arXiv subjects

Kihun Nam

Publications and source records attributed to Kihun Nam.

16 recordsLinked to original sources

arXiv2026

Quadratic Mean-Field BSDEs and Exponential Utility Maximization

In this paper, we study a class of real-valued mean-field backward stochastic differential equations (BSDEs) with generators of quadratic growth in the control variable and the mean-field term. Under this assumption, together with a bounded terminal condition, we establish the existence and uniqueness of solutions. Our approach departs from classical fixed-point arguments and instead combines Malliavin calculus with refined BMO and stability estimates. The result bridges the gap between the quadratic BSDE results of [Ann. Probab. 45 (2017), pp.~3795--3828] and Hao et al. [Ann. Appl. Probab. 35 (2025), pp.~2128--2174]. Moreover, motivated by the structure of the mean-field exponential utility maximization problem introduced in our paper, we extend our framework to terminal conditions without continuity or the Markovian assumption. We establish the existence and uniqueness of solutions under a smallness terminla value on the terminal conditions. We then apply this extended theory to solve a mean-field exponential utility maximization problem, which developing the classical framework of Hu et al. [Ann. Appl. Probab. 15 (2005), pp.~1691--1712] to a fully coupled quadratic mean-field setting.

math.OC↗

arXiv2025

No arbitrage assumption implies the differentiability of derivative pricing function

In this article, we show necessary and sufficient conditions for a function to transform a continuous Markov semimartingale to a semimartingale. As a result, the no-arbitrage principle guarantees the differentiability of asset prices with respect to the underlying noise, if the asset prices are continuous and the underlying noise is a continuous Markov semimartingale.

math.PR↗

arXiv2025

Macroscopic Market Making

We propose a macroscopic market making model à la Avellaneda-Stoikov, using continuous processes for orders instead of discrete point processes. The model intends to bridge the gap between market making and optimal execution problems, while shedding light on the influence of order flows on the optimal strategies. We demonstrate our model through three problems. The study provides a comprehensive analysis from Markovian to non-Markovian noises and from linear to non-linear intensity functions, encompassing both bounded and unbounded coefficients. Mathematically, the contribution lies in the existence and uniqueness of the optimal control, guaranteed by the well-posedness of the strong solution to the Hamilton-Jacobi-Bellman equation and the (non-)Lipschitz forward-backward stochastic differential equation. Finally, the model's applications to price impact and optimal execution are discussed.

q-fin.MF↗

arXiv2025

Strong solutions of mean-field FBSDEs and their applications to multi-population mean-field games

We study the existence of strong solutions for mean-field forward-backward stochastic differential equations (FBSDEs) with measurable coefficients and their implication on the Nash equilibrium of a multi-population mean-field game. More specifically, we allow the coefficients to be discontinuous in the forward process and non-Lipschitz continuous concerning their time-sectional distribution. Using the Pontryagin stochastic maximum principle and the martingale approach, we apply our existence result to a multi-population mean-field game (MPMFG) model where the interacting agents in the system are grouped into multiple populations. Each population shares the same objective function, and we take changes in population sizes into consideration.

math.PR↗

arXiv2025

Switching to a Green and sustainable finance setting: a mean field game approach

We consider a continuum of carbon-emitting firms who seek to maximise their stock price, and a regulator (e.g., Government) who wishes for the economy to flourish, whilst simultaneously punishing firms who behave non-green. Interpreting the regulator as a major player and the firms as the minor players, we model this setting through a mean field game with major and minor players. We extend the stochastic maximum principle derived by Carmona & Zhu [A probabilistic approach to mean field games with major and minor players. Annals of Applied Probability, 2016, 94, 745--788] by relaxing the assumptions on the forms of the minimisers for the Hamiltonians, allowing them to depend on more arguments. This allows the major and representative minor player to interact in a more natural fashion, thereby permitting us to consider more realistic models for our green and sustainable finance problem. Through our stochastic maximum principle, we derive explicit Nash equilibria for a number of examples.

math.PR↗

arXiv2023

Data-driven Multiperiod Robust Mean-Variance Optimization

We study robust mean-variance optimization in multiperiod portfolio selection by allowing the true probability measure to be inside a Wasserstein ball centered at the empirical probability measure. Given the confidence level, the radius of the Wasserstein ball is determined by the empirical data. The numerical simulations of the US stock market provide a promising result compared to other popular strategies.

q-fin.MF↗

arXiv2023

Robust Wasserstein Optimization and its Application in Mean-CVaR

We refer to recent inference methodology and formulate a framework for solving the distributionally robust optimization problem, where the true probability measure is inside a Wasserstein ball around the empirical measure and the radius of the Wasserstein ball is determined by the empirical data. We transform the robust optimization into a non-robust optimization with a penalty term and provide the selection of the Wasserstein ambiguity set's size. Moreover, we apply this framework to the robust mean-CVaR optimization problem and the numerical experiments of the US stock market show impressive results compared to other popular strategies.

q-fin.MF↗

arXiv2022

Global Well-posedness of Non-Markovian Multidimensional Superquadratic BSDE

(Working Paper) Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay between the local well-posedness of fully coupled path-dependent forward backward stochastic differential equations (FBSDEs) and backward iterations of the superquadratic BSDE. The superquadratic BSDE studied in this article includes quadratic BSDEs appear in stochastic differential game and price impact model. We also study the well-posedness of superquadratic FBSDEs and FBSDEs with measurable coefficients using the corresponding BSDE results. Our result also provides the well-posedness of a system of path-dependent quasilinear partial differential equations where the nonlinearity has superquadratic growth in the gradient of the solution.

math.PR↗

arXiv2021

Coupled FBSDEs with Measurable Coefficients and its Application to Parabolic PDEs

Using purely probabilistic methods, we prove the existence and the uniqueness of solutions fora system of coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous coefficients. As a corollary, we obtain the well-posedness of semilinear parabolic partial differential equations (PDEs) $$ \begin{aligned} &\mathcal{L} u(t,x)+F(t,x,u,\partial_x u)=0;\qquad u(T,x)=h(x)\\ &\mathcal{L}:=\partial_t+\frac{1}{2}\sum_{i,j=1}^m(σσ^\intercal)_{ij}(t,x)\partial^2_{x_ix_j} \end{aligned} $$ in the natural domain of the second-order linear parabolic operator $\mathcal{L}$. We allow $F$ and $h$ to be discontinuous with respect to $x$. Finally, we apply the result to optimal policy-making for pandemics and pricing of carbon emission financial derivatives.

math.PR↗

arXiv2019

Correlated time-changed Lévy Processes

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their main result hinges on the stopping time property of the time changes, but all of the models CW proposed for the time changes do not satisfy this assumption. In this paper, when the time changes are adapted, but not necessarily stopping times, we provide analogous results to CW. We show that our approach can be applied to all models in CW.

math.PR↗

arXiv2019

Time-changed \levy processes and option pricing: a critical comment

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their framework hinges on the stopping time property of the time changes. By analyzing the measurability of the time changes with respect to the underlying filtration, we show that all models CW proposed for the time changes fail to satisfy this assumption.

q-fin.MF↗

arXiv2018

Variations of dynamic random networks: localization approach

We study the variation of exchangeable graph-valued process ${\bf Γ}$ and its graph limit. We used a constructive method using localization technique. Our method provides a specific estimation of variation for exchangeable graph-valued process ${\bf Γ}$ and its graph limit for different types of metric. As a result, we extend the bounded variation of graph limit as in Crane (2016).

math.PR↗

arXiv2017

Locally Lipschitz BSDE driven by a continuous martingale: path-derivative approach

Using a new notion of path-derivative, we study well-posedness of backward stochastic differential equation driven by a continuous martingale $M$ when $f(s,γ,y,z)$ is locally Lipschitz in $(y,z)$: \[Y_{t}=ξ(M_{[0,T]})+\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\rm tr}[M,M]_{s}-\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t}\] Here, $M_{[0,t]}$ is the path of $M$ from $0$ to $t$ and $m$ is defined by $[M,M]_{t}=\int_{0}^{t}m_{s}m_{s}^{*}d{\rm tr}[M,M]_{s}$. When the BSDE is one-dimensional, we could show the existence and uniqueness of solution. On the contrary, when the BSDE is multidimensional, we show existence and uniqueness only when $[M,M]_{T}$ is small enough: otherwise, we provide a counterexample that has blowing-up solution. Then, we investigate the applications to utility maximization problems.

math.PR↗

arXiv2017

BSE's, BSDE's and fixed point problems

In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed point problem in a space of random vectors. This makes it possible to employ general fixed point arguments to find solutions. For instance, Banach's contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean-Vlasov type equations and equations with coefficients of superlinear growth.

math.PR↗

arXiv2015

Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the form $$ Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s $$ with bounded terminal conditions $ξ$ and drivers $f$ that grow at most quadratically in $Z_s$. We consider three different cases. In the first one the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver $f$ in $Z_s$ is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending the solution recursively.

math.PR↗

arXiv2013

BSDEs with terminal conditions that have bounded Malliavin derivative

We show existence and uniqueness of solutions to BSDEs of the form $$ Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ in the case where the terminal condition $ξ$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $ξ$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type.

math.PR↗