn recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored.

SummaryIn this article, we study the open‐loop and closed‐loop solvability for indefinite mean‐field stochastic linear quadratic (MF‐SLQ) optimal control problem and its application in finance, where the controlled stochastic system is driven by a Brownian motion and a Poisson random martingale measure and also disturbed by some stochastic processes. The intrinsic property of stochastic systems results in the inequivalence of those two solvabilities, which is different from deterministic case. Based on a well‐posedness result of the problem, it is shown that the uniform convexity of cost functional is sufficient for the open‐loop solvability of the problem. By a matrix minimum principle, the necessity condition of regular solvability for a decoupled Riccati equation is established, meanwhile, the closed‐loop solvability is turned out to be equivalent to the regular solvability of Riccati equations with some constraints on the solutions of disturbances equations, moreover, the optimal closed‐loop strategy is characterized by regular solutions of Riccati equations and adapted solutions of disturbances equations. And then a mean‐variance model is considered to solve optimal portfolio selection strategy problem of the insurance company with liability. Our study fills a gap in the field of solvability for mean‐field stochastic optimal control with random jumps and provides a different way for solving mean‐variance portfolio selection model.

Let \(\{\mu _{t}^{(i)}\}_{t\ge 0}\) (\(i=1,2\)) be continuous convolution semigroups (c.c.s.) of probability measures on \(\mathbf{Aff(1)}\) (the affine group on the real line). Suppose that \(\mu _{1}^{(1)}=\mu _{1}^{(2)}\). Assume furthermore that \(\{\mu _{t}^{(1)}\}_{t\ge 0}\) is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then \(\mu _{t}^{(1)}=\mu _{t}^{(2)}\) for all \(t\ge 0\). We end up with a possible application in mathematical finance.

Recently, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) was developed, which generalizes the concepts of standard Brownian motion (Bm), fractional Brownian motion (fBm) and G-Brownian motion (GBm) under the framework of sublinear expectation. The fGBm can exhibit long-range dependence and feature the volatility uncertainty of financial markets simultaneously. Thus it can be a better alternative stochastic process in the financial applications. In this paper, some financial markets driven by the fGBm were considered and the corresponding arbitrage opportunities were discussed. Specifically, the financial market M that consists a risk free asset and a risky asset will admit some arbitrage opportunity if the stochastic integrals with respect to the fGBm were established in the path-wise sense, and the arbitrage opportunity can be excluded if the stochastic integrals were established in the Wick calculus sense. What is more, the arbitrage opportunity for the financial market ( M , C ) , consisting of the original market M and some contingent claim C were also investigated in the general case, and some interval of free arbitrage prices for the claim were derived by applying the generalized fractional G-Girsanov theorem and fractional G-Clark-Ocone theorem. This study generalized the well-known existing results of arbitrage theory in financial markets driven by the standard Bm, fBm and GBm.

A stochastic process is a time-dependent random variable. Stochastic processes such as Brownian motion and Ito processes develop in continuous time. This means that time is a real variable that can assume any real value. In many financial modeling applications, however, it is convenient to constrain time to assume only discrete values. A time series is a discrete-time stochastic process; that is, it is a collection of random variables Xi indexed with the integers …–n, … ,–2,–1,0,1,2, …,n, … Keywords: discrete-time stochastic processes; filtration; finite-dimensional distributions; autopredictive model; adjustment models; data generation process; DGP; stationary time series; multivariate time series; correlated random walks; Factor models; cointegrated model; infinite moving-average representation; causal time series

We are interested in the quasi-stationarity for the time-inhomogeneous Markov process$$X_t = \frac{B_t}{(t+1)^\kappa},$$where (Bt)t≥0is a one-dimensional Brownian motion andκ∈ (0,∞). We first show that the law ofXtconditioned not to go out from (−1, 1) until timetconverges weakly towards the Dirac measureδ0whenκ>½, whentgoes to infinity. Then, we show that this conditional probability measure converges weakly towards the quasi-stationary distribution for an Ornstein-Uhlenbeck process whenκ=½. Finally, whenκ<½, it is shown that the conditional probability measure converges towards the quasi-stationary distribution for a Brownian motion. We also prove the existence of aQ-process and a quasi-ergodic distribution forκ=½ andκ<½.

The purpose of this appendix is to explain the concept of weak convergence on C[0,1], the space of continuous functions on the unit interval [0,1]. It explains what is behind the formulae involving Brownian motion and stochastic integrals, which appear in the discussion of the limit distributions in cointegration theory. We discuss the existence of stochastic processes like Brownian motion, and find distributions of suitable functionals of Brownian motion using the continuous mapping theorem.